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Sex Surrogates
"Sex therapy is the treatment of sexual dysfunction" and a sex surrogate "is a professional substitute trained to help the patient overcome [sexual] inhibitions."1
Sex surrogates may be physically intimate with their patients and can even have intercourse with a patient under a sex therapist's care. The sex therapist and sex surrogate may work together to treat the handicapped, both physically and mentally, those who have PTSD, including veterans and sexual abuse survivors, and those who have anxiety and other issues about sex.
The International Professional Surrogates Association trains and certifies surrogate partners in Surrogate Partner Therapy "based on the successful methods of Masters and Johnson," according to their website, which also notes that the "legal status of surrogate partners is undefined in most of the United States and most countries around the world. This means that there are generally no laws regulating the profession."2 Other terms include medically assisted sex and partner surrogates.
The sourced articles below should provide more information on people who have sex for therapeutic healing.
Footnotes: 1. medical-dictionary.thefreedictionary.com, accessed 5/21/2018; 2. IPSA International Professional Surrogates Association, "Legal Status – Legal and Ethical," surrogatetherapy.org, accessed 5/21/2018
What Is Surrogate Partner Therapy?
"IPSA's Surrogate Partner Therapy is based on the successful methods of Masters and Johnson. In this therapy, a client, a therapist and a surrogate partner form a three-person therapeutic team. The surrogate participates with the client in structured and unstructured experiences that are designed to build client self-awareness and skills in the areas of physical and emotional intimacy. These therapeutic experiences include partner work in relaxation, effective communication, sensual and sexual touching, and social skills training."
Online Resource, surrogatetherapy.org, accessed on 5/21/2018
"Sex Surrogates" Sleep With Their Clients To Help Their Sex Lives…And No, It's Not Prostitution
"Sex surrogates work closely with sex therapists to help clients overcome sexual problems such as erectile dysfunction or sex trauma through actual physical intimacy — that means sex — with people who are suffering in their sex lives."
Marianne Garvey, bravotv.com, 2/20/2018
The Disabled Sexual Surrogate
"… In a disability context first arising in California many years ago as part of the Independent Living Movement, this type of therapy is now more commonly referred to as 'partner surrogacy' to remove the exclusive connotation of sex. Surrogacy is not exclusively about genital stimulation. It is about helping individuals discover their inherent erotic natures and acquire the tools they need to manifest their erotic natures with themselves and their partners."
Lawrence Shapiro, Journal Reproductive Health Matters, Taylor & Francis Ltd, tandfonline.com, 5/23/2017
Sex Surrogates Helping Patients Treat Anxiety, PTSD And Erectile Dysfunction Through Sexual Intimacy
"It's called sex surrogacy, which is a controversial form of sex therapy. It entails pairing up patients with trained sexual partners and is used to help people with problems including physical disabilities, anxiety, autism, schizophrenia, PTSD, vaginal spasms and erectile dysfunction. Practically all of them are nervous about intimacy."
Rachel Hosie, independent.co.uk, 5/17/2017
What It's Like To Pay Someone To Have Sex With You For Therapy
"What is Surrogate Partner Therapy, you ask? … Surrogate Partner Therapy can help with painful intercourse for women, body image, and learning to become comfortable with your sexuality, body, and helps deal with specialties such as disabilities etc., but in my experience and observation, surrogate partner therapy is so much more than just sex therapy. … Consider the man who suffers from erectile dysfunction or premature/rapid ejaculation."
Moushumi Ghose, yourtango.com, 4/24/2017
How Sexual Surrogates Are Helping Transgender Clients
"The sexual surrogacy model was first introduced in Human Sexual Inadequacy, a text penned by the famed sex research duo William Masters and Virginia Johnson back in 1970. The therapist helps the patient unearth issues impacting their sexual performance, while the surrogate relies on a more intimate approach to help them hammer it out. … 'You don't go through 20 or 30 years of being uncomfortable with your body and then, through hormones and surgery, be able to experience your body in all its richness. It just doesn't work quite that way,' she [Laura A. Jacobs] said. 'It can take a while to reconnect with your body and really experience yourself as a sexual being.' Surrogates, she explained, can help in that process."
Carrie Weisman, vice.com, 4/7/2017
Q & A: Just What Is A Sex Surrogate's Job Description?
"Question: Do some sexual surrogates have intercourse with their clients? Yes, some do. It depends on what the sex therapist, client, and surrogate have discussed and agreed on. A surrogate partner is not a prostitute. Also, a surrogate partner is not at a client's beck and call, willing to do anything or everything a client wishes for. … The legal status of surrogate partner work is unclear in many states and not all sex therapists (even in states where it's explicitly legal) work with surrogates, but some do."
Debby Herbenick, kinseyconfidential.org, 6/24/2016
Body Talk: Sexual Surrogates' Healing Touch
"'Surrogate Partner Therapy,' or SPT, is perhaps the most radical form of somatic therapy focused on intimacy and sexual expression. Surrogate partners engage in 'two-way touch,' receiving as well as giving. They provide a truly unique space: an opportunity to explore intimacy without the additional concerns that arise in romantic partnerships."
Elizabeth Costello, sfweekly.com, 3/30/2016
Sexual Surrogates Help Many Who Suffer Alone
"Sexual surrogacy clients, consist mostly of men, although some are women. They generally do not find sex enjoyable and their anxiety over sexual performance is often crippling. They may feel isolated and stuck in a perpetual cycle of sexual failures. Lack of confidence results in shortened romantic endeavours and the prospect of loneliness, with the sexual surrogate providing their last hope."
Robert T Muller, PhD, psychologytoday.com, 5/27/2013
Sex Surrogates Seem Like Prostitutes But Are Helping Hands
"Sex surrogacy emerged in the 1970s and flourished for more than a decade before the HIV/AIDS epidemic put a focus on safe sex practices, according to Barbara Keesling, who for 12 years worked as a sex surrogate and is now a professor at California State University at Fullerton. … The surrogates are credentialed and trained by the International Professional Surrogates Association (IPSA) and can be either male or female."
Susan Donaldson James, abcnews.go.com, 8/2/2011
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\section{Introduction}

The recent progress in nano-technology has resulted in the ever increasing possibility of producing a great variety of physical systems, like quantum dots, wires, etc., many of which can be simulated by one-dimensional quantum models. Two kinds of problems are usually considered: the formation of bound states in a potential, corresponding to the trapping of a particle in a given medium, and the presence of resonances in the scattering by this potential. Here we are concerned with the second kind of problems.

In one-dimensional scattering, an incident wave is partly reflected and partly transmitted, unless the potential increases infinitely so as to constitute an impenetrable barrier, in which case the wave is totally reflected. Nevertheless, this reflection occurs after a certain dwelling, that is, with a time-delay \cite{caac} which depends on the energy of the incident wave. Peaks in the time delay may appear for certain energies, revealing the existence of resonances \cite{merz,nuss,oroz} in the scattering process.

The first indication of the presence of resonances in the scattering by an infinitely rising potential appeared in the study, by Yaris {\em et al} \cite{yari} and by Caliceti {\em et al} \cite{cali}, of the one-dimensional cubic anharmonic oscillator of Hamiltonian
\begin{equation}
H=-\,\frac{{\rm d}^2}{{\rm d}x^2}+\frac{x^2}{4}-\lambda\,x^3, \qquad \lambda>0\,,  \label{i1}
\end{equation}
and the discovery that it possesses complex eigenvalues and localized eigenfunctions. The properties of these solutions, that could be interpreted as resonances, were discussed by Alvarez \cite{alv1}. These pioneering works opened the way to further studies of resonances in anharmonic oscillators
\begin{equation}
H=-\,\frac{{\rm d}^2}{{\rm d}x^2}+\frac{x^2}{4}-g\,x^N,  \label{i2}
\end{equation}
of degree $N=3$ \cite{jen1,jen2} or of arbitrary even and odd values of $N$ \cite{zin1,zin2,jen3,prl}.

Resonances in the cubic anharmonic oscillator were intuitively associated \cite{alv1} to quantum tunneling through the potential barrier formed by the combination of the $x^2$ and $x^3$ terms in the Hamiltonian (\ref{i1}). However, this interpretation, which underlies also more recent studies of anharmonic oscillators \cite{jen1,prl}, does not seem to be satisfactory. Let us consider, for instance, the case of the Hamiltonian (\ref{i1}) with $\lambda=256$. The corresponding potential presents a barrier of maximum height $1/28311552$ between its zeros located at $x=0$ and $x=1/1024$. This imperceptible barrier cannot explain the marked localization of the wave function $\psi(x)$ of the first resonance, shown in Fig. 1. We find more plausible to associate such a localization with the cubic term in the potential. In fact, resonances have been found even in absence of the $x^2$ term in the Hamiltonian (\ref{i1}), i. e., in the case of a purely cubic potential \cite{fer2}. A reasonable explanation of the existence of such localized solutions would be the following: thinking in terms of density of probability of presence of a particle, represented by $|\psi(x)|^2$, the exponentially decreasing behavior as $x$ goes to $-\infty$ is, of course, related to the classical impenetrability of that region, whereas the decreasing probability values found for positive increasing $x$ can be associated to the fact that a classical particle in a cubic potential reaches $+\infty$ in a finite time.
\begin{figure}[t]
\resizebox{11cm}{!}{\includegraphics{cubicosc01.eps}}
\caption{Squared modulus of the normalized wave functions of the first (continuous line) and second (dashed line) resonances of the Hamiltonian (\ref{i1}) with $\lambda=256$. The corresponding eigenvalues are $8.59629958291-6.24556570865$ i and $30.5501842198-22.1959889066$ i. The procedure used to compute the eigenvalues and eigenfunctions is described in  \cite{fer2}. The strong localization of the wave function of the first resonance hardly could be attributed to the existence of an extremely small barrier, of width less than $10^{-4}$ and maximum height less than $10^{-8}$, in the potential $x^2/4-256\,x^3$.} \label{cub1}
\end{figure}

Recently, resonances have been searched in one-dimensional potentials rising linearly \cite{ahm2}, quadratically \cite{ahm2,nfer}, or exponentially \cite{ahm2,ahm1,ahm3} with the distance. The algebraic solvability, in terms of well known special functions, of the Schr\"odinger equation for those potentials allows to obtain algebraic expressions for the reflection amplitude and for the time delay as a function of the energy. Resonances are clearly shown in this way.

Our main purpose in the present paper is to discuss scattering by a family of divergent-at-infinity odd potentials given by
\begin{equation}
V_a(x)=-{\rm sgn}(x)\,|x|^a \label{extra1}
\end{equation}
with positive rational values of $a$,
\begin{equation}
a=p/q\,,   \qquad p, q,\quad {\rm positive \;\; integers}.  \label{i4}
\end{equation}
An alternative notation, more convenient for considering complex scaling of the variable, is
\begin{equation}
V_a(x)=\left\{\begin{array}{lll}(-x)^a,&\qquad \mbox{for}&\quad x<0, \\ -\,x^a, &\qquad \mbox{for}&\quad x>0. \end{array}\right. \label{i3}
\end{equation}
Figure 2 illustrates the shape of these potentials for the three cases of $0<a<1$, $a=1$, and $a>1$.
Besides evaluating the phase shift experienced by a wave impinging from the right on one of those potentials, at arbitrary real energies, we are interested in the possible existence of Gamow states, that is, eigenfunctions of the Hamiltonians
\begin{equation}
H_a=-\,\frac{{\rm d}^2}{{\rm d}x^2}+V_a(x)   \label{i5}
\end{equation}
corresponding to complex eigenvalues and behaving as a pure outgoing wave for $x\to +\infty$.

Up to here, we have used the term ``resonances" to refer to such kind of solutions. This is common practice. However, in what follows, we prefer to designate them as ``Gamow states"  and to reserve the term ``resonances" for scattering states, with real energy, such that the time delay in the reflection is considerably enhanced. The occurrence of a resonance is an indication of the existence of a Gamow state of complex energy, whose real part is nearly equal to the resonant energy and whose imaginary part is small. A Gamow energy of relatively large imaginary part does not imply the existence of resonant scattering.

Very precise numerical eigenvalues of  Hamiltonians of the form $p^2/2-x^K$, with integer $K$, have been recently obtained by Fern\'andez and Garcia \cite{fega} using the complex rotation and the Riccati-Pad\'e methods. These authors, however, have not considered scattering problems.
\begin{figure}[t]
\resizebox{11cm}{!}{\includegraphics{potentials.eps}}
\caption{Three examples of potentials of the family $\{V_a(x)\}$, as given in (\ref{i3}), corresponding to each one of the three ranges of values of the parameter $0<a<1$, $a=1$, and $a>1$.} \label{potentials}
\end{figure}

Potentials of the family (\ref{i3}) present an impenetrable barrier for an incident (from the right) wave, which should be totally reflected. Therefore, the wave function representing the scattering process for one of these potentials appears, at large distances, as
\begin{equation}
\psi(x) \sim \left\{\begin{array}{ll} 0\,, &\quad x\to- \infty, \\\psi^{\rm{incoming}}(x) - S(E)\,\psi^{\rm{outgoing}}(x), & \quad x\to +\infty\,, \end{array}\right .  \label{i8}
\end{equation}
The scattering function $S(E)$, dependent on the energy $E$ of the particle associated to the incoming wave, contains all the information about the interaction of the wave with the potential. Its determination is the first purpose of this paper.

