text:
\section{Introduction: file preparation and submission}

The \verb"iopart" \LaTeXe\ article class file is provided to help authors prepare articles for submission to IOP Publishing journals.
  This document gives advice on preparing your submission, and specific instructions on how to use \verb"iopart.cls" to follow this advice.  You
  do not have to use \verb"iopart.cls"; articles prepared using any other common class and style files can also be submitted.
    It is not necessary to mimic the appearance of a published article.

The advice
on \LaTeX\ file preparation in this document applies to
the journals listed in table~\ref{jlab1}.  If your journal is not listed please go to the journal website via \verb"http://iopscience.iop.org/journals" for specific
submission instructions.

\begin{table}
\caption{\label{jlab1}Journals to which this document applies, and macros for the abbreviated journal names in {\tt iopart.cls}. Macros for other journal titles are listed in appendix\,A.}
\footnotesize
\begin{tabular}{@{}llll}
\br
Short form of journal title&Macro name&Short form of journal title&Macro name\\
\mr
2D Mater.&\verb"\TDM"&Mater. Res. Express&\verb"\MRE"\\
Biofabrication&\verb"\BF"&Meas. Sci. Technol.$^c$&\verb"\MST"\\
Bioinspir. Biomim.&\verb"\BB"&Methods Appl. Fluoresc.&\verb"\MAF"\\
Biomed. Mater.&\verb"\BMM"&Modelling Simul. Mater. Sci. Eng.&\verb"\MSMSE"\\
Class. Quantum Grav.&\verb"\CQG"&Nucl. Fusion&\verb"\NF"\\
Comput. Sci. Disc.&\verb"\CSD"&New J. Phys.&\verb"\NJP"\\
Environ. Res. Lett.&\verb"\ERL"&Nonlinearity$^{a,b}$&\verb"\NL"\\
Eur. J. Phys.&\verb"\EJP"&Nanotechnology&\verb"\NT"\\
Inverse Problems&\verb"\IP"&Phys. Biol.$^c$&\verb"\PB"\\
J. Breath Res.&\verb"\JBR"&Phys. Educ.$^a$&\verb"\PED"\\
J. Geophys. Eng.$^d$&\verb"\JGE"&Physiol. Meas.$^{c,d,e}$&\verb"\PM"\\
J. Micromech. Microeng.&\verb"\JMM"&Phys. Med. Biol.$^{c,d,e}$&\verb"\PMB"\\
J. Neural Eng.$^c$&\verb"\JNE"&Plasma Phys. Control. Fusion&\verb"\PPCF"\\
J. Opt.&\verb"\JOPT"&Phys. Scr.&\verb"\PS"\\
J. Phys. A: Math. Theor.&\verb"\jpa"&Plasma Sources Sci. Technol.&\verb"\PSST"\\
J. Phys. B: At. Mol. Opt. Phys.&\verb"\jpb"&Rep. Prog. Phys.$^{e}$&\verb"\RPP"\\
J. Phys: Condens. Matter&\verb"\JPCM"&Semicond. Sci. Technol.&\verb"\SST"\\
J. Phys. D: Appl. Phys.&\verb"\JPD"&Smart Mater. Struct.&\verb"\SMS"\\
J. Phys. G: Nucl. Part. Phys.&\verb"\jpg"&Supercond. Sci. Technol.&\verb"\SUST"\\
J. Radiol. Prot.$^a$&\verb"\JRP"&Surf. Topogr.: Metrol. Prop.&\verb"\STMP"\\
Metrologia&\verb"\MET"&Transl. Mater. Res.&\verb"\TMR"\\
\br
\end{tabular}\\
$^{a}$UK spelling is required; $^{b}$MSC classification numbers are required; $^{c}$titles of articles are required in journal references; $^{d}$Harvard-style references must be used (see section \ref{except}); $^{e}$final page numbers of articles are required in journal references.

\end{table}
\normalsize

Any special submission requirements for the journals are indicated with footnotes in table~\ref{jlab1}.
Journals which require references in a particular format will need special care if you are using BibTeX, and you might need to use a \verb".bst" file
that gives slightly non-standard output in order to supply any extra information required.  It is not
necessary to give references in the exact style of references used in published articles, as long as all of
the required information is present.

Also note that there is an incompatibility
between \verb"amsmath.sty" and \verb"iopart.cls" which cannot be completely worked around.  If your article relies
on commands in \verb"amsmath.sty" that are not available in \verb"iopart.cls", you may wish to consider using a different
class file.

Whatever journal you are submitting to, please look at recent published articles (preferably
articles in your subject area) to familiarize yourself with the features of the journal.  We do not demand
that your \LaTeX\ file closely resembles a published article---a generic `preprint' appearance of the sort
commonly seen on \verb"arXiv.org" is fine---but your submission should be presented
in a way that makes it easy for the referees to form an opinion of whether it is suitable for the journal.
The generic advice in this document---on what to include in an abstract, how best to present complicated
mathematical expressions, and so on---applies whatever class file you are using.

\subsection{What you will need to supply}
Submissions to our journals are handled via the ScholarOne web-based submission system.  When you submit
a new article to us you need only submit a PDF of your article.  When you submit a revised version,
we ask you to submit the source files as well.  Upon acceptance for publication we will use the source files to produce a proof of your article in the journal style.