The Schr\"odinger equation to be solved is, in appropriate units for lengths and energies,
\begin{equation}
-\,\frac{{\rm d}^2\psi(x)}{{\rm d}x^2}+V_a(x)\,\psi(x)=E\,\psi(x)\,,  \qquad -\infty < x < +\infty, \label{i9}
\end{equation}
with $V_a(x)$ as given in (\ref{i3}). The two differential equations resulting for the cases of $x\leq 0$ and $x\geq 0$ can be treated simultaneously when written in the form
\begin{equation}
-\,\frac{{\rm d}^2\phi^{(\sigma)}(x)}{{\rm d}x^2}-\sigma \,x^a\,\phi^{(\sigma)}(x)=E\,\phi^{(\sigma)}(x)\,,  \quad 0\leq x < +\infty,\quad \sigma=\pm1\,, \label{i10}
\end{equation}
understanding that $\phi^{(+1)}(x)$ coincides with  $\psi(x)$ on the positive real semiaxis whereas $\phi^{(-1)}(x)$ is the mirror image, with respect to the vertical axis $x$=0, of $\psi(x)$ on the negative real semiaxis.

There are two particular values of the parameter $a$, namely $a=1$ and $a=2$, such that the corresponding Eq.~(\ref{i9}) becomes algebraically solvable. In the more general case, scattering states of the Hamiltonians  $H_a$, given
in (\ref{i5}), appear as global solutions of the Schr\"odinger equation (\ref{i9}) behaving at large distances as in Eq.~(\ref{i8}). Our method to obtain such global solutions finds inspiration in a procedure proposed by Naundorf \cite{nau1,nau2}
forty years ago. The method, later considerably improved \cite{gom1}, has been used for the solution of several quantum problems \cite{gom2,gom3,sesm}. For rational values $a>2$ a general formalism can be written. For values $0<a\leq 2$ the method is equally applicable, but the resulting expressions depend on $a$ in a manner that cannot be represented by a common form.

Section 2 recalls the algebraic solution of (\ref{i9}) for the two values of $a=1$ and $a=2$, mentioned above. Most sections of the paper are devoted to the general case of rational $a$ as given in (\ref{i4}), with the restriction $a>2$. Bases of solutions of Eq.~(\ref{i10}), useful either near the origin or at very large distances, are given in Section 3. Writing global solutions requires the knowledge of the connection factors relating the elements of the basis at the origin with those of the basis for large $x$. Section 4 explains the procedure for evaluating the connection factors. Global solutions with the appropriate behavior at infinity are then written in Section 5. This results in an expression for the scattering function and, consequently, for the phase shift in terms of the connection factors. Eigenvalues of a family of $\mathcal{PT}$-symmetric Hamiltonians $H_a^{\mathcal{PT}}$, related to $H_a$ by means of a Symanzik scaling of the variable, are reported in Section 6. The results suggest that the symmetry becomes broken for $a<3$. The mentioned Symanzik scaling, applied to the eigenvalues of $H_a^{\mathcal{PT}}$, allows to obtain immediately the Gamow energies for $H_a$.  This is done in Section 7. The possible existence of resonances associated to the Gamow energies is analyzed, also in Section 7, by computing the time delay in the scattering by $V_a$. The particular case of $a=1/2$, not included in the general case of $a>2$ and as a representative of the values of $a\in (0, 1)$, is treated in Section 8. Some final comments can be found in Section 9.

\section{Algebraically solvable cases}

For $a=1$ in our potential $V_a(x)$ given in (\ref{i3}), i. e., for the so called linear potential,
\begin{equation}
V_1(x)=-x\,, \qquad -\infty<x<+\infty\,, \label{i11}
\end{equation}
the Schr\"odinger equation (\ref{i9}), with the notation
\begin{equation}
z\equiv -(x+E)\,, \qquad u(z)\equiv \psi(x),   \label{i12}
\end{equation}
becomes the Airy equation \cite[Eq.~10.4.1]{abra}
\begin{equation}
u^{\prime\prime}(z)-z\,u(z) = 0,  \label{i13}
\end{equation}
whose solution $u(z)=$Ai$(z)$ vanishes as $z\to +\infty$ (i. e., $x\to -\infty$) and oscillates for $z\to -\infty$ (i. e., $x\to +\infty$), as expected for the physical solution. Written in terms of Hankel functions \cite[Eq.~10.4.15]{abra}, this
 physical solution, up to an arbitrary multiplicative constant, becomes
\begin{equation}
{\rm Ai}(-(x+E))=\frac{1}{2}\left(\frac{x+E}{3}\right)^{1/2}\left[{\rm e}^{{\rm i}\pi/6}H_{1/3}^{(1)}(\zeta)+{\rm e}^{-{\rm i}\pi/6}H_{1/3}^{(2)}(\zeta)\right],   \label{i14}
\end{equation}
where \begin{equation}
\zeta=\frac{2}{3}(x+E)^{3/2}\,.   \label{i15}
\end{equation}
Choosing the arbitrary multiplicative constant equal to $2\pi^{1/2}{\rm e}^{-{\rm i}\pi/4}$, and  retaining only the dominant terms in the asymptotic expansions of the Hankel functions \cite[Eqs. 9.2.7 and 9.2.8]{abra}, we obtain for the physical solution
\begin{eqnarray}
\psi_{\rm phys}(x) & = & 2\pi^{1/2}{\rm e}^{-{\rm i}\pi/4}\,{\rm Ai}(-(x+E)) \nonumber \\
&  \sim & - {\rm e}^{{\rm i}\pi/2}\,\varphi_3(x)+\varphi_4(x)\,, \qquad  {\rm for} \; x\to +\infty\,  \label{i16}
\end{eqnarray}
with
\begin{eqnarray}
\varphi_3(x) & = (x+E)^{-1/4}\exp\left({\rm i}\,\frac{2}{3}\,(x+E)^{3/2}\right)\,, \\  \varphi_4(x) & = (x+E)^{-1/4}\exp\left(-{\rm i}\,\frac{2}{3}\,(x+E)^{3/2}\right)\,.  \label{i17}
\end{eqnarray}
The Barton's criterion \cite{bart} to define incoming and outgoing waves in
divergent-at-infinity potentials shows that $\varphi_3$ and $\varphi_4$
correspond  respectively to an outgoing (to the right) and an incoming
(from the right) waves. Therefore, comparison of the asymptotic expression
of $\psi_{\rm phys}(x)$, as given in Eq.~(\ref{i16}), with that expected,
according to Eq.~(\ref{i8}), reveals that the scattering function is a constant,
\begin{equation}
S(E)= {\rm e}^{{\rm i}\pi/2}\,.   \label{i18}
\end{equation}
This lack of structure of the scattering function in the case of a one-dimensional linear potential $ V_1(x)$ has already been noticed by Ahmed {\em et al} \cite{ahm2}, who have considered a family of potentials including $V_1(x)$ as a particular case. Such lack of structure
was to be expected, since a displacement of the energy accompanied by a corresponding translation in the variable $x$ leaves invariant the Schr\"odinger equation.

The other value of $a$ leading to an algebraically solvable problem, namely $a=2$, was already considered in a previous
 paper \cite{fer1}. The solutions of Eqs. (\ref{i10}) for $a=2$
can be expressed in terms of confluent hypergeometric functions, whose asymptotic behavior is well known. The scattering function turns out to be in this case
\begin{equation}
S(E) =  \,e^{{\rm i}\pi/2}\,\frac{\mathcal{N}(E)}{\mathcal{D}(E)},   \label{i19}
\end{equation}
with the notation
\begin{eqnarray}
\mathcal{N}(E) & = & \frac{{\rm e}^{{\rm i}\pi/8}}{\Gamma\left(\frac{3-E}{4}\right)\,\Gamma\left(\frac{1+{\rm i}E}{4}\right)}
+ \frac{{\rm e}^{-{\rm i}\pi/8}}{\Gamma\left(\frac{1-E}{4}\right)\,\Gamma\left(\frac{3+{\rm i}E}{4}\right)},  \label{i20} \\
\mathcal{D}(E) & = & \frac{{\rm e}^{-{\rm i}\pi/8}}{\Gamma\left(\frac{3-E}{4}\right)\,\Gamma\left(\frac{1-{\rm i}E}{4}\right)}
+ \frac{{\rm e}^{{\rm i}\pi/8}}{\Gamma\left(\frac{1-E}{4}\right)\,\Gamma\left(\frac{3-{\rm i}E}{4}\right)}.  \label{i21}
\end{eqnarray}
A  graphical representation of this scattering function and a thorough discussion of its properties was given in \cite{fer1}.

\section{General case of $a>2$. Local solutions of the Schr\"odinger equation}

Now we consider the general case of $a$ as given in (\ref{i4}). The restriction to $a>2$ is not essential for the application of our method of solution of the Schr\"odinger equation, but it allows us to write the equations below in a common form. For $a\leq 2$, certain expressions, like the Thom\'e solutions to be defined below, cannot be written in a form independent of $a$. This is illustrated in Section 8, where the case of $a=1/2$ is considered.

To avoid dealing with fractional powers, we introduce the  change of variables
\begin{equation}
x\equiv t^{2q},\qquad \phi^{(\sigma)}(x)\equiv t^{q-1/2}\,w^{(\sigma)}(t), \qquad t\in [0,+\infty). \label{ii1}
\end{equation}
The differential equation (\ref{i10}) turns, in this way, into
\begin{equation}
t^2\,\frac{{\rm d}^2w^{(\sigma)}(t)}{{\rm d}t^2}+\left(4q^2\,\sigma\,t^{2p+4q}+4q^2\,E\,t^{4q}-q^2+1/4\right)w^{(\sigma)}(t)=0 ~.  \label{ii2}
\end{equation}
Two kinds of solutions of this equation are to be considered. The first kind is a pair of (Frobenius) solutions written as series expansions, convergent for finite $t$,
\begin{equation}
w_i^{(\sigma)}(t)=t^{\nu_i}\,\sum_{n=0}^\infty c_{n,i}^{(\sigma)}\,t^n,  \qquad i=1, 2,  \label{ii3}
\end{equation}
with indices
\begin{equation}
\nu_1= -q+1/2, \qquad \nu_2=q+1/2,  \label{ii4}
\end{equation}
and coefficients given by the recurrence relation
\[
c_{0,i}^{(\sigma)}=1, \qquad c_{2q,1}^{(\sigma)}=0,
\]
\begin{equation}
n(n+2\nu_i-1)c_{n,i}^{(\sigma)}=-\,4q^2E\,c_{n-4q,i}^{(\sigma)}-4q^2\sigma\,c_{n-2p-4q,i}^{(\sigma)} \,.  \label{ii5}
\end{equation}
The corresponding solutions
\begin{equation}
\phi_i^{(\sigma)}(x)\equiv t^{q-1/2}\,w_i^{(\sigma)}(t)\,,  \qquad i=1, 2,   \label{ii6}
\end{equation}
of Eq.~(\ref{i10}) are regular for finite $x$ and constitute an adequate basis to express the physical solution of the Schr\"odinger equation as linear combination
\begin{equation}
\psi_{\rm phys}(x)=\left\{\begin{array}{lll}A_1^{(-1)}\,\phi_1^{(-1)}(-x)+A_2^{(-1)}\,\phi_2^{(-1)}(-x),&\qquad \mbox{for}&\quad x<0, \\
 A_1^{(+1)}\,\phi_1^{(+1)}(x)+A_2^{(+1)}\,\phi_2^{(+1)}(x), &\qquad \mbox{for}&\quad x>0.\end{array}\right . \label{ii7}
\end{equation}
 The continuity at $x=0$ of $\psi_{\rm phys}(x)$ and of its derivative is guaranteed if we take
\begin{equation}
A_1^{(-1)}=A_1^{(+1)}=A_1\,, \qquad A_2^{(-1)}=-A_2^{(+1)}=-A_2\,, \label{ii8}
\end{equation}
with constants $A_1$ and $A_2$ to be determined as explained below.