\subsubsection{Text.}When you send us the source files for a revised version of your submission,
you should send us the \LaTeX\ source code of your paper with all figures read in by
the source code (see section \ref{figinc}).  Articles can be prepared using almost any version of \TeX\ or \LaTeX{},
not just \LaTeX\ with the class file \verb"iopart.cls".  You may split your \LaTeX\ file into several parts, but please show
which is the `master' \LaTeX\ file that reads in all of the other ones by naming it appropriately.  The `master'
\LaTeX\ file must read in all other \LaTeX\ and figure files from the current directory.  {\it Do not read in files from a different directory, e.g. \verb"\includegraphics{/figures/figure1.eps}" or
\verb"\include{../usr/home/smith/myfiles/macros.tex}"---we store submitted files
all together in a single directory with no subdirectories}.
\begin{itemize}
\item {\bf Using \LaTeX\ packages.} Most \LaTeXe\ packages can be used if they are
available in common distributions of \LaTeXe; however, if it is essential to use
a non-standard package then any extra files needed to process the article must
also be supplied.  Try to avoid using any packages that manipulate or change the standard
\LaTeX\ fonts: published articles use fonts in the Times family, but we prefer that you
use \LaTeX\ default Computer Modern fonts in your submission.  The use of \LaTeX\ 2.09, and of plain
\TeX\ and variants such as AMSTeX is acceptable, but a complete PDF of your submission should be supplied in these cases.
\end{itemize}
\subsubsection{Figures.} Figures should ideally be included in an article as encapsulated PostScript files
(see section \ref{figinc}) or created using standard \LaTeX\ drawing commands.
 Please name all figure files using the guidelines in section \ref{fname}.
We accept submissions that use pdf\TeX\ to include
PDF or bitmap figures, but please ensure that you send us a PDF that uses PDF version 1.4 or lower
(to avoid problems in the ScholarOne system).
You can do this by putting \verb"\pdfminorversion=4" at the very start of your TeX file.

\label{fig1}All figures should be included within the body of the text
at an appropriate point or grouped together with their captions at the end of the article. A standard graphics inclusion package such as \verb"graphicx" should be used for figure inclusion, and the package should be declared in the usual
way, for example with \verb"\usepackage{graphicx}", after the \verb"\documentclass" command.
Authors should avoid using special effects generated by including verbatim
PostScript code in the submitted \LaTeX\ file. Wherever possible, please try to use standard \LaTeX\ tools
and packages.

\subsubsection{References.\label{bibby}}
You can produce your bibliography in the standard \LaTeX\ way using the \verb"\bibitem" command. Alternatively
you can use BibTeX: our preferred  \verb".bst" styles are:

\begin{itemize}
\item For the numerical (Vancouver) reference style we recommend that authors use
 \verb"unsrt.bst"; this does not quite follow the style of published articles in our
 journals but this is not a problem.  Alternatively \verb"iopart-num.bst" created by Mark A Caprio
 produces a reference style that closely matches that in published articles.  The file is available from
\verb"http://ctan.org/tex-archive/biblio/bibtex/contrib/iopart-num/" .
\item For alphabetical (Harvard) style references we recommend that authors use the \verb"harvard.sty"
in conjunction with the \verb"jphysicsB.bst" BibTeX style file.  These, and accompanying documentation, can be downloaded
from \penalty-10000 \verb"http://www.ctan.org/tex-archive/macros/latex/contrib/harvard/".
Note that the \verb"jphysicsB.bst" bibliography style does not include article titles
in references to journal articles.
To include the titles of journal articles you can use the style \verb"dcu.bst" which is included
in the \verb"harvard.sty" package.  The output differs a little from the final journal reference
style, but all of the necessary information is present and the reference list will be formatted
into journal house style as part of the production process if your article is accepted for publication.
\end{itemize}

\noindent Please make sure that you include your \verb".bib" bibliographic database file(s) and any
\verb".bst" style file(s) you have used.

\subsection{\label{copyright}Copyrighted material and ethical policy} If you wish to make use of previously published material for which you do not own the copyright then you must seek permission from the copyright holder, usually both the author and the publisher.  It is your responsibility to obtain copyright permissions and this should be done prior to submitting your article. If you have obtained permission, please provide full details of the permission granted---for example, copies of the text of any e-mails or a copy of any letters you may have received. Figure captions must include an acknowledgment of the original source of the material even when permission to reuse has been obtained.  Please read our ethical policy before writing your article.