The second kind of solutions of the differential equation (\ref{ii2}) to be considered is the pair of formal (Thom\'e) solutions
\begin{equation}
w_j^{(\sigma)}(t)=\exp \left[\alpha_j^{(\sigma)}\,t^{p+2q}/(p+2q)\right]\,t^{\mu_j}\,\sum_{m=0}^\infty a_{m,j}^{(\sigma)}\,t^{-m}, \qquad j=3, 4, \label{ii9}
\end{equation}
with exponents
\begin{equation}
\alpha_3^{(\sigma)}=-\alpha_4^{(\sigma)}=-2q(-\sigma)^{1/2}\,, \qquad \mu_3=\mu_4=-(p+2q-1)/2, \label{ii10}
\end{equation}
and coefficients given by
\begin{eqnarray}
a_{0,j}^{(\sigma)} =  1, \qquad
2\alpha_j^{(\sigma)}ma_{m,j}^{(\sigma)} & = & 4q^2E\,a_{m-p+2q,j}^{(\sigma)} \nonumber  \\
 & &\hspace{-30pt} +\,\left((m\! +\!\mu\! -\! 1)(m\! +\! \mu)-q^2+1/4\right)a_{m-p-2q,j}^{(\sigma)}\,.  \label{ii11}
\end{eqnarray}
The series of negative powers of $t$ in the right-hand side of Eq.~(\ref{ii9}) is
 non-convergent in general, but it becomes asymptotic for $t\to+\infty$. The symbol $(-\sigma)^{1/2}$ entering the definition (\ref{ii10}) of the exponents $\alpha_j^{(\sigma)}$ is two-valued. To avoid ambiguities, we adopt the convention
\begin{equation}
(-\sigma)^{1/2} = \left\{\begin{array}{lll} 1  \qquad {\rm when} \quad \sigma=-1\,,  \\
 -{\rm i}  \qquad {\rm when} \quad \sigma=+1\,. \end{array}.\right.  \label{ii12}
\end{equation}
Then, with the notation
\begin{equation}
\phi_j^{(\sigma)}(x)\equiv t^{q-1/2}\,w_j^{(\sigma)}(t), \qquad \sigma=\pm 1, \quad j=3, 4, \label{ii13}
\end{equation}
it is evident that, for real energy $E$, the wave function $\psi(x)$ corresponding to $\phi_3^{(-1)}$ or $\phi_4^{(-1)}$ respectively vanishes or increases exponentialy as $x\to -\infty$. On the other hand, bearing in mind that the flux of probability associated to a wave function $\psi(x)$ is given (in appropriate units) by
\begin{equation}
j(x) = -\,{\rm i}\left(\psi(x)^*\,\frac{{\rm d}\psi(x)}{{\rm d}x}-\frac{{\rm d}\psi(x)^*}{{\rm d}x}\,\psi(x)\right),  \label{ii14}
\end{equation}
where the asterisk stands for complex conjugation, one realizes immediately that $\phi_3^{(+1)}(x)$ and $\phi_4^{(+1)}(x)$, which for real $E$ are complex conjugate of each other, correspond respectively to outgoing (to the right) and incoming (from the right) waves.

\section{Connection factors}

Written the physical solution of the Schr\"odinger equation in the form (\ref{ii7}), the constants $A_1$ and $A_2$ are to be chosen so as to ensure that $\psi_{\rm phys}$ is regular at $x\to\pm\infty$.
The behavior of the Frobenius solutions, $\phi_i^{(\sigma)}$ ($i=1, 2$), as $x\to\pm\infty$ can be written if their connection factors $T_{i,j}^{(\sigma)}$ with the Thom\'e solutions, defined by the relations
\begin{equation}
\phi_i^{(\sigma)}(x)\sim T_{i,3}^{(\sigma)}\,\phi_3^{(\sigma)}(x)+T_{i,4}^{(\sigma)}\,\phi_4^{(\sigma)}(x)\,, \qquad x\to \infty, \qquad i=1, 2\,,  \label{iii1}
\end{equation}
are known. The computation of the connection factors, for each given value of the energy $E$, is the crucial point in our method of solution of the Schr\"odinger equation. We use a procedure, whose theoretical basis was given in \cite{gom1}.

Obviously, the connection factors can be written as the quotient of two Wronskians
\begin{equation}
 T_{i,3}^{(\sigma)}=\frac{\mathcal{W}\left[ \phi_i^{(\sigma)},\,\phi_4^{(\sigma)}\right]}{\mathcal{W}\left[ \phi_3^{(\sigma)},\,\phi_4^{(\sigma)}\right]},\qquad
 T_{i,4}^{(\sigma)}=\frac{\mathcal{W}\left[ \phi_i^{(\sigma)},\,\phi_3^{(\sigma)}\right]}{\mathcal{W}\left[ \phi_4^{(\sigma)},\,\phi_3^{(\sigma)}\right]},\qquad  i=1,2, \label{iii2}
\end{equation}
of the solutions discussed above. The denominators, easily obtained by direct evaluation, are
\begin{equation}
\mathcal{W}\left[\phi_3^{(\sigma)},\,\phi_4^{(\sigma)}\right]=-\mathcal{W}\left[\phi_4^{(\sigma)},\,\phi_3^{(\sigma)}\right]=\alpha_4^{(\sigma)}/q\,.
\label{iii3}
\end{equation}
The computation of the numerators is more involved. In principle,
\begin{equation}
\mathcal{W}\left[ \phi_i^{(\sigma)},\,\phi_j^{(\sigma)}\right]=\frac{1}{2q}\,\mathcal{W}\left[ w_i^{(\sigma)},\,w_j^{(\sigma)}\right]\,,
\label{iii4}
\end{equation}
the right-hand side being independent of $t$. However, their direct evaluation, with the expressions (\ref{ii3}) and (\ref{ii9}) for $w_i$ and $w_j$, produces a doubly infinite series of positive and negative powers of $t$, from which it is not clear how to obtain its constant value. Our procedure to overcome this difficulty consists of comparing the asymptotic expansions of two conveniently chosen functions, $f(t)$ and $h(t)$, such that
\begin{equation}
f(t)=\mathcal{W}\left[w_i^{(\sigma)},\,w_j^{(\sigma)}\right]\,h(t)\,.  \label{iii5}
\end{equation}
The uniqueness of the asymptotic expansion of any function \cite[Sec. 1.3]{erde}, for a given asymptotic sequence, allows one to obtain the value of $\mathcal{W}[w_i^{(\sigma)},\,w_j^{(\sigma)}]$. For the function $f(t)$ we choose the Wronskian of two auxiliary functions
\begin{eqnarray}
u_{i,j}^{(\sigma)}(t)=&\exp\left( -\alpha_j^{(\sigma)}\,t^{p+2q}/(2p+4q)\right)\,w_{i}^{(\sigma)}(t),  \nonumber \\
u_j^{(\sigma)}(t)=&\exp\left( -\alpha_j^{(\sigma)}\,t^{p+2q}/(2p+4q)\right)\,w_j^{(\sigma)}(t),   \nonumber \\
  &  i=1, 2,  \qquad  j=3, 4 .  \label{iii6}
\end{eqnarray}
The corresponding function $h(t)$ becomes evident from the obvious relation
\begin{equation}
\mathcal{W}\left[u_{i,j}^{(\sigma)},u_j^{(\sigma)}\right]=\mathcal{W}\left[w_{i}^{(\sigma)},w_j^{(\sigma)}\right] \,\exp\left(-\,\alpha_j^{(\sigma)}\,t^{p+2q}/(p+2q)\right). \label{iii7}
\end{equation}
A formal expansion of the left-hand side of this equation can be obtained by direct computation, bearing in mind the definitions (\ref{iii6}) and the expressions (\ref{ii3}) and (\ref{ii9}). The result can be written in the form
\begin{equation}
\mathcal{W}\left[u_{i,j}^{(\sigma)},u_j^{(\sigma)}\right] \sim \sum_{n=-\infty}^\infty \eta_{n,i,j}^{(\sigma)}\,t^{n+\nu_i+\mu_j},   \label{iii8}
\end{equation}
with the notation
\begin{equation}
  \eta_{n,i,j}^{(\sigma)} = \sum_{m=0}^\infty a_{m,j}^{(\sigma)}\Big(\alpha_j^{(\sigma)}\,c_{n+m+1-p-2q,i}^{(\sigma)}
-(n\!+\!2m\!+\!1\!+\!\nu_i\!-\!\mu_j)\,c_{n+m+1,i}^{(\sigma)}\Big).    \label{iii9}
\end{equation}
In order to write a similar formal expansion for the right-hand side of the relation (\ref{iii7}), we write the Wronskian of $w_{i}^{(\sigma)}$ and $w_j^{(\sigma)}$ as the sum of $p+2q$ unknown constants, $\{g_{L,i,j}^{(\sigma)}\}$ ($L=0, 1, \ldots ,p+2q-1$),
\begin{equation}
\mathcal{W}\left[w_i^{(\sigma)},w_j^{(\sigma)}\right]=\sum_{L=0}^{p+2q-1} g_{L,i,j}^{(\sigma)}\,.  \label{iii10}
\end{equation}
This allows to write Eq.~(\ref{iii7}) in the form
\begin{equation}
\sum_{n=-\infty}^\infty \eta_{n,i,j}^{(\sigma)}\,t^{n+\nu_i+\mu_j}\sim\sum_{L=0}^{p+2q-1} \left(g_{L,i,j}^{(\sigma)}\,\exp\left(-\alpha_j^{(\sigma)}\,t^{p+2q}/(p+2q)\right)\right)\,.
\label{iii11}
\end{equation}
Now, recalling the Heaviside's exponential series \cite[Sec. 2.12]{hard},
\begin{equation}
\exp(z)\sim\sum_{n=-\infty}^{\infty}\frac{z^{n+\delta}}{\Gamma(n+1+\delta)},  \qquad  |\arg(z)|<\pi\,,   \label{iii12}
\end{equation}
valid for arbitrary $\delta$, we can write adequate formal expansions
\begin{equation}
\exp\left(-\alpha_j^{(\sigma)}\,t^{p+2q}/(p+2q)\right)\sim\sum_{n=-\infty}^{\infty}
\frac{\left(-\alpha_j^{(\sigma)}\,t^{p+2q}/(p+2q)\right)^{n+\delta_{L,i,j}}}{\Gamma(n+1+\delta_{L,i,j})},    \label{iii13}
\end{equation}
with
\begin{equation}
\delta_{L,i,j}=(\nu_i+\mu_j+L)/(p+2q), \qquad L=0, 1, \ldots, p+2q-1, \label{iii14}
\end{equation}
for the exponentials appearing in each one of the $p+2q$ terms in the right-hand side of Eq.~(\ref{iii11}), which becomes
\begin{equation}
\sum_{m=-\infty}^\infty \eta_{m,i,j}^{(\sigma)}t^{m+\nu_i+\mu_j} \sim \sum_{L=0}^{p+2q-1} g_{L,i,j}^{(\sigma)}\sum_{n=-\infty}^{\infty}\frac{\left(-\alpha_j^{(\sigma)}t^{p+2q}/(p+2q)\right)^{n+\delta_{L,i,j}}}
{\Gamma(n+1+\delta_{L,i,j})}.  \label{iii15}
\end{equation}
Comparison of the coefficients of $t^{(p+2q)n+\nu_i+\mu_j+L}$ ($L=0,1,\ldots,p+2q-1$) in
the two sides of this equation allows one to determine the unknown constants $g_{L,i,j}^{(\sigma)}$ and, according to (\ref{iii10}), to obtain
\begin{equation}
\mathcal{W}\left[w_{i}^{(\sigma)},w_j^{(\sigma)}\right]=\sum_{L=0}^{p+2q-1}\frac{\Gamma(n+1+\delta_{L,i,j})}
{\left(-\alpha_j^{(\sigma)}/(p+2q)\right)^{n+\delta_{L,i,j}}}\; \eta_{(p+2q)n+L,i,j}^{(\sigma)}\,.    \label{iii16}
\end{equation}
The condition $|\arg(z)|<\pi$ for the validity of the representation (\ref{iii12}) entails,  for the validity of the expansions (\ref{iii13}), the condition $|\arg(-\alpha_j^{(\sigma)})|<\pi$. This can be fulfilled, with an adequate representation of the minus sign in front of $\alpha$ by ${\rm e}^{{\rm i}\pi}$ or ${\rm e}^{-{\rm i}\pi}$, for all values of the indices $\sigma$ and $j$, except for $\sigma=-1$ and $j=4$, since $\arg\left(\alpha_4^{(-1)}\right)=0$. In this case, the positive real semi-axis is a Stokes ray for $T_{i,3}^{(-1)}$ and, as usual, this connection factor should be obtained as the average of its values in the adjacent Stokes sectors. The same effect is obtained if we take for the Wronskian of $w_{i}^{(-1)}$ and $w_4^{(-1)}$ on the ray $\arg t=0$ the average of its expressions for $t$ just above and below the positive real semiaxis. We obtain, in this way,
\begin{eqnarray}
\mathcal{W}\left[w_{i}^{(-1)},w_4^{(-1)}\right] & = & (-1)^n\sum_{L=0}^{p+2q-1}\Bigg[\frac{\cos (\delta_{L,i,4}\pi)\,\Gamma(n+1+\delta_{L,i,4})}
{\left(\alpha_4^{(-1)}/(p+2q)\right)^{n+\delta_{L,i,4}}} \nonumber \\
& & \hspace{100pt}\times\,\eta_{(p+2q)n+L,i,4}^{(-1)}\Bigg]. \label{iii17}
\end{eqnarray}

Once the necessary  Wronskians have been calculated, it is immediate to obtain, from Eq.~(\ref{iii2}), the connection factors.
The values of the $\mathcal{W}[w_{i}^{(\sigma)},w_j^{(\sigma)}]$ are independent of the integer $n$ appearing in the right-hand sides of their expressions (\ref{iii16}) and (\ref{iii17}). Different choices for $n$ must lead to the same values of the Wronskians. Another
useful test of the accuracy of the computed values of the connection factors is the relation
\begin{equation}
T_{1,3}^{(\sigma)}\,T_{2,4}^{(\sigma)}-T_{2,3}^{(\sigma)}\,T_{1,4}^{(\sigma)} =
\frac{\mathcal{W}\left[\phi_1^{(\sigma)},\phi_2^{(\sigma)}\right]}{\mathcal{W}\left[\phi_3^{(\sigma)},\phi_4^{(\sigma)}\right]} =
q \,(\alpha_4^{(\sigma)})^{-1},   \label{iii18}
\end{equation}
which must be satisfied.

\section{Global solution of the Schr\"odinger equation}

Written the physical solution in the form (\ref{ii7}), its behavior in the far regions is given, in view of the relations (\ref{ii8}) and (\ref{iii1}), by
\begin{equation}
  \psi_{\rm phys}(x)\sim\left\{\begin{array}{ll}\left(A_1T_{1,3}^{(-1)}-A_2T_{2,3}^{(-1)}\right)\,\phi_3^{(-1)}(-x) \\ \hspace{20pt} +\left(A_1T_{1,4}^{(-1)}-A_2T_{2,4}^{(-1)}\right)\,\phi_4^{(-1)}(-x),& \mbox{as}\quad x\to -\infty, \\
\left(A_1T_{1,3}^{(+1)}+A_2T_{2,3}^{(+1)}\right)\,\phi_3^{(+1)}(x) \\  \hspace{20pt} +\left(A_1T_{1,4}^{(+1)}+A_2T_{2,4}^{(+1)}\right)\,\phi_4^{(+1)}(x), &\mbox{as}\quad x\to +\infty,\end{array}\right. \label{iv1}
\end{equation}
The necessary cancelation of the divergent term as $x\to -\infty$ imposes on the coefficients $A_1$ and $A_2$ the restriction
\begin{equation}
A_1\,T_{1,4}^{(-1)}-A_2\,T_{2,4}^{(-1)}=0\,.  \label{iv2}
\end{equation}
This condition fixes the coefficients $A_1$ and $A_2$ up to a common arbitrary multiplicative constant. We may, therefore, add the requirement
\begin{equation}
A_1\,T_{1,4}^{(+1)}+A_2\,T_{2,4}^{(+1)}=1\,,  \label{iv3}
\end{equation}
unless the connection factors be such that
\begin{equation}
T_{1,4}^{(-1)}\,T_{2,4}^{(+1)}+T_{2,4}^{(-1)}\,T_{1,4}^{(+1)}=0\,,  \label{iv4}
\end{equation}
in which case
\begin{equation}
A_1\,T_{1,4}^{(+1)}+A_2\,T_{2,4}^{(+1)}=0  \label{iv5}
\end{equation}
and $\psi_{\rm phys}(x)$ becomes a pure outgoing wave as $x\to\infty$.
Leaving aside, for the moment, this possibility, which corresponds to a Gamow state, we obtain for the coefficients $A_1$ and $A_2$ the expressions
\begin{equation}
A_1=\frac{T_{2,4}^{(-1)}}{T_{1,4}^{(-1)}\,T_{2,4}^{(+1)}+T_{2,4}^{(-1)}\,T_{1,4}^{(+1)}}\,, \quad     A_2=\frac{T_{1,4}^{(-1)}}{T_{1,4}^{(-1)}\,T_{2,4}^{(+1)}+T_{2,4}^{(-1)}\,T_{1,4}^{(+1)}}\,. \label{iv6}
\end{equation}
Comparison of the expression (\ref{iv1}) with the expected behavior, Eq.~(\ref{i8}), of the physical solution gives for the scattering function
\begin{equation}
S(E)=-\, \frac{N(E)}{D(E)}\,,  \label{iv7}
\end{equation}
where
\begin{equation}
\begin{array}{ll}N(E) = T_{1,4}^{(-1)}\,T_{2,3}^{(+1)}+T_{2,4}^{(-1)}\,T_{1,3}^{(+1)}\,, \\
D(E) = T_{1,4}^{(-1)}\,T_{2,4}^{(+1)}+T_{2,4}^{(-1)}\,T_{1,4}^{(+1)}\,. \end{array}   \label{iv8}
\end{equation}
From the structure of the connection factors one can see that
\begin{equation}
N(E^*)=[D(E)]^*\,, \qquad {\rm and} \qquad  D(E^*)=[N(E)]^*\,,   \label{iv9}
\end{equation}
which imply the fulfilment of the familiar unitarity condition
\begin{equation}
S(E)[S(E^*)]^*=1\,.   \label{iv10}
\end{equation}
This expression becomes $|S(E)|=1$ when the energy $E$ is real. In this case, it is possible to define a (real) phase shift, $\delta(E)$, such that
\begin{equation}
S(E)=\exp[2{\rm i}\,\delta(E)]\,.   \label{iv11}
\end{equation}

We show in Fig. 3 the phase shift for energies in the interval $-5<E<15$ for several values of $a$. When comparing the steepness of the different curves, one should bear in mind that the variable $E$ represented on the horizontal axis is not the physical energy, but a dimensionless parameter proportional to it. If the physical problem is that of a particle of mass $m$ and a potential $V_0x^a$, the proportionality constant is, obviously,  $(2m/\hbar^2)^{a/(a+2)}V_0^{-2/(a+2)}$.
\begin{figure}
\resizebox{11cm}{!}{\includegraphics{phaseshifta=vv.eps}}
\caption{Phase shift of a wave scattered by the potential (\ref{i3}) with some values of $a$, namely $a=$2.25, 2.5, 2.75, 3, 4, 5, 6 and 7 (continuous lines), and a=0.5 (dotted line).}
\label{phsha=v}
\end{figure}

\section{Related $\mathcal{PT}$-symmetric potentials}

Now we turn our attention to the Gamow states. Their energies are the (complex) values of $E$ for which Eq.~(\ref{iv4}) is fulfilled. Alternatively, we may benefit from the close connection between the eigenstates of Hamiltonians (\ref{i5}) and those of the $\mathcal{PT}$-symmetric ones
\begin{equation}
H_a^{\mathcal{PT}}=-\,\frac{{\rm d}^2}{{\rm d}x^2}+V_a^{\mathcal{PT}}(x)\,,   \label{i6}
\end{equation}
with
\begin{equation}
V_a^{\mathcal{PT}}(x)=-{\rm i}\,{\rm sgn}(x)\,|x|^a\,,
\end{equation}
or, equivalently,
\begin{equation}
V_a^{\mathcal{PT}}(x)=\left\{\begin{array}{lll}{\rm i}\,(-x)^a,&\qquad \mbox{for}&\quad x<0, \\ -\,{\rm i}\,x^a, &\qquad \mbox{for}&\quad x>0,\end{array}\right.\,. \label{i7}
\end{equation}
Such connection is well known from the birth of the $\mathcal{PT}$-symmetric quantum theory marked by the seminal paper of Bender and Boettcher \cite{ben1}. (A tutorial introduction to $\mathcal{PT}$-symmetry and a later complete update can be found in two other papers by Bender \cite{ben2,ben3}.) In fact, the change of variable (Symanzik scaling)
\begin{equation}
x=\exp[{\rm i}\pi/(2a+4)]\,\hat{x}, \label{v1}
\end{equation}
converts the Schr\"odinger equation (\ref{i9}) in an analogous one, for the variable $\hat{x}$, with the potential
\begin{equation}
V_a^{\mathcal{PT}}(\hat{x})=\left\{\begin{array}{lll}{\rm i}\,(-\hat{x})^a,&\qquad \mbox{for}&\quad\hat{x}<0, \\ -{\rm i}\,\hat{x}^a, &\qquad \mbox{for}&\quad \hat{x}>0,\end{array}\right. \label{v2}
\end{equation}
and energy
\begin{equation}
E^{\mathcal{PT}}= \exp({\rm i}\pi/(a+2))\,E\,.  \label{v3}
\end{equation}
Besides, under such complex scaling, the solution corresponding to a Gamow state (vanishing incoming wave) of the Hamiltonian (\ref{i5}) becomes a bound state (vanishing wave function at infinity) in the potential (\ref{v2}). In other words, the positions in the lower complex $E$ half-plane of the poles of the scattering function (Gamow states) for the potential (\ref{i1}) are reached by a rotation of angle $-\pi/(a+2)$ of the poles on the real $E^{\mathcal{PT}}$ axis (bound states) of the scattering function for the potential (\ref{i7}).

We have, therefore, two strategies at our disposal. The first one is to find the complex $E$ zeros of the left-hand side of Eq.~(\ref{iv4}). The second one requires the solution of
the Schr\"odinger equation for the Hamiltonian (\ref{i6}) (analogous to Eq.~(\ref{i10}), with $\sigma$ replaced by i$\sigma$ and $E$ by $E^{\mathcal{PT}}$), to compute their connection factors $T_{i,j}^{\mathcal{PT}(\sigma)}$ by using the procedure described in Sections 3 and 4, to find the zeros $E^{\mathcal{PT}}$ of
\[
T_{1,4}^{\mathcal{PT}(-1)}\,T_{2,4}^{\mathcal{PT}(+1)}+T_{2,4}^{\mathcal{PT}(-1)}\,T_{1,4}^{\mathcal{PT}(+1)}=0\,,
\]
and, finally, to apply the inverse scaling
\begin{equation}
E= \exp(-{\rm i}\pi/(a+2))\,E^{\mathcal{PT}}.  \label{v4}
\end{equation}
to obtain the Gamow energies. In both strategies one has to find zeros of a function computed numerically. But it is much easier to determine real zeros (in the case of unbroken $\mathcal{PT}$-symmetry) than complex ones. For this reason we have adopted the second strategy. As a bonus, we unveil the breakdown of the $\mathcal{PT}$-symmetry, that is, the existence of complex eigenvalues, in the family of Hamiltonians $H_a^\mathcal{PT}$ for $a<3$.

We report in Table 1 the five lowest eigenvalues of the $PT$-symmetric Hamiltonians (\ref{i6}), for some values of $a$. For comparison, we have added the eigenvalues of $H_2^{\mathcal{PT}}$, for which the Schr\"odinger equation becomes algebraically solvable \cite{fer3}. The three lowest eigenvalues of $H_5^{\mathcal{PT}}$ and $H_7^{\mathcal{PT}}$, obtained by using the Riccati-Pad\'{e} method, are shown in Table 6 of \cite{fega}. The total agreement with our results makes evident the power of the method.
\begin{table}
\caption {The first five eigenvalues $E^{\mathcal{PT}}$ of the Hamiltonians (\ref{i6}) for several values of the parameter $a$. The $\mathcal{PT}$-symmetry of $H_a$ for $a\geq 3$ is unbroken, as its spectrum is entirely real. For the considered values of $a<3$ the number of real eigenvalues is finite and pairs of complex eigenvalues appear: the $\mathcal{PT}$-symmetry becomes broken. (In the case of $a=2.75$, the first complex conjugate pair is constituted by the 6th and 7th eigenvalues, as explained in the text.) The mechanism of convergence of real eigenvalues into complex ones, as $a$ decreases from 3, is that shown in \cite{ben1} and \cite{ben2}.}
\begin{tabular}{|l|rrrrr|}
\hline
$a$ & 1st eigenv. & 2nd eigenv. & 3rd eigenv. & 4th eigenv. & 5th eigenv.  \\
\hline
\footnotesize 2 & \footnotesize 1.2580917622 & \multicolumn{2}{c}{\footnotesize 4.99131440 $\pm$ 0.78048559 i} & \multicolumn{2}{c|}{\footnotesize 8.61814319 $\pm$ 3.36325532 i} \\
\footnotesize 2.125 & \footnotesize 1.2339159563 & \footnotesize 4.8245841893 & \footnotesize 5.5401494915 & \multicolumn{2}{c|}{\footnotesize 9.30471749 $\pm$ 3.04996251 i} \\
\footnotesize 2.2 & \footnotesize 1.2217833975 & \footnotesize 4.5449222689 & \footnotesize 6.0175212001 & \multicolumn{2}{c|}{\footnotesize 9.71611618 $\pm$ 2.82567175 i} \\
\footnotesize 2.25 & \footnotesize 1.2145298920 & \footnotesize 4.4495432370 & \footnotesize 6.2362764414 & \multicolumn{2}{c|} {\footnotesize 9.98930212 $\pm$ 2.65983187 i} \\
\footnotesize 2.5 & \footnotesize 1.1862679202 & \footnotesize 4.2168282738 & \footnotesize 6.9332328859 & \multicolumn{2}{c|} {\footnotesize 11.31541913 $\pm$ 1.56603260 i$\;$} \\
\footnotesize 2.75 & \footnotesize 1.1678900359 & \footnotesize 4.1350219842 & \footnotesize 7.3113845045 &\footnotesize 11.2251446520 & \footnotesize 13.7120060272 \\
\footnotesize 3 & \footnotesize 1.1562670720 & \footnotesize 4.1092287528 & \footnotesize 7.5622738550 & \footnotesize 11.3144218202 & \footnotesize 15.2915537504 \\
\footnotesize 4 & \footnotesize 1.1468501770 & \footnotesize 4.1943612352 & \footnotesize 8.3020614579 & \footnotesize 12.9101152346 & \footnotesize 18.1061761121 \\
\footnotesize 5 & \footnotesize 1.1647704080 & \footnotesize 4.3637843677 & \footnotesize 8.9551669982 & \footnotesize 14.4177548303 & \footnotesize 20.6101375101 \\
\footnotesize 6 & \footnotesize 1.1928491725 & \footnotesize 4.5449872159 & \footnotesize 9.5456178719 & \footnotesize 15.7278538420 & \footnotesize 22.8656231475 \\
\footnotesize 7 & \footnotesize 1.2247116893 & \footnotesize 4.7214625354 & \footnotesize 10.0754495631 & \footnotesize 16.8724570744 & \footnotesize 24.8575115670 \\
\hline
\end{tabular}
\end{table}