\subsection{Naming your files}
\subsubsection{General.}
Please name all your files, both figures and text, as follows:
\begin{itemize}
\item Use only characters from the set a to z, A to Z, 0 to 9 and underscore (\_).
\item Do not use spaces or punctuation characters in file names.
\item Do not use any accented characters such as
\'a, \^e, \~n, \"o.
\item Include an extension to indicate the file type (e.g., \verb".tex", \verb".eps", \verb".txt", etc).
\item Use consistent upper and lower case in filenames and in your \LaTeX\ file.
If your \LaTeX\ file contains the line \verb"\includegraphics{fig1.eps}" the figure file must be called
\verb"fig1.eps" and not \verb"Fig1.eps" or \verb"fig1.EPS".  If you are on a Unix system, please ensure that
there are no pairs of figures whose names differ only in capitalization, such as \verb"fig_2a.eps" and \verb"fig_2A.eps",
as Windows systems will be unable to keep the two files in the same directory.
\end{itemize}
When you submit your article files, they are manipulated
and copied many times across multiple databases and file systems. Including non-standard
characters in your filenames will cause problems when processing your article.
\subsubsection{\label{fname}Naming your figure files.} In addition to the above points, please give each figure file a name which indicates the number of the figure it contains; for example, \verb"figure1.eps", \verb"figure2a.eps", etc. If the figure file contains a figure with multiple parts, for example figure 2(a) to 2(e), give it a name such as \verb"figure2a_2e.eps", and so forth.
\subsection{How to send your files}
Please send your submission via the ScholarOne submission system.  Go to the journal home
page, and use the `Submit an article' link on the right-hand side.

\section{Preparing your article}

\subsection{Sample coding for the start of an article}
\label{startsample}
The code for the start of a title page of a typical paper in the \verb"iopart.cls" style might read:
\small\begin{verbatim}
\documentclass[12pt]{iopart}
\begin{document}
\title[The anomalous magnetic moment of the
neutrino]{The anomalous magnetic moment of the
neutrino and its relation to the solar neutrino problem}

\author{P J Smith$^1$, T M Collins$^2$,
R J Jones$^3$\footnote{Present address:
Department of Physics, University of Bristol, Tyndalls Park Road,
Bristol BS8 1TS, UK.} and Janet Williams$^3$}

\address{$^1$ Mathematics Faculty, Open University,
Milton Keynes MK7~6AA, UK}
\address{$^2$ Department of Mathematics,
Imperial College, Prince Consort Road, London SW7~2BZ, UK}
\address{$^3$ Department of Computer Science,
University College London, Gower Street, London WC1E~6BT, UK}
\ead{williams@ucl.ac.uk}

\begin{abstract}
...
\end{abstract}
\keywords{magnetic moment, solar neutrinos, astrophysics}
\submitto{\jpg}
\maketitle
\end{verbatim}
\normalsize

At the start of the \LaTeX\ source code please include
commented material to identify the journal, author, and (if you are sending a revised
version or a resubmission) the reference number that the journal
has given to the submission. The first non-commented line should be
\verb"\documentclass[12pt]{iopart}"  to load the preprint class
file.  The normal text will be in the Computer Modern 12pt font.
It is possible to specify 10pt font size by passing the option \verb"[10pt]" to the class file.
Although it is possible to choose a font other than Computer Modern by loading external packages, this is not recommended.

The article text begins after \verb"\begin{document}".
Authors of very long articles may find it convenient to separate
their article into a series of \LaTeX\ files each containing one section, and each of which is called
in turn by the primary file.  The files for each section should be read in from the current directory;
please name the primary file clearly so that we know to run \LaTeX\ on this file.

Authors may use any common \LaTeX\ \verb".sty" files.
Authors may also define their own macros and definitions either in the main article \LaTeX\ file
or in a separate \verb".tex" or \verb".sty" file that is read in by the
main file, provided they do not overwrite existing definitions.
It is helpful to the production staff if complicated author-defined macros are explained in a \LaTeX\ comment.
The article class \verb"iopart.cls" can be used with other package files such
as those loading the AMS extension fonts
\verb"msam" and \verb"msbm", which provide the
blackboard bold alphabet and various extra maths symbols as well as symbols useful in figure
captions.  An extra style file \verb"iopams.sty" is provided to load these
packages and provide extra definitions for bold Greek letters.

\subsection{\label{dblcol}Double-column layout}
The \verb"iopart.cls" class file produces single-column output by default, but a two-column layout can be obtained by
using \verb"\documentclass[10pt]" at the start of the file and \verb"\ioptwocol" after the \verb"\maketitle" command.  Two-column output will begin
on a new page (unlike in published double-column articles, where the two-column material
starts on the same page as the abstract).

In general we prefer to receive submissions in single-column format even for journals
published in double-column style; however, the \verb"\ioptwocol" option may be useful to test figure sizes
and equation breaks for these journals.  When setting material
in two columns you can use the asterisked versions of \LaTeX\ commands such as \verb"\begin{figure*} ... \end{figure*}"
to set figures and tables across two columns.  If you have any problems or any queries about producing two-column output, please contact us at \verb"submissions@iop.org".

\section{The title and abstract page}
If you use \verb"iopart.cls", the code for setting the title page information is slightly different from
the normal default in \LaTeX.  If you are using a different class file, you do not need to mimic the appearance of
an \verb"iopart.cls" title page, but please ensure that all of the necessary information is present.

\subsection{Titles and article types}
The title is set using the command
\verb"\title{#1}", where \verb"#1" is the title of the article. The
first letter
of the title should be capitalized with the rest in lower case.
The title appears in bold case, but mathematical expressions within the title may be left in light-face type.

If the title is too long to use as a running head at the top of each page (apart from the
first) a short
form can be provided as an optional argument (in square brackets)
before the full title, i.e.\ \verb"\title[Short title]{Full title}".