Table 1 also illustrates the breakdown of $\mathcal{PT}$-symmetry for $a$ below a critical value, $a_{\rm cr}$, which we conjecture to be equal to 3. For values, $a\geq 3$, the spectrum of $H_a$ is entirely real, according to  \cite{ben1} and \cite{shin}. In the case of $a=3$, for instance, besides the first five eigenvalues shown in table 1, we have obtained the sequence
\[
\begin{array}{ll}19.4515291307\,, \quad 23.7667404355\,, \quad 28.2175249730\,, \quad  32.7890827819\,, \\
                 37.4698253605\,, \quad 42.2504052192\,, \quad 47.1231055739\,,  \quad 52.0814360527\,, \quad {\rm etc.} \end{array}
\]
For $a=2.75$, instead, only the first five eigenvalues, shown in Table 1, are real; then
there is the  complex conjugate pair 18.91286864 $\pm$ 2.84850650 i\,. For $a=2.5$, only three real eigenvalues remain. And so on. The symmetry-breakdown mechanism is entirely analogous to that reported by Bender and Boettcher \cite{ben1} for the family of Hamiltonians
\begin{equation}
H_N^{\rm BB} = -\frac{{\rm d}^2}{{\rm d}x^2}-({\rm i}\,x)^N\,, \label{v5}
\end{equation}
whose $\mathcal{PT}$-symmetry becomes broken for $N<2$. (Notice that our family $\{H_a^\mathcal{PT}\}$ of Hamiltonians is different from the $\{H_N^{\rm BB}\}$ of Bender and Boettcher. However, $V_3^{\mathcal{PT}}(x)=\mathcal{P}\,V_3^{BB}(x)$ and the spectra of $H_3^{\mathcal{PT}}$ and $H_3^{BB}$ are the same. For other values of $a$ and $N$, the potentials $V_a^\mathcal{PT}$ and $V_N^{\rm BB}$ may be the same (for $a=N=5, 9, \ldots$) or $\mathcal{P}$-transformed of each other (for $a=N=7, 11, \ldots$), but the spectra of $H_a^\mathcal{PT}$ and $H_N^{\rm BB}$, with odd integer $a=N>3$, should not be expected to be coincident, since the self-adjoint extensions considered in the two cases are different: we are imposing to the eigenfunctions of $H_a^\mathcal{PT}$ boundary conditions on the real axis, whereas, in the case of $H_N^{\rm BB}$ with $N\geq 4$, the real axis is replaced by a contour in the complex plane.)

The fact that complex $\mathcal{PT}$-symmetric Hamiltonians possess real eigenvalues and, consequently, non-decaying eigenstates has been attributed \cite{nobl} to the exact balance of the positive and negative imaginary parts of the potential, that is, to the equilibrated source and drain of probability associated to those imaginary parts. But the reason of the breakdown of the  $\mathcal{PT}$-symmetry is far from being completely understood. It has been conjectured \cite{ben4} that, except in very exceptional cases, the analyticity of the potential is a necessary condition for the entire spectrum of a complex $\mathcal{PT}$-symmetric Hamiltonian to be real. In a recent paper \cite{ahm4}, Z.~Ahmed {\em et al} have presented a family of solvable potentials that seem to support the conjecture. Our results may be useful for an eventual discussion of that still unproved conjecture.

Another interesting issue raised in Ref.~\cite{nobl} is what could be called the ``existence of a classically allowed region" in complex potentials. Noble {\em et al} have shown that the eigenfunctions of $\mathcal{PT}$-symmetric anharmonic oscillators are concentrated in a region where the eigenenergy is above the modulus of the complex potential. The same fact can be observed in our case of purely imaginary potentials. For instance, in the case of $V_3^{\mathcal{PT}}=-{\rm i}\,x^3$, the first eigenenergy  $E_1=1.156267072$ is larger than the modulus of the potential for $-1.05\lesssim x \lesssim 1.05$. In Fig.~4, which shows the squared modulus of the normalized eigenfunction, a concentration of the probability in the mentioned classically allowed region is clearly seen. For a quantitative comparison, we report in Table 2 the values of $|\psi_1(x)|^2$ at several points.
\begin{figure}
\resizebox{11cm}{!}{\includegraphics{mod2wfPT.eps}}
\caption{Squared modulus of the normalized first eigenfunction of the Hamiltonian $H_3^{\mathcal{PT}}=-{\rm d}^2/{\rm d}x^2-{\rm i}\,x^3$. The bulk of the probability lies in the region $-1.05<x<1.05$, which would be the classically allowed region, according to Ref.~\cite{nobl}.}
\label{m2wf}
\end{figure}
\begin{table}
\begin{center}
\caption {Numerical values of the squared modulus of the  normalized first eigenfunction of the Hamiltonian $H_3^{\mathcal{PT}}=-{\rm d}^2/{\rm d}x^2-{\rm i}\,x^3$, shown in Fig.~4. The probability is drastically reduced out of the interval (-1.05,1.05), considered by Noble {\em et al} \cite{nobl} as the classically allowed region.}
\begin{tabular}{|l|l|}
\hline
$\pm\,x$ & $|\psi_1(x)|^2$  \\
\hline
0 & 0.541405 \\
0.5 & 0.434904 \\
1 & 0.219137 \\
1.05 & 0.198931 \\
1.1 & 0.179604 \\
1.5 & 0.064408 \\
2 & 0.010063 \\
2.5 & 0.000760 \\
\hline
\end{tabular}
\end{center}
\end{table}

\section{Gamow states}

Once the eigenvalues of the Hamiltonians $H_a^{\mathcal{PT}}$ have been computed, the Gamow energies of the Hamiltonians  $H_a$ are immediately obtained. A table analogous to Table 1, with each entry multiplied by a factor $\exp(-{\rm i}\pi/(a+2))$, would result. Comparison can be made with the values obtained by Fern\'andez and Garcia \cite[Table 4]{fega} for the first six Gamow states of the anharmonic oscillators
\begin{equation}
\frac{1}{2}p^2-x^K\,,  \qquad {\rm with} \quad K = 3 \; {\rm and}\; 5.  \label{vi1}
\end{equation}
Due to the different notation for the kinetic energy term in the Hamiltonians (\ref{i5}) and (\ref{vi1}), our Gamow energies (i. e., the values given in our Table 1 multiplied by $\exp[-{\rm i}\pi/(a+2)]$) for $a=K$ should be multiplied by a factor $2^{-K/(K+2)}$ to be compared with the results given in  \cite{fega}. The agreement is total.

The wave functions of the Gamow states can be written immediately. For small or moderate values of $x$, the expressions in Eq.~(\ref{ii7}), with coefficients related as in Eq.~(\ref{ii8}), are applicable. The fulfilment of Eq.~(\ref{iv5}) requires to take
\begin{equation}
A_1=A\,T_{2,4}^{(+1)}\,,\qquad A_2=-\,A\,T_{1,4}^{(+1)}\,  \label{vi2}
\end{equation}
where $A$ represents an arbitrary constant, which may be chosen conveniently
to fix the norm and the phase of the wave function. For large values of $x$ one should use the asymptotic expressions
\begin{eqnarray}
  \psi_{\rm phys}(x) & \sim & \frac{1}{2}\,A\,\frac{T_{2,4}^{(+1)}}{T_{2,4}^{(-1)}}\,\exp\left[-\frac{(-x)^{1+a/2}}{1+a/2}\right](-x)^{-a/4}\sum_{m=0}^\infty a_{m,3}^{(-1)}\,(-x)^{-m/(2q)}, \nonumber \\  & &  \hspace{220pt} x<0\,,   \label{vi3}  \\
  \psi_{\rm phys}(x) & \sim &  \frac{{\rm i}}{2}\,A\,\exp\left[{\rm i}\,\frac{x^{1+a/2}}{1+a/2}\right]x^{-a/4}\sum_{m=0}^\infty a_{m,3}^{(+1)}\,x^{-m/(2q)},   \qquad x>0\,,   \label{vi4}
\end{eqnarray}
where use has been made of Eqs. (\ref{iii18}) and (\ref{iv4}).

Figures 5 and 6 illustrate the kinds of graphics obtained for the wave functions of the Gamow states of the Hamiltonians $H_a$. We have represented the two lowest eigenstates of $H_6$; the results for other values of $a>2$ are similar. The wave functions are normalized in the form
\begin{equation}
\int_{-\infty}^{+\infty}|\psi(x)|^2\,{\rm d}x = 1\,.  \label{vi5}
\end{equation}
For increasing negative values of $x$, the wave function vanishes according to Eq.~(\ref{vi3}). On the large positive $x$ side, the real and imaginary parts of the wave function oscillate according to Eq.~(\ref{vi4}). The amplitude and the wavelength of the oscillations decrease as $x^{-a/4}$ and $x^{-a/2}$, respectively, as $x$ increases.

It remains to investigate the influence of the existence of Gamow states on the scattering at real energies. With this purpose, we have considered the quantity
\begin{equation}
\Delta\tau=2\,\frac{{\rm d}\,\delta (E)}{{\rm d}E}\,,  \label{vi6}
\end{equation}
proportional to the time delay \cite{caac} suffered by the scattered wave, for the range of energies $-5\leq E \leq 15$. Approximate values of $\Delta\tau$ were calculated by numerical derivation with respect to $E$ of the phase shifts obtained in Section 5. The results for several values of $a>2$ are shown in Fig.~7. Notice that, as said at the end of Section 5, $E$ is not the physical energy, but proportional to it. Consequently, when comparing the different curves shown in the figure, the dependence on $a$ of the horizontal and vertical scales should be borne in mind. In all curves a resonance (peak), associated to the first Gamow state, is clearly marked. It becomes sharper for larger $a$, as was to be expected from the fact that the pole of $S(E)$ in the lower complex half-plane, located at the Gamow energy, approaches the real axis as $a$ increases. For $a\geq 4$, a less marked second resonance, associated to the second Gamow state, becomes apparent. These results suggest that, although the number of Gamow states of $H_a$ seems to be infinite, only a finite number of them have effective influence on the scattering at real energies. The number of true resonances and their mean life increase with $a$.
\begin{figure}
\resizebox{11cm}{!}{\includegraphics{firstGamow.eps}}
\caption{Wave function of the first Gamow state of the Hamiltonian $H_6$. The squared modulus (full) and the real (dashed) and imaginary (dotted) parts of the wave function are shown. The  multiplicative constant has been chosen in such a way that the function becomes normalized as in equation (\ref{vi5}) and positive at $x=0$.}
\label{wfG1}
\end{figure}
\begin{figure}
\resizebox{11cm}{!}{\includegraphics{secondGamow.eps}}
\caption{Wave function of the second Gamow state of $H_6$. The comments in the caption of the preceding figure are applicable here.}
\label{wfG2}
\end{figure}
\begin{figure}
\resizebox{11cm}{!}{\includegraphics{timedelaya=vv.eps}}
\caption{Time delay suffered by a wave scattered by potentials of the form (\ref{i3}) with $a=$2.5, 3, 4, 7 (continuous lines), and 0.5 (dotted line).}
\label{tidea=v}
\end{figure}

\section{Particular case of $a=1/2$}

The formalism developed above, in sections 3 to 7, is directly applicable in the case of $a>2$. Our method of solution of the Schr\"odinger equation, based on the computation of the connection factors of the Frobenius solutions with the Thom\'e ones, is also applicable in the case of $0<a\leq 2$, although general expressions, valid for any $a$, cannot be written. Here we outline the procedure to be followed in the case of $a=1/2$. This is an example that, besides of constituting an illustration of the applicability of our method, gives results very different from those obtained for $a>2$.