For article types other than papers, \verb"iopart.cls"
has a generic heading \verb"\article[Short title]{TYPE}{Full title}"
and some specific definitions given in table~\ref{arttype}. In each case (apart from Letters
to the Editor and Fast Track Communications) an
optional argument can be used immediately after the control sequence name
to specify the short title; where no short title is given, the full title
will be used as the running head.  Not every article type has its own macro---use \verb"\article" for
any not listed.  A full list of the types of articles published by a journal is given
in the submission information available via the journal home page.
The generic heading could be used for
articles such as those presented at a conference or workshop, e.g.
\small\begin{verbatim}
\article[Short title]{Workshop on High-Energy Physics}{Title}
\end{verbatim}\normalsize
Footnotes to titles may be given by using \verb"\footnote{Text of footnote.}" immediately after the title.
Acknowledgment of funding should be included in the acknowledgments section rather than in a footnote.

\begin{table}
\caption{\label{arttype}Types of article defined in the {\tt iopart.cls}
class file.}
\footnotesize\rm
\begin{tabular*}{\textwidth}{@{}l*{15}{@{\extracolsep{0pt plus12pt}}l}}
\br
Command& Article type\\
\mr
\verb"\title{#1}"&Paper (no surtitle on first page)\\
\verb"\ftc{#1}"&Fast Track Communication\\
\verb"\review{#1}"&Review\\
\verb"\topical{#1}"&Topical Review\\
\verb"\comment{#1}"&Comment\\
\verb"\note{#1}"&Note\\
\verb"\paper{#1}"&Paper (no surtitle on first page)\\
\verb"\prelim{#1}"&Preliminary Communication\\
\verb"\rapid{#1}"&Rapid Communication\\
\verb"\letter{#1}"&Letter to the Editor\\
\verb"\article{#1}{#2}"&Other articles\\\ & (use this for any other type of article; surtitle is whatever is entered as {\tt
\#1})\\
\br
\end{tabular*}
\end{table}

\subsection{Authors' names and addresses}
For the authors' names type \verb"\author{#1}",
where \verb"#1" is the
list of all authors' names. Western-style names should be written as initials then
family name, with a comma after all but the last
two names, which are separated by `and'. Initials should {\it not} be followed by full stops. First (given) names may be used if
desired.  Names in Chinese, Japanese and Korean styles should be written as you want them to appear in the published article. Authors in all IOP Publishing journals have the option to include their names in Chinese, Japanese or Korean characters in addition to the English name: see appendix B for details.

If the authors are at different addresses a superscripted number, e.g. $^1$, \verb"$^1$", should be used after each
name to reference the author to his/her address.
If an author has additional information to appear as a footnote, such as
a permanent address, a normal \LaTeX\ footnote command
should be given after the family name and address marker
with this extra information.

The authors' affiliations follow the list of authors.
Each address is set by using
\verb"\address{#1}" with the address as the single parameter in braces.
If there is more
than one address then the appropriate superscripted number, followed by a space, should come at the start of
the address.

E-mail addresses are added by inserting the
command \verb"\ead{#1}" after the postal address(es) where \verb"#1" is the e-mail address.
See section~\ref{startsample} for sample coding. For more than one e-mail address, please use the command
\verb"\eads{\mailto{#1}, \mailto{#2}}" with \verb"\mailto" surrounding each e-mail address.  Please ensure
that, at the very least, you state the e-mail address of the corresponding author.

\subsection{The abstract}
The abstract follows the addresses and
should give readers concise information about the content
of the article and indicate the main results obtained and conclusions
drawn. It should be self-contained---there should be no references to
figures, tables, equations, bibliographic references etc.  It should be enclosed between \verb"\begin{abstract}"
and \verb"\end{abstract}" commands.  The abstract should normally be restricted
to a single paragraph of around 200 words.

\subsection{Subject classification numbers}
We no longer ask authors to supply Physics and Astronomy Classification System (PACS)
classification numbers.  For submissions to {\it Nonlinearity}\/ we ask that you should
supply Mathematics Subject Classification (MSC) codes.  MSC numbers are included after the abstract
using \verb"\ams{#1}".

The command
\verb"\submitto{#1}" can be inserted, where \verb"#1" is the journal name written in full or the appropriate control sequence as
given in table~\ref{jlab1}. This command is not essential to the running of the file and can be omitted.

\subsection{Keywords}
Keywords are required for all submissions. Authors should supply a minimum of three (maximum seven) keywords appropriate to their article as a new paragraph starting \verb"\noindent{\it Keywords\/}:" after the end of the abstract.

\subsection{Making a separate title page}
To keep the header material on a separate page from the
body of the text insert \verb"\maketitle" (or \verb"\newpage") before the start of the text.
If \verb"\maketitle" is not included the text of the
article will start immediately after the abstract.

\section{The text}
\subsection{Sections, subsections and subsubsections}
The text of articles may be divided into sections, subsections and, where necessary,
subsubsections. To start a new section, end the previous paragraph and
then include \verb"\section" followed by the section heading within braces.
Numbering of sections is done {\it automatically} in the headings:
sections will be numbered 1, 2, 3, etc, subsections will be numbered
2.1, 2.2,  3.1, etc, and subsubsections will be numbered 2.3.1, 2.3.2,
etc.  Cross references to other sections in the text should, where
possible, be made using
labels (see section~\ref{xrefs}) but can also
be made manually. See section~\ref{eqnum} for information on the numbering of displayed equations. Subsections and subsubsections are
similar to sections but
the commands are \verb"\subsection" and \verb"\subsubsection" respectively.
Sections have a bold heading, subsections an italic heading and
subsubsections an italic heading with the text following on directly.
\small\begin{verbatim}
\section{This is the section title}
\subsection{This is the subsection title}
\end{verbatim}\normalsize

The first section is normally an introduction,  which should state clearly
the object of the work, its scope and the main advances reported, with
brief references to relevant results by other workers. In long papers it is
helpful to indicate the way in which the paper is arranged and the results
presented.

Footnotes should be avoided whenever possible and can often be included in the text as phrases or sentences in parentheses. If required, they should be used only for brief notes that do not fit conveniently into the text. The use of
displayed mathematics in footnotes should be avoided wherever possible and no equations within a footnote should be numbered.
The standard \LaTeX\ macro \verb"\footnote" should be used.  Note that in \verb"iopart.cls" the \verb"\footnote" command
produces footnotes indexed by a variety of different symbols,
whereas in published articles we use numbered footnotes.  This
is not a problem: we will convert symbol-indexed footnotes to numbered ones during the production process.

\subsection{Acknowledgments}
Authors wishing to acknowledge assistance or encouragement from
colleagues, special work by technical staff or financial support from
organizations should do so in an unnumbered `Acknowledgments' section
immediately following the last numbered section of the paper. In \verb"iopart.cls" the
command \verb"\ack" sets the acknowledgments heading as an unnumbered
section.

Please ensure that you include all of the sources of funding and the funding contract reference numbers that you are contractually obliged to acknowledge. We often receive requests to add such information very late in the production process, or even after the article is published, and we cannot always do this. Please collect all of the necessary information from your co-authors and sponsors as early as possible.

\subsection{Appendices}
Technical detail that it is necessary to include, but that interrupts
the flow of the article, may be consigned to an appendix.
Any appendices should be included at the end of the main text of the paper, after the acknowledgments section (if any) but before the reference list.
If there are
two or more appendices they should be called Appendix A, Appendix B, etc.
Numbered equations will be in the form (A.1), (A.2), etc,
figures will appear as figure A1, figure B1, etc and tables as table A1,
table B1, etc.

The command \verb"
\section{Introduction}

Most quantum mechanical systems do not possess an analytic solution, even in one dimension. Merely brute force numerical methods for such problems can be costly; hence, symmetry principles that enhance the efficiency of such a process have historically proved useful. In this regard, inspired by \cite{lin_bootstraps_2020}, the bootstrapping methodology of conformal field theory\footnote{To be familiarised with state of the art for CFT bootstrap techniques, the reader is directed for example to \cite{poland_conformal_2019,bissi_selected_2022} and the references within. This list is in no way exhaustive.} was recently adapted in \cite{han_bootstrapping_2020} in a bid to numerically solve such classes of systems that are not analytically soluble or are hard to do so even numerically.
\medskip

The central philosophy of their bootstrapping approach is as follows: given a set of initial data for a Hamiltonian quantum system (such as energy eigenvalues), other statistical moments of quantum operators can be computed using symmetries of the system. These moments of different order are, in turn, related to each other via recursion relations, effectively reducing the search space for independent data. To determine whether the initial data is physically viable (i.e. whether it is consistent with a square-integrable eigenstate of the whole Schr\"{o}dinger equation), one then enforces an adequate set of constraints upon the computed statistical moments, such as requiring the positivity of quadratic operators. By computing more and more statistical moments and testing them against the given constraints, the goal is that the set of allowed initial data points will converge to the system's actual solution.
\medskip

The bootstrapping methodology of \cite{han_bootstrapping_2020}, implemented in that work as a tool to solve matrix models and matrix quantum mechanics, has recently been applied to numerically solve various quantum mechanical problems \cite{aikawa_bootstrap_2022,aikawa_application_2022,berenstein_bootstrapping_2021,berenstein_bootstrapping_2022,berenstein_anomalous_2022,bhattacharya_numerical_2021,hu_different_2022,khan_bootstrapping_2022,morita_bootstraping_2022,tchoumakov_bootstrapping_2021,nakayama_bootstrapping_2022,du_bootstrapping_2022,bai_bootstrapping_2022,li_null_2022}. In particular, the procedure has been used to find the energy spectrum of physical systems, and has proved most effective at finding lower energy modes. Furthermore, it has been observed that the convergence speed of the bootstrap is improved by computing statistical moments that have the same symmetries as the total system. That is, the efficiency of the bootstrap increases by, for example, considering creation and annihilation operators for a diagonalisable system \cite{aikawa_bootstrap_2022}, or periodic moments for a periodic potential \cite{aikawa_application_2022,berenstein_bootstrapping_2022,tchoumakov_bootstrapping_2021}.
\medskip

With such a potentially powerful algorithm by our side, we must test out the effectiveness of the system by employing it to solve already tractable sectors. Ideally, one hopes to use the bootstrap as a trained black-box to readily spew out the spectrum of a quantum system with certain symmetries, and the training requires it to go through many familiar problems. Our present work is just another small step towards that goal.