The formalism developed in Eqs. (\ref{ii1}) to (\ref{ii8}) is valid also in this case, with $p=1$ and $q=2$. The Thom\'e solutions, however, are not given by Eq.~(\ref{ii9}), but by
\begin{equation}
w_j^{(\sigma)}(t)=\exp \left[\alpha_j^{(\sigma)}\,\frac{t^5}{5}+\beta_j^{(\sigma)}\,\frac{t^3}{3}+\gamma_j^{(\sigma)}\,t\right]\,t^{\mu_j}\,\sum_{m=0}^\infty a_{m,j}^{(\sigma)}\,t^{-m}, \qquad j=3, 4,  \label{vii1}
\end{equation}
with exponents $\alpha_j^{(\sigma)}$ and $\mu_j$ as given in Eqs. (\ref{ii10}) and (\ref{ii12}), new exponents
\begin{equation}
 \beta_j^{(\sigma)}=-\,8E/\alpha_j^{(\sigma)}\,, \qquad  \gamma_j^{(\sigma)}=-\,(\beta_j^{(\sigma)})^2/(2\alpha_j^{(\sigma)})\,, \label{vii2}
\end{equation}
and coefficients $a_{m,j}^{(\sigma)}$ obeying the recurrence relation
\begin{eqnarray}
a_{0,j}^{(\sigma)} =  1, \qquad
2\alpha_j^{(\sigma)}m\,a_{m,j}^{(\sigma)} & = & 2\beta_j^{(\sigma)}\gamma_j^{(\sigma)}\,a_{m-1,j}^{(\sigma)}-2(m\! -\! 1)\beta_j^{(\sigma)}\,a_{m-2,j}^{(\sigma)} \nonumber  \\
 & &\hspace{-150pt} +\,(\gamma_j^{(\sigma)})^2\,a_{m-3,j}^{(\sigma)}-2(m\! -\! 2)\gamma_j^{(\sigma)}\,a_{m-4,j}+\left((m\!-\! 3)(m\! -\! 2)-15/4\right)a_{m-5,j}^{(\sigma)}\,.  \label{vii3}
\end{eqnarray}
Then, Eqs. (\ref{iii1}) to (\ref{iii8}), with $p=1$ and $q=2$, are applicable. Equation (\ref{iii9}), instead, should be replaced by
\begin{eqnarray}
\eta_{n,i,j}^{(\sigma)} &=& \sum_{m=0}^\infty a_{m,j}^{(\sigma)}\Big(\alpha_j^{(\sigma)}\,\hat{c}_{n+m-4,i,j}^{(\sigma)}
+2\beta_j^{(\sigma)}\,\hat{c}_{n+m-2,i,j}^{(\sigma)}+2\gamma_j^{(\sigma)}\,\hat{c}_{n+m,i,j}^{(\sigma)} \nonumber  \\
 & &\hspace{80pt} -\,(n+2m+3+\nu_i)\,\hat{c}_{n+m+1,i,j}^{(\sigma)}\Big),    \label{vii4}
\end{eqnarray}
where the $\hat{c}_{r,i,j}^{(\sigma)}$ are the coefficients of the series expansion of $\exp\left(\beta_j^{(\sigma)}t^3/3+\gamma_j^{(\sigma)}t\right)\,w_i^{(\sigma)}(t)$,
\begin{equation}
\exp\left(\beta_j^{(\sigma)}t^3/3+\gamma_j^{(\sigma)}t\right)\,w_i^{(\sigma)}=\sum_{n=0}^\infty \hat{c}_{n,i,j}^{(\sigma)}\,t^{n+\nu_i}\,.  \label{vii5}
\end{equation}
These coefficients can be calculated by means of the recurrence relation
\[
\hat{c}_{0,i,j}^{(\sigma)}=1\,, \qquad  \hat{c}_{4,1,j}^{(\sigma)} = \gamma_j^{(\sigma)}\left(\beta_j^{(\sigma)}/3+(\gamma_j^{(\sigma)})^3/4!\right)\,,
\]
\begin{eqnarray}
n(n+2\nu_i-1)\hat{c}_{n,i,j}^{(\sigma)} & = & 2(n-1+\nu_i)\gamma_j^{(\sigma)}\,\hat{c}_{n-1,i,j}^{(\sigma)}   -(\gamma_j^{(\sigma)})^2\,\hat{c}_{n-2,i,j}^{(\sigma)} \nonumber  \\
 & & \hspace{-50pt}+\,2(n-2+\nu_i)\beta_j^{(\sigma)}\,\hat{c}_{n-3,i,j}^{(\sigma)}
 -2\beta_j^{(\sigma)}\gamma_j^{(\sigma)}\,\hat{c}_{n-4,i,j}^{(\sigma)}   \nonumber  \\
 & & \hspace{-50pt}-\,(\beta_j^{(\sigma)})^2\,\hat{c}_{n-6,i,j}^{(\sigma)}
 -16E\,\hat{c}_{n-8,i,j}^{(\sigma)}-16\sigma\,\hat{c}_{n-10,i,j}^{(\sigma)}\,.  \label{vii6}
\end{eqnarray}
(Details of how to come at Eqs. (\ref{vii4}) and (\ref{vii6}) can be found in Ref. \cite[Appendix B]{gom2}.) The procedure, then, continues as in the case of $a>2$, Eqs. (\ref{iii10}) to (\ref{iv11}).

The results obtained for the phase shift and the time delay are shown in Figs. 3 and 7, respectively. They are very different from those found for $a>2$. Instead of increasing monotonously with the energy, the phase shift for $a=1/2$ decreases slightly at negative energies $E\lesssim -0.5$ and then increases with a steeper and steeper slope. The corresponding time delay does not show any indication of the existence of resonances.

\section{Final comments}

We have shown, in the preceding sections, how it is possible to write a scattering function (of the complex energy) which represents the interaction of a wave with a potential of the form given in Eq.~(\ref{i3}). This scattering function obeys the usual unitarity condition, which allows the definition, at real energies, of the phase shift. As it occurs in the case of short-range potentials, poles in the lower complex-energy half-plane correspond to Gamow states. The time delay, obtained as the derivative of the phase shift with respect to the energy, has served to analyze the possible existence of real-energy resonant scattering. Our conclusion is that there exist only a few resonances, at energies associated to the real part of the first Gamow poles. The larger the value of the parameter $a$, the more and sharper resonances are found.

Although it was not the main purpose of this paper, we have found a breakdown of the $\mathcal{PT}$-symmetry in the family of Hamiltonians (\ref{i6}) for $a<a_{\rm cr}\leq 3$. Real eigenvalues and the first complex ones are given explicitly in Table 1 for some rational values of $a$ in
the interval $2<a<3$.

Obtainig global solutions of the Schr\"odinger equation is essential for our determination of the scattering function. In principle, two local solutions valid for $x\to +\infty$ or two other for $x\to -\infty$ can be immediately written: they are the Thom\'e solutions, given in Eq.~(\ref{ii9}). Those for $x\to+\infty$ represent, respectively, incoming  and outgoing waves, whereas those for $x\to-\infty$ correspond, respectively, to exponentially decreasing or increasing waves. To comply with the physical conditions, the combination of incoming and outgoing waves, which involves the scattering function $S(E)$, should match with the solution which vanishes as $x\to-\infty$: this determines $S(E)$. But the matching cannot be imposed directly, but trough other local solutions valid for finite $|x|$: the two Frobenius solutions, given in Eq.~(\ref{ii3}). Frobenius and Thom\'e solutions are linked by the connection factors, as shown in Eq.~(\ref{iii1}). The problem of solving the Schr\"odinger equation with the appropriate boundary conditions becomes, in this way, that of computing the connection factors. We have exploited, with this purpose, a procedure which has proved to be useful for the solution of other quantum mechanical problems. The procedure, developed in Section 4, is rigorous and does not require  approximations like perturbation expansions or truncation of the Hilbert space. However, numerical computation is unavoidable to obtain the coefficients of the local solutions, by application of the corresponding recurrence relations, and for summing the series defining the connection factors and the function of the energy whose zeros are the eigenvalues. As a reward, explicit expressions of the wave functions, in the form of convergent series or asymptotic expansions, are obtained.

\section*{Acknowledgments}The comments of two anonymous reviewers have greatly contributed to improve the presentation of this work. Financial support from Conselho Nacional de Desenvolvimento Cient\'{\i}fico e Tecnol\'{o}gico  (CNPq, Brazil) and from  Gobierno de Arag\'on and Fondo Social Europeo (Project E24/1) and Ministerio de Ciencia e Innovaci\'on (Project MTM2012-33575) is gratefully acknowledged.

redpajama_set_name: RedPajamaArXiv


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text:
#include <net-snmp/net-snmp-config.h>
#include <net-snmp/net-snmp-includes.h>
#include <net-snmp/agent/net-snmp-agent-includes.h>

#ifndef NSH_PRIVATE_H_
#define NSH_PRIVATE_H_

#define FN_HANDLE(_n) ( nsh_handle_ ## _n )
#define FN_INDEX(_n)  ( _n ## _index )

#define ENTRY_SIZE(s,e) sizeof(((s*)0)->e)

#ifndef UNUSED
#define UNUSED(x) UNUSED_ ## x __attribute__ ((unused))
#endif

#define NSH_DFLT_MIN_COLUMN 1
#define NSH_DFLT_TIMEOUT    5

typedef netsnmp_variable_list* (*nsh_get_first_cb)(void                  **loop_context,
						   void                  **data_context,
						   netsnmp_variable_list *put_index_data,
						   netsnmp_iterator_info *data);

typedef netsnmp_variable_list* (*nsh_get_next_cb)(void                  **loop_context,
						  void                  **data_context,
						  netsnmp_variable_list *put_index_data,
						  netsnmp_iterator_info *data);

typedef void (*nsh_free_cb)(netsnmp_cache *cache,
			    void          *vmagic);

struct nsh_table_index_t;

typedef struct nsh_table_reg_t {
	nsh_get_first_cb          get_first;
	nsh_get_next_cb           get_next;
	nsh_free_cb               free;
	struct nsh_table_index_t* idx;
	int                       num_idx;
	int                       min_column;
	int                       max_column;
	int                       timeout;
} nsh_table_reg_t;

int __nsh_scalar_handler(u_char type,
			 int id,
			 int (*get_cb)(void *val, int len, int id),
			 int get_sz,
			 long get_arg,
			 int (*set_cb)(void *val, int id),
			 netsnmp_mib_handler *UNUSED(handler),
			 netsnmp_handler_registration *UNUSED(reginfo),
			 netsnmp_agent_request_info *reqinfo,
			 netsnmp_request_info *requests);

/**
 * _nsh_register_scalar - Register a OID scalar handler
 * @_name   : OID define. The defined must be named oid_@name and on
 *            format 'x,y,z...'.
 * @_access : HANDLER_CAN_RONLY | HANDLER_CAN_RWRITE
 *
 * This function will register a scalar handler callback for the OID @_oid.
 * The name of the callback function is based on the variable name @_oid,
 * i.e. the same name shall also be used as input to nsh_scalar_handler().
 */
#define _nsh_register_scalar(_name, _access)				\
({									\
	oid _o[] = { oid_ ## _name };					\
	netsnmp_handler_registration *_reginfo;				\
	int _ret = MIB_REGISTERED_OK;					\
									\
	_reginfo = netsnmp_create_handler_registration(			\
		#_name, FN_HANDLE(_name), _o, OID_LENGTH(_o), _access);	\
	if (_reginfo)							\
		_ret = netsnmp_register_scalar(_reginfo);		\
	_ret;								\
})

/**
 * _nsh_scalar_handler - Create a scalar handler
 * @_oid      : OID define, see nsh_register_scalar().
 * @_type     : OID type:
 *              %ASN_COUNTER, %ASN_INTEGER, %ASN_TIMETICKS, %ASN_UNSIGNED,
 *              %ASN_IPADDRESS, %ASN_OCTET_STR or %ASN_OBJECT_ID
 * @_id:      : ID in a scalar group, ignored if grouping is not used.
 *              A group of scalar will share the same @_get_cb function and @_id
 *              is used to distinguish the OID requests from each other.
 * @_get_cb   : Callback function to get OID value.
 *              If %NULL and @_type is not set to %ASN_OCTET_STR or
 *              %ASN_OBJECT_ID, the value of @_get_arg will be returned.
 * @_get_sz   : Length of returned value.
 *              When @_type is %ASN_OBJECT_ID this will be the size of OID.
 *              When @_type is %ASN_OCTET_STR this will be the size of the array
 *              or the maximum string size if @_get_arg is set.
 * @_get_arg  : Used as return value if %_get_cb is %NULL.
 *              If set and @_type is %ASN_OCTET_STR, the value returned is
 *              treated as a string and will be of length strlen() instead of
 *              @_get_sz.
 * @_set_cb   : Callback function to set OID value.
 *              If %NULL OID is readonly.
 *
 * This function will create a scalar handler callback. The callback function
 * name will be based on the name of the @_oid parameter, so the same @_oid name
 * shall be used here as when registering the OID with nsh_register_scalar().
 *
 * Returns:
 * %SNMP_ERR_NOERROR on success.
 */
#define _nsh_scalar_handler(_oid, _type, _id,				\
			   _get_cb,  _get_sz, _get_arg, _set_cb)       	\
static int FN_HANDLE(_oid)(netsnmp_mib_handler *handler,	       	\
			 netsnmp_handler_registration *reginfo,		\
			 netsnmp_agent_request_info *reqinfo,		\
			 netsnmp_request_info *requests)		\
{									\
	return __nsh_scalar_handler(_type, _id,				\
				    _get_cb, _get_sz, _get_arg,		\
				    _set_cb,				\
				    handler,				\
				    reginfo, reqinfo, requests);	\
}