\medskip
More specifically, as we have already mentioned, work has recently begun in applying the bootstrapping procedure to quantum systems with periodic potentials, which have several applications, such as in solid state physics \cite{harper_single_1955,aubry_analyticity_1980,marsal_topological_2020}. In particular, \cite{aikawa_application_2022,berenstein_bootstrapping_2022,tchoumakov_bootstrapping_2021} have considered a cosine potential (the Mathieu problem), and successfully reproduced the expected energy spectrum. However, reconstructing the dispersion relation for the quasimomentum has proved challenging. In \cite{tchoumakov_bootstrapping_2021}, the authors had success using a statistical approach, approximating the dispersion relation from a probability distribution for momentum reconstructed from a finite number of moments.
\medskip

 In this paper, we continue the investigation of the quantum mechanical bootstrap as applied to periodic potentials. Starting with a generic periodic potential, we concentrate on the Kronig-Penney model \cite{kronig_quantum_1931}, a simplified model of a particle in a one-dimensional lattice where a series of periodic delta spikes model the crystal structure (the Dirac comb), which has applications from models of graphene \cite{masir_magnetic_2009} to ultra-cold atoms \cite{negretti_generalized_2014}, to name a few. We choose a general spatially periodic operator basis to compute our recursion relation and reconstruct the probability density function using that relation\footnote{Note this analytical trick of using moment recursion relations was already discussed many decades ago in \cite{banerjee_transition_1977}.}. A numerical search for allowed energy eigenvalues restricted by the positivity constraint for the generic operator basis gives rise to the discrete band structure associated with the problem. We find the convergence of bootstrapped energy bands to the analytic ones depends explicitly on the structure of the operator basis chosen. We also sketch a way to extract the exact dispersion relation for the band structure using our bootstrap data.
\medskip

The rest of the paper is organised as follows: In Section \ref{SecBootIdentities}, we review the general bootstrapping identities and use them to construct a recursion relation first for a general periodic potential and then specialise in the Kronig-Penney model. In Section \ref{SecNumericImplementation} we review the positivity constraint for periodic operators, numerically reproduce the band gaps of the known solution to the Kronig-Penney model, and discuss how to obtain the full energy spectrum from the positivity constraint for a periodic potential. In Section \ref{SecDispersionRelation}, we find that an analytic approach can reconstruct the exact dispersion relation of the model. We conclude with a teaser of the road ahead. The appendices contain some extra details and neat examples of anomalies appearing in recursion relations for completeness.

\section{Bootstrapping identities for a periodic potential}
\label{SecBootIdentities}
We begin with a summary of how to take stock of the initial data associated with a bootstrap problem, mostly following the treatise of \cite{han_bootstrapping_2020}.
We begin with the general one-dimensional Hamiltonian with a space-dependent potential
\begin{eqnarray}
H = p^2 + V(x),
\end{eqnarray}
defined over the interval $\lbrace x_1 \leq x \leq x_2\rbrace$, where $\left[ \hat{x}, \hat{p} \right] = i$, and we use throughout the convention $\hbar = 1$ and $2m = 1$. Consider an eigenstate $|\psi \rangle$ of the Hamiltonian $\hat{H}$ with energy $E$, defined in the position basis $|x\rangle$ as
\begin{eqnarray}
| \psi \rangle = \int_{x_1}^{x_2} dx \psi(x) |x \rangle.
\end{eqnarray}
In the eigenstate $| \psi \rangle$, the expectation values of position-space functions $f(x)$ are defined as
\begin{eqnarray}
\left \langle f(x) \right \rangle = \int_{x_1}^{x_2} dx f(x) \rho(x),
\end{eqnarray}
where $\rho(x) = \left| \psi(x) \right|^2$ is the wavefunction density. The first step of the bootstrapping procedure is to construct a relation for moments of the distribution $\rho(x)$ using two identities. The first identity is that, for an operator $\hat{O}$, the following average vanishes, albeit up to an anomaly:
\begin{eqnarray}
\left \langle \left[ \hat{H}, \hat{O} \right] \right \rangle + \mathcal{A}_{\hat{O}} = 0, \label{EqConstraint1}
\end{eqnarray}
where $\mathcal{A}_{\hat{O}}$ is the anomaly defined in \cite{berenstein_anomalous_2022} as
\begin{eqnarray}
\mathcal{A}_O = \left \langle \left( H^{\dagger} - H \right) O \right \rangle_{\psi} = \int_{x_1}^{x_2} dx \left[ \left( \hat{H} \psi(x) \right)^{\dagger} - \psi^{\dagger} (x) \hat{H} \right] \hat{O} \psi(x). \label{EqAnomalyDefinition}
\end{eqnarray}
When expectation values are computed over a finite domain, the anomaly generates contact terms in the recursion relation \cite{berenstein_anomalous_2022}. Mathematically, it corresponds to the difference between computing the expectation values by approaching the boundaries of the domain from the left or the right.
\medskip

Using equation (\ref{EqConstraint1}), and choosing two classes of operators, namely $\hat{O} = f(x)$ and $\hat{O} = f(x)p$, we obtain the following relations:
\numparts
\begin{eqnarray}
 - \langle f'' \rangle - 2i \langle f' p \rangle + \mathcal{A}_{f} = 0, \label{EqFCon1}\\
    - \langle f'' p \rangle - 2i \langle f' p^2 \rangle + i \langle f V' \rangle + \mathcal{A}_{fp} = 0, \label{EqFPCon1}
\end{eqnarray}
\endnumparts
having used the identity $\left[ p, f \right] = -i f'$ and the notation $f^{(n)} = \partial^n f/ \partial x^n$. The second identity we need is that as $|\psi \rangle$ is an energy eigenstate of the Hamiltonian,
\begin{eqnarray}
    \left \langle \hat{O} \hat{H}\right \rangle = E \left \langle \hat{O} \right \rangle, \label{EqConstraint2}
\end{eqnarray}
from which (for $\hat{O} = f$) we obtain the relation
\begin{eqnarray}
    \langle fp^2 \rangle + \langle f V \rangle - E \langle f \rangle = 0. \label{EqFCon2}
\end{eqnarray}
By relating $f$ to its derivatives, from equations (\ref{EqFCon1}), (\ref{EqFPCon1}) and (\ref{EqFCon2}), we can construct a relation between moments of the probability distribution.

\subsection{A general periodic potential}
For a periodic potential of period $a$, the domain of interest becomes $\lbrace - a/2 \leq x \leq a/2 \rbrace$ and the wavefunction satisfies the Dirichlet boundary condition $\psi(-a/2) = \psi(a/2)$. A stronger constraint is actually imposed by the Bloch theorem, which states that the wavefunctions of energy eigenstates of a periodic lattice can be written as $\psi(x) = \exp \left( i k x \right) u(x)$. Here, $k$ is a wavevector bounded by $\lbrace -\pi/a \leq k \leq \pi/a \rbrace$, and $u(x) = u(x+a)$ i.e. a function with the same periodicity of the lattice.
\medskip

For our choice of $f$, it will prove convenient to consider a spatially periodic function. In particular, we pick $f = t_n = \exp \left( i 2\pi n x/ a \right)$, where $t_n$ is a shorthand we use throughout this work. Using the Bloch ansatz, we compute the anomalies mentioned in the last section:
\begin{eqnarray}
    \mathcal{A}_{\hat{t}_n} = \mathcal{A}_{\hat{t}_n p} = 0.
\end{eqnarray}
Additionally, $t_n$ is conveniently related to derivatives of itself via $t_n' = \frac{i 2\pi n} {a} t_n$, and we obtain for $n \not = 0$:
\numparts
\begin{eqnarray}
    \langle t_n p \rangle + n \frac{\pi}{a} \langle t_n \rangle = 0, \label{EqTP1}\\
    4n^2 \left( \frac{\pi}{a} \right)^2 \langle t_n p \rangle + 4n \frac{\pi}{a} \langle t_n p^2 \rangle + i \langle t_n V' \rangle = 0,  \\
    \langle t_n p^2 \rangle + \langle t_n V \rangle - E \langle t_n \rangle = 0. \label{EqTP2}
\end{eqnarray}
\endnumparts
For $n = 0$, we have $\langle t_0 \rangle = \langle 1 \rangle$, and the only non-trivial bootstrapping identity at this level is
\begin{eqnarray}
    \langle V'\rangle = 0, \label{Eqn0Identity}
\end{eqnarray}
which is the statement that the average derivative of a periodic potential is zero. For $n \not = 0$; however, the recursion relation must then be derived from the expression
\begin{eqnarray}\label{genrec}
    4n \frac{\pi}{a} \left[ E - n^2 \left( \frac{\pi}{a} \right)^2 \right] \langle t_n \rangle = 4n \frac{\pi}{a} \langle t_n V \rangle - i \langle t_n V' \rangle . \label{EqMaster}
\end{eqnarray}
Equation (\ref{EqMaster}) will mostly be used as the master relation for bootstrapping periodic potentials throughout our work. There are two ways to deal with the derivative of the potential that appears here. As it is periodic, one can complete a Fourier decomposition, and write
\begin{eqnarray}
    V(x) = \sum_{m=-\infty}^{\infty} V_m t_m,
\end{eqnarray}
where the modes are given by
\begin{eqnarray}
    V_m = \frac{1}{a}\int_{-a/2}^{a/2} dx V(x) e^{-i2\pi n x /a} = \frac{1}{a}\int_{-a/2}^{a/2} dx V(x) t_{-m}. \label{EqFourierModes}
\end{eqnarray}
Such a decomposition yields
\begin{eqnarray}
    n \left[ E - n^2 \left( \frac{\pi}{a} \right)^2 \right] \langle t_n \rangle = \sum_{m=-\infty}^{\infty} \left( 2n + m \right) V_m \langle t_{n+m} \rangle. \label{EqRecurrenceFourier}
\end{eqnarray}
Such a formulation of the recursion relation is practical for a potential composed of a finite number of $\lbrace V_m \rbrace$ modes, such as the cosine potential discussed in \cite{tchoumakov_bootstrapping_2021,berenstein_bootstrapping_2022}. In that instance, $V_m \propto \delta_{m,\pm 1}$, and the recurrence relation links each moment to only two others. Indeed, one can reproduce the exact recurrence relation discussed in \cite{berenstein_bootstrapping_2022,tchoumakov_bootstrapping_2021} using our general formula above.
\medskip

However, equation (\ref{EqRecurrenceFourier}) becomes much less useful for potential with a large number of Fourier modes, like in the case for a delta function (where the set of $\lbrace V_m \rbrace$ is infinite). We thus consider a second approach, one that involves reconstructing the probability distribution function from the moments. Starting with (\ref{genrec}) and expanding the expectation value as an integral and integrating by parts we obtain
\begin{eqnarray}
    \langle t_n V' \rangle = - \langle t_n' V \rangle - \int_{-a/2}^{a/2} dx \rho'(x) t_n V,
\end{eqnarray}
where all other contributions have vanished due to the periodicity of $t_n$, $V(x)$, and $\rho(x)$. The recursion relation (\ref{genrec}) then becomes
\begin{eqnarray}
    4n \frac{\pi}{a} \left[ E - n^2 \left( \frac{\pi}{a} \right)^2 \right] \langle t_n \rangle = 2n \frac{\pi}{a} \langle t_n V \rangle + i \int_{-a/2}^{a/2} dx \rho'(x) t_n V. \label{EqRecursionRhoDashed}
\end{eqnarray}
To proceed further, we get a handle on $\rho'(x)$ by using integration by parts to obtain from equation (\ref{Eqn0Identity})
\begin{eqnarray}
    0 = \int_{-a/2}^{a/2} dx \rho(x) V' = - \int_{-a/2}^{a/2} dx \rho'(x) V, \label{EqRhoDashedConstraint}
\end{eqnarray}
which will allow us to compute the recursion relation in the case studied in this work.
\subsection{The recursion relation in the Kronig-Penney Model}
\label{SectionRecursioninKP}
We now turn our attention to the central point of the paper, the Kronig-Penney model \cite{kronig_quantum_1931}, where an infinite array of rectangular barriers approximates a 1D crystalline lattice. In the limit that these barriers become infinitely narrow, they are treated as a series of periodic delta spikes of uniform height, and the potential becomes
\begin{eqnarray}
    V(x) = A \sum_{m=-\infty}^{\infty} \delta \left(x - m a \right), \label{EqKPPotential}
\end{eqnarray}
where $a$ is the spacing between the barriers (the period of the lattice) and $A$ parameterises the height of the potential barriers. Analytic solutions to this system is a textbook problem and solving the Schr\"{o}dinger equation in Fourier space yields the familiar dispersion relation \cite{singh_kronigpenney_1983}
\begin{eqnarray}
    \cos \left( k a \right) = \cos \left( a \sqrt{E} \right) + \frac{A}{2\sqrt{E}} \sin \left( a \sqrt{E} \right), \label{EqKPDispersion}
\end{eqnarray}
where $k$ is the Bloch wavevector introduced earlier. For $V(x)$ as defined in equation (\ref{EqKPPotential}), we can explicitly compute the right hand side of equation (\ref{EqRhoDashedConstraint}) and obtain $\rho'(0) = 0$, which implies that the integral on the right hand side of equation (\ref{EqRecursionRhoDashed}) vanishes; i.e. $\int_{-a/2}^{a/2} dx \rho'(x) t_n V = 0$. We additionally compute that
\begin{eqnarray}
    \langle t_n V \rangle = A \rho(0),
\end{eqnarray}
so that equation (\ref{EqRecursionRhoDashed}) gives rise to,
\begin{eqnarray}\label{rho0}
    \langle t_n \rangle = \frac{A\rho(0)}{2} \frac{1}{E - n^2 \pi^2/a^2}, \label{EqTN}
\end{eqnarray}
which is valid for $n \not =0 $ and $E \not = n^2 \pi^2/a^2$. In the special case that $E_m = m^2 \pi^2/a^2$ for some $m \in \mathbb{Z}/\lbrace 0 \rbrace$, we can explicitly find that $\rho(0) = 0$, together with $\langle t_n \rangle = 0$, provided we have $n \not = \pm m$ or $n = 0$. This particular limit is exactly the one corresponding to the infinite square well, as will be discussed in detail in \ref{AppInfSquareWell}. Observe that for $E < n^2\pi^2/a^2$, the denominator of (\ref{rho0}) is always negative, and the sign of $\langle t_n \rangle$ is always opposite to that of $\rho(0)$ i.e. if $\rho(0) > 0$, as we would expect, $\langle t_n \rangle$ is negative for all non-zero $n$.
\medskip

Having solved the recursion relation for the moments, we can now reconstruct the wavefunction density $\rho(x)$. As we did with $V(x)$, we write down a Fourier expansion of the probability density:
\begin{eqnarray}
    \rho(x) = \sum_{m=-\infty}^{\infty} \rho_m t_m,
\end{eqnarray}
where the modes are again given by,
\begin{eqnarray}
    \rho_m = \frac{1}{a} \int_{-a/2}^{a/2} dx \rho(x) t_{-m} = \frac{1}{a} \langle t_{-m} \rangle.
\end{eqnarray}
Substituting equation (\ref{EqTN}) for $m \not =0$ and $\langle t_0 \rangle = 1$, we obtain the functional form of the probability density:
\begin{eqnarray}
    \rho(x) = \