#endif /* NSH_PRIVATE_H_ */

/**
 * Local Variables:
 *  c-file-style: "linux"
 * End:
 */

redpajama_set_name: RedPajamaGithub


Robert Altman's McCabe & Mrs. Miller is a film that feels preserved in amber. It's a slow-burn piece of dramatic cinema that takes a rather bleak look at the inevitability of death, and I can't seem to find one negative thing to say about it after numerous viewings. Serving as yet a further reminder of how versatile and unpredictable a filmmaker Robert Altman was, especially when compared with our current cookie-cutter studio mentality, a work like McCabe and Mrs. Miller feels nearly impossible to contemplate being made today. The laconic pace is purposeful, character motivation is sometimes fuzzy and dictated by living in the moment rather than through a complete worldview, and the way that Altman and co-writer Brian McKay were able to weave various layers into their story in an organic fashion is worthy of study.
Adapted from the novel McCabe by Edmund Naughton, the film stars Warren Beatty and Julie Christie, both flawless in two of the best performances in their legendary careers, subverting expectations and creating two lasting portraits of tough-love individuals who are realists more than anything else. I love how this film isn't really a "western" in the traditional sense, but it's got the atmosphere and personality of one at times, with hardscrabble production values and Altman's now-famous use of overlapping dialogue in full effect, resulting in a film that feels akin to a dream. The phenomenal supporting cast includes William Devane, Michael Murphy, Shelley Duvall, Antony Holland, Hugh Millais, Rene Auberjonois, and Keith Carradine in his screen debut.
Vilmos Zsigmond's warmly fuzzy widescreen cinematography goes for wide shots mixed with slow zooms, creating a sense of openness while still retaining a certain level of intimacy, and never losing sight of the desolate and chilly landscape. The "flashing" technique that Altman and Zsigmond favored during post-production created a halo effect to the images, eliminating any color saturation, resulting in a picture quality that's gauzy and opaque. Roger Deakins must have watched this movie 1,000 times before shooting The Assassination of Jesse James by the Coward Robert Ford.
Leonard Cohen's sparse musical score is yet another key ingredient to Altman's masterpiece, with the two artists joining forces as Cohen was a big fan of Altman's idiosyncratic comedy Brewster McCloud, and Altman was particularly taken with Cohen's dark and brooding style from his debut album. The evocative production design by Leon Erickson was in perfect tandem with Ilse Richter's lived-in costumes; there isn't one department of the filmmaking process that wasn't operating at peak strength during this production. The aesthetic chill in McCabe and Mrs. Miller extends to the film's unsentimental themes as well, creating total package cinema in a way that most filmmakers could only have dreamed of achieving. Especially the film's haunting, snowy final moments which are fully appropriate is particularly downbeat.
Few other storytellers would have been able to pull of the tricky balancing act that is McCabe & Mrs. Miller, yet Altman, ever the sly storyteller, managed to keep you engaged to a narrative that offers little in the way of conventional cinematic pleasure, and instead invites you to watch and listen as two exceedingly selfish yet practical people try and figure out how to survive life in some of the most unpredictable and unsparing of circumstances. Second unit director and lead editor Louis Lombardo had a nearly Herculean task in terms of assembling the picture; apparently the editing process lasted nine months. And because the film involves multiple characters, both big and small, there's a mosaic quality that begins to take shape, as the central relationship between Beatty and Christie doesn't kick into full-gear until half of the movie's running time has elapsed. Altman even asked all of the background performers to "find their characters" on their own, having them chose their own costumes, and create their own backstories so as to feel more realistic.
A film like this would be very hard to get made today unless it were a passion project for a very influential star and/or filmmaker, as it's more about mood and relationships than it is about concrete plotting with easy to define character traits for the various personalities that comprise the story. In 2010, McCabe and Mrs. Miller was selected for preservation in the United States National Film Registry by the Library of Congress, and last year, the film became available on Blu-ray from the Criterion Collection. This supreme work of art deserves a spot in any film fan's physical media collection.
Nicely down. I agree, it's a perfect film. So is The Long Goodbye. They just work. Very poetic.. Thanks.


text:
Q: Don't return repeated ids and give priority to the second table I have the following tables:
CREATE TABLE usuarios ( id INT, nome varchar );
INSERT INTO usuarios VALUES
(1, 'Teste'),
(2, 'Teste1'),
(3, 'Teste2'),
(4, 'Teste3'),
(5, 'Teste4'),

CREATE TABLE FichaColab( id INT, nomcompleto varchar );
INSERT INTO FichaColab VALUES
(1, 'Teste Teste'),
(3, 'Teste2 Teste2'),
(5, 'Teste4 Teste4'),

I intend to get all the names from the first table, but if the id exists in the second table, return the name of the second table instead of the name of the first.
Here is the result I want:

id
nome

1
Teste Teste

2
Teste1

3
Teste2 Teste2

4
Teste3

5
Teste4 Teste4

I'm trying like this:
SELECT usuarios.id, usuarios.nome
FROM usuarios
UNION
SELECT FichaColab.Id, nomcompleto
FROM FichaColab

But this way returns everything from the two tables and repeats the ids and can't.

A: This should do the job:
select * from (
select FichaColab.* from FichaColab left join  usuarios on usuarios.id = FichaColab.id --everithing in table FichaColab
union
select usuarios.* from FichaColab right join usuarios on usuarios.id = FichaColab.id where FichaColab.id is null --everithing from usuarios where no record in FichaColab exists
) as unsorted
order by ID --sort it

Demo

A: You can use a LEFT JOIN between the two tables, then COALESCE on the name, by giving priority on the full name.
SELECT u.id, COALESCE(fc.nomcompleto, u.nome) AS nome
FROM      usuarios   u
LEFT JOIN FichaColab fc ON u.id = fc.id

Check the demo here.

redpajama_set_name: RedPajamaStackExchange


text:
Home Flashbox Movies & TV News & Updates Videos
Watch Bride Battle Husband's Insane Family in New 'Ready or Not' Trailer
Film stars Samara Weaving, Mark O'Brien, Adam Brody, Andie MacDowell, Henry Czerny
Jon Blistein Jun 18, 2019
Samara Weaving as Grace in 'Ready or Not' trailer.
A young bride is forced to survive her new family's outrageous tradition in the bloody and funny trailer for Ready or Not, out August 23rd.
The film stars Samara Weaving as Grace, who marries into the wealthy and strange Le Domas family. But after her wedding to Alex Le Domas (Mark O'Brien), she's told the marriage won't be complete until she plays a little game. While Grace suspects the round of "Hide and Seek" she's roped into is just a quirky ritual, she soon realizes that the La Domas family is actually trying to hunt and kill her.
A surreal cat-and-mouse chase ensues, with Alex ostensibly trying to help Grace survive while the rest of the La Domas clan remains hellbent on sacrificing her in a mysterious ritual that apparently keeps them all alive. The trailer teases a clever blend of gruesome mayhem and black humor, best exemplified in the various maids the La Domas family keeps accidentally killing during their hunt.
Ready or Not also stars Adam Brody, Henry Czerny, Andie MacDowell, Kristian Bruun, Melanie Scrofano, Elyse Levesque and Nicky Guadagni. Matt Bettinelli-Olpin and Tyler Gillett directed the film.
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Movie Review: Star Wars: The Force Awakens
Peter Travers Dec 24

redpajama_set_name: RedPajamaCommonCrawl


text:
MA3201 Ringer og moduler > MA3201 Rings and modules
MA3201 Rings and modules
14.08: First lecture Thursday.20/8, at 12:15 in K28 (likker i Kjemiblokk 1, 2.etg).
Lecturer: sverre o. smalø, email: sverresm at math ntnu no, SBII, room 850, tlf. 735 91750
Office hours: Thursdays 14:15-15:00, SBII, room 850. I am awailable all days between 9.00 to 15.00 from now on and until December 14th. (Only by appointment for Saturdays and Sundays)
Teaching hours:
Lectures: Thursdays 12:15-14:00, K28
Friday 14:15-16:00, MA24
Some of the Friday-lectures will be used for exercise sessions.
The problem session on Friday October 2nd will be given by Markus Schidmeier, with problems from from Exercises 2-6.
The problem session on Thursday 22nd of October will be given by Magnus Hellstrøm-Finnsen. The problem discussed will be from Exercises 7-8 + one given in class.
Thursday October 29th: The Jordan-Hölder theorem for modules which are both Noetherian and Artinian was given.
Friday October 30th: The exam problem 1 and 2 from dec. 11th 2007 was given together with problem 4 from the exam November 30th 2005.
Thursday Nov. 5th: The plan is to do problem 4 from Dec. 11th, 2007; and 1, 2 and 3 from Nov.30th, 2005.
Friday Nov. 6th: Problems from Dec. 2010 and Dec. 2011.
Textbook and syllabus
Authors: P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul
Title: Basic Abstract Algebra
Edition: Second Edition
Preliminary syllabus:
Chapter 9 All sections
Chapter 10 All sections
Chapter 14 14.1-14.5
Plan/log
19.08 Introduction, definitions, various examples Chap 9.1, 9.2, 9.3
23.08 Polynomial rings and formal power series, subrings, characteristic of a ring, algebras, ideas and homomorphisms Chap 9.4, 9.5, 10.1, 10.2
26.08 More on Ideals and homomorphisms, left ideals, right ideals, ideals in the full matrix algebra, sums and direct sums of ideals, products of rings, Chap 10.1, 10.2, 10.3
30.08 Minimal, maximal and prime ideals, the real numbers as the rings of Cauchy-sequences modulo the ideal of sequences converging to zero Chap 10.4
02.09 Maximal ideals are prime, the maximal ideals of Z, C[x] and R[X], description of prime ideals for commutative rings. Nilpotent and nil ideals, Zorns's lemma and the existence of maximal ideals. Chap 10.5,10.6
06.09 Modules and Vectorspaces Chap 14.1, 14.2
09.09 The rest of Chapter 14.2, and most of 14.3 Chap 14.2, 14.3
09.13 Some exercises, the rest of 14.3 and started on 14.4 Chap 14.3, 14.4
09.16 Rest of 14.4 and started on 14.5 Chap 14.4, 14.5
09.20 Finished 14.4 started on 19.2 Chap 14.5, 19.2
09.23 Finished therem 19.2.5 and 19.2.6, gave examples, and finished section 19.1 Chap 19.1, 19.2
09.27 We will take a look at the exercises, so look through the exercises given as exercise 1, 2 and 3, and make and efford to try to solve those problems
09.30 Endomorphism ring of a direct sum as a matrix ring, Th 19.2.7, 19.2.8,19.2.9, 19.2.10 Chap 19.1, 19.2
10.04 Exercise 2 and 3
10.07 Th 19.2.11, 19.2.12, 19.2.14, Examples 2.15 a, and b. Chap 19.2
10.14 Wedderburn Artin theorem Chap 19.3
10.21 Comments on Weddernburn Artin theorem and start on Smith normal form Chapter 20.1
10.28 Smith normal form Chapter 20.1
Final exam: 15.12.2015, written, 4 hours, 9:00-13:00
Problem sessions(tentative)
Exercise 1 Chapter 9, page 173-176: 1, 2, 3, 4, 5, 7, 9, 11 Chapter 10, page 187: 1, 2, 4, 7
Exercise 2 Chapter 10, page 194-195: 1, 2, 3, 4, 8, 10, 11, 12 page 202-203: 1, 2, 3, 4, 7
Exercise 3 Chapter 10, page 209: 1, 2, 4, 6 page 210: 1, 2
Exercise 4 Chapter 14, page 252-253: 1, 3, 4, 5, 8, 10, 11 page 260: 2, 3, 4, 6, 7
Exercise 5 Chapter 14, page 262-263: 1, 2, 3, 5, 6 page 268: 1, 2, 4, 5, 7, 8, 9
Exercise 6 Chapter 19, page 368: 1, 2 page 381: 1, 4, 6
Exercise 7 Chapter 19, page 381-382 2, 7, 8, 9, 11
Exercise 8 Chapter 19, page 388 3, 5 chapter 20, page 401 1,2,3
Exercise 9 Exam Dec 11 2007 Problems 1, 2, 3 Nov 30 2005 Problems 1, 4
Exercise 10 Dec 11 2007 Problem 4 Nov 30 2005 Problem 2, 3 Dec 11 2009 Problem 2, 3
Exercise 11 Dec 13 2010 Dec 2011 dec 2012 dec 2014 dec 2014
2015-11-25, Sverre Olaf Smalø


Firstly, I'll explain why this post isn't going up in the morning as it usually would.
I just haven't had the time to write it up until now! It's been such a busy day, a busy week to tell the truth. A little bit stressful but here's how it went down . . .
. . . I got the timetable for my A Level mocks. They're in the first week of Decemember and it just so happens they fall on the day before and the day of my birthday. Great! Revision began this week which meant a few late nights but I'm hoping it'll be worth it!
. . . I was asked to make a video for my school's open day. Basically, it has to include footage from all departments and aspects of the school and show it off in it's best light so I've been looking through a bunch of clips this week. One of my deputy headmaster doing the Cha Cha Slide had me in stitches . . .
. . . I paid a visit to the new shop in my local city. It's beautifully decorated and sells all manner of things - cute home & kitchen ware, Cath Kidston handbags, quirky craft bits as well as tea and cupcakes! I could have stayed all day inside that place. I've also been having a few sight problems this week so I also went to the opticians to book an appointment so hopefully that will be sorted out soon!
. . . this little blog reached 200 followers on Bloglovin'! It's been a massive milestone in my mind for ages - I'm so happy and I just want to thank you all so much. It's your comments and contribution that make this whole thing worthwhile.
So that was a little roundup of what happened during my last seven days!


Women today are accomplished, successful and beautiful, but still are not content in their personal lives. They always complain that their husbands or boyfriends don't give them the attention and love they used to. So, what happened and where did the love and care vanish? Let's analyze this situation.
It happens often that women are too content or busy in their lives, and fail to understand why their husbands are unhappy and disoriented. Women fail to realize that they need to rekindle the romance every now and then. But, sadly many do not know how to do so. So, here is a compiled a list of things that one can do to make the men in their lives happy.
Be honest and understanding with him. Make him feel as if he is the only guy in the world for you, by expressing your love for him.
Buy him a small gift like a card saying how much you love him, or just make a late night phone call telling him that you just wanted to hear his voice, and the likes.
Do not complain when he stays out with his friends. Men need their time with their friends, so, do not hassle him too much when he plans a boys night out. Don't be jealous and clingy, as men strongly dislike this.
No matter how long you have been with your guy, you still need to flirt with him. It makes him feel wanted and important. Holding hands and exchanging playful remarks, helps keep your relationship playful and romantic.
Men are disorganized but cleanliness is one thing that turns them on. They love that their room and house is spot free, it automatically brightens up the mood and ensures that the next few hours are all going for the lady's praise.
If you don't give your partner enough compliments start now. This is a very common misconception that only girls love compliments. Guys love hearing praises, tell him that you are lucky to have him.
Put him on cloud 9 by admiring things that are important to him, like his bike, new haircut, or even his boxing gloves.
The saying, 'a way to man's heart is through his stomach', may sound old-fashioned, but it's 100% true. You don't have to be an expert in cooking just making him something small will make you the queen of his heart for centuries to come.
Be truthful about your feelings and thoughts, as men are not supernatural creatures that can read minds.
When it comes to a serious relationship, they expect their woman to be brutally honest with them. So, if you have any insecurities, or are feeling low, giving him a 'I am fine' answer will never help.
Support him in his ambition, as you are an important part of his life. Make small gestures, as they go a long way in strengthening and cementing a relationship.
Always be there for him when he is low, as men are very emotional even if they do not show it.


text:
Mats Møller Dæhli (Oslo, 1995. március 2. –) norvég válogatott labdarúgó, a német Nürnberg középpályása.

Pályafutása

Klubcsapatokban
Dæhli a norvég fővárosban, Osloban született. Az ifjúsági pályafutását a helyi Lyn és Stabæk csapatában kezdte, majd az angol Manchester United akadémiájánál folytatta.

2013-ban mutatkozott be a Molde első osztályban szereplő felnőtt keretében. 2014-ben az angol első osztályban érdekelt Cardiff City, míg 2015-ben a német Freiburg szerződtette. 2017 és 2018 között a St. Pauli csapatát erősítette kölcsönben. A lehetőséggel élve 2018-ban a St. Paulihoz igazolt. 2020-ban a belga Genkhez csatlakozott. A 2020–21-es szezon második felében a Nürnbergnél szerepelt kölcsönben. 2021. július 1-jén négyéves szerződést kötött a Nürnberg együttesével. Először a 2021. július 25-ei, Erzgebirge Aue ellen 0–0-ás döntetlennel zárult mérkőzésen lépett pályára. Első gólját 2021. július 30-án, a Paderborn ellen 2–2-es döntetlennel végződő találkozón szerezte meg.

A válogatottban
Dæhli az U15-östől az U23-asig több korosztályú válogatottban is képviselte Norvégiát.

2013-ban debütált a felnőtt válogatottban. Először a 2013. november 15-ei, Dánia ellen 2–1-re elvesztett mérkőzés 60. percében, Per Ciljan Skjelbredet váltva lépett pályára. Első válogatott gólját 2014. október 10-én, Málta ellen 3–0-ra megnyert EB-selejtezőn szerezte meg.

Statisztikák
2023. március 10. szerint

A válogatottban

Góljai a válogatottban

Sikerei, díjai
Molde
 Norvég Kupa
 Győztes (1): 2013

Freiburg
 2. Bundesliga
 Feljutó (1): 2015–16

Jegyzetek

További információk
Transfermarkt

1995-ben született személyek
Norvég labdarúgók
Labdarúgó-középpályások
A Molde FK labdarúgói
A Cardiff City labdarúgói
Az SC Freiburg labdarúgói
Az FC St. Pauli labdarúgói
A KRC Genk labdarúgói
Az 1. FC Nürnberg labdarúgói
Az Eliteserien labdarúgói
A Premier League labdarúgói
A Bundesliga labdarúgói
A 2. Bundesliga labdarúgói
Élő személyek

redpajama_set_name: RedPajamaWikipedia


How could I have such a thought, I loved my son more than anything in this world. I was disgusted and ashamed of myself. Weeks turned into months. I went to see therapists explaining my horrible thoughts and they all smiled and nodded along sympathetically. I was just depressed they told me, 'you have a lot going on'. How naive I was.
I really wanted to get off the meds, despite helping with the depression symptoms I saw them as a sign of inner weakness and mental vulnerability. Not something a man is ready to admit to. I fought the intrusive thoughts, resisted them, abhorred them and yes punished myself… I was as far from loving myself as humanly possible… I may have been on another planet from the planet of loving kindness.
I have always been an inquisitive person. I turned to the Internet to try and make sense of it all.What could I do to get rid of these harmful thoughts… They were like an anchor slowly dragging me to the bottom of a very dark place. I stuck it out through a course of mindfullness. How can you be mindful though of something that bombards you hundreds of times a day… Sooner than later it just wears you out, right?
I remember describing to a therapist that I had a tic but it was all in my head…. and throughout 3 years of on -off therapy never did any of my therapists mention obsessive compulsive disorder.
I moved to the U.S in 2013… My mental problems moved with me. I was going to be a father again, my family was going to be complete but I was terrified of my intrusive thoughts doubling up on me….would I have these thoughts towards both of my children… Would my mental disorder mercilessly strangle the joy a parent should have raising his children.
But that didn't help take the sting out of this recurring 'I hate my son' thought that kept on chipping away at me. It ebbed and flowed.. Some weeks were OK followed by what felt like mini mental breakdowns… Back to my family doctor..change the dosage..improve a little, another mini breakdown. Repeat.
My symptoms were ever so gradually getting worse… Like a slow tide retreating from the shoreline. I felt like it was make or break for me. I had visions of being sent away to a psych ward and being committed. You know, like you see in the movies.
In January of this year I found a psychiatrist. A good psychiatrist. Within 1 session he told me he was 99% sure I had OCD. I thought OCD meant you needed to wash your hands all the time or need to hoard stuff like you see on TV I remember asking him. Turns out there are as many variations of OCD as there are shades of blue in the sky. But you knew that already.
I was referred to a CBT psychologist. My psychiatrist told me he was the 'Rolls Royce 'of psychologists here in the city where I live. I sure hoped he was. I could do with a Rolls Royce, especially the Psychologist kind. I've been doing ERP for a few months. Never found it particularly challenging but it has made a difference. For me, understanding the biology behind OCD has been very helpful, to know that a deficiency in your brain mechanisms was causing this was like a gentle push in the right direction, finally. Books yes…I devoured them, Brain Lock, Mindfulness for OCD, meditation…. My wish list on Amazon was 40 books deep!
Some of you might call it Pure O… I'm not sure where my OCD sits. Is there such a thing as hate OCD? Sure, I've had some of the others too but they haven't been sticky like my 'I hate you' thought. It's always been cognitive for me but to be honest I have a fear that it will turn into physical symptoms… But thats just the OCD talking right?
It's great to have finally found The OCD stories. For the longest time I was convinced I was the only person on earth who had this type of mental problem. Maybe I would be taken away by some agency where they would stick me in one of those MRI tubes and run hundreds of tests on me and diagnose me with some rare mental disease.. The first reported case ever. How crazy is that. Maybe the disease would bear my name for eternity.
I'm starting to look into ACT. One thing I can't seem to get over is … Why the 'I hate you thought' and why my son? It could have been anything else in this world… Anything. I guess OCD sniffs out your soft spot, sticks a thorn into that spot and like a torturer, manipulates that thorn … Pushes it in harder, withdraws… Pushes again.
Realizing also that OCD has always been the protein on the plate (the Sirloin steak) and that depression was just the side serving has helped me start my long road to 'recovery' . My depression waxes and wanes but the intrusive thought is always there. I wake up in the morning and my brain goes searching for it. I'm sure it's the same for a lot of you.
For a long time I thought I was weak because of my OCD… but it's actually the other way round. For all of us dealing with OCD no matter how severe and in whatever shape or form, know that your inner fortitude must be incredibly strong to deal with this monster every day. I am not a religious person but whether you believe in a God or not… I like to think that God only gives you what you can handle in life… No more.
It hasn't been all uphill. My OCD has taught me so much about myself. I was lost spiritually… For a long time. I am connecting back with my Self through meditation and other spiritual practices. My mental struggles have been the doorway to finding my spirituality again. I feel like I have just started down my path and it's a long path. But I'm interested to experience everything along the way…. Good and bad.
For those of you reading this, who may have been dealing with this disease for years or it's just raising its ugly black head in your life now, I hope something I've written resonates with you.
There is light at the end of the tunnel. Don't give up striving for finding that peace within yourself …that means making peace with your OCD. Lean into it, stare it down. Find the treatment that works for you, don't be afraid to try different treatments if one doesn't seem to help you. Work at it. Every day. That will be your guard dog.
I loved your "MMA" fighter analogy, and could also relate to your "thorn in the soft spot" comment. OCD always aims for what you care about most, that's how it gets you to pay attention. Best of luck in this exciting time for you!
TC , are you in Arizona ?
I found a lot of similarities in your story that I have experienced with harm ocd. I read brain lock too and it has helped a ton. Thanks for giving the optimistic words to keep on going and to never give up?
Thanks for your kind comment Lauren!
Thanks for this great story. I saw myself in a lot of what you had to say. Nice to know I'm not alone.
Hey, sorry to hear that. It does get better.


text:
Q: Excel VBA - select last used columns I want to select the last column and 2nd last column in my spreadsheet.
I've found my last column by:
LastColumn = Sheets("Sheet1").Cells.Find("*", [a1], , , xlByColumns, xlPrevious).Column
    On Error GoTo 0

Then I'm trying to use the LastColumn to select from the 2nd last column to the last column and copy and paste.
Sheets("Sheet1").Columns(LastColumn - 1:LastColumn).Select

but it doesn't seem to work. Can someone please help and to what I may have gone wrong.
Thanks

A: Sheets("Sheet1").Columns(LastColumn - 1).Select
EDIT: thanks Doug for pointing out my error...
Sheets("Sheet1").Columns(LastColumn - 1).Resize(,2).Select


It's almost time for the kids to go back to school! These leggings are sure to be a hit in your classroom. Perfect for teachers! Black leggings with chalk style pattern of a schoolhouse, books, pencils, scissors, and abc's with little red apples. 92% polyester, 8% spandex. Made in China. Women's One size fits 3-14.
THIS IS A PRE-ORDER. Estimated ship date is: July 27th.


text:
Corker Successfully Urge Administration Officials to Examine Delays in Compensation for Sick Nuclear Workers
U.S. Senators Lamar Alexander (R-TN) and Bob Corker (R-TN) today announced that U.S. Administration officials have found the funds needed to avoid further delays in processing claims to compensate Tennessee nuclear weapons workers facing illnesses. Alexander and Corker said the announcement – made in a letter by Administration officials after the senators asked for an examination into the delays – is a victory in the push for compensating sick nuclear workers from the Oak Ridge Department of Energy complex, Ft. Campbell in Clarksville, contractor facilities in Erwin and other locations. "This is a win for Tennessee and a step in the right direction in making sure our Cold War heroes get the compensation they deserve," said Alexander. "Tennessee has twice the number of claims than any other state, and we have to be a leader on making sure they are processed more quickly and effectively. This announcement is a good step in keeping the promises made to Tennessee workers who were hurt while helping defend our country. I'm glad the Administration has focused on the problem and has found a way to fix it." "I'm pleased the Department of Labor and the CDC have corrected the funding shortfall in the EEOICPA program so that our nation's Cold War veterans, harmed in the service of their country, will receive the fair treatment and just compensation Congress intended," said Corker. "Tennessee's workers and workers across the country deserve to have their claims processed expeditiously, and I will continue to work with my colleagues to ensure that adequate funds are available to meet that objective." The letter from Administration officials is in response to one written in June by Alexander, Corker and 14 other Senators in a bipartisan effort to urge Health and Human Service (HHS) Secretary Michael Leavitt and Department of Labor (DOL) Secretary Elaine Chao to fix delays in processing the compensation claims. The Administration letter announces that the U.S. Department of Labor and the Centers for Disease Control and Prevention (CDC) working together have identified funds to enable the National Institute for Occupation Safety and Health (NIOSH) to continue the processing of claims at full speed. Tennessee has more than 23,000 health claims from more than 9,000 individual workers under the Energy Employees Occupational Illness Compensation Program (EEOICPA). It is the "Part B" component of EEOICPA that was at risk of running out of funds this year. EEOICPA is responsible for identifying former nuclear weapons workers suffering from workplace-related illnesses who are eligible for compensation. There are currently 6,500 Tennessee claims still waiting for a final decision under the EEOICPA system, and 2,900 of these claims are under "Part B." In 2000, Congress created EEOICPA to provide appropriate compensation and medical benefits to workers who contracted radiation-induced cancers, beryllium diseases or silicosis during the course of their work for the Department of Energy or its contractors. Alexander cosponsored legislation that became law in 2004 that transferred the responsibility of claims processing from the Department of Energy to the Department of Labor in order to enhance and speed-up the processing of these claims. The letter to Leavitt and Chao – signed by 16 senators and the second from the group to Chao this year concerning EEOICPA – calls attention to serious funding shortfalls and continues to urge that these shortfalls be "quickly corrected." This week the senators received a letter from Administration officials confirming that funds are being reprogrammed for Part B of EEOICPA, which is the portion of EEOICPA for which HHS is responsible. Copies of both letters are attached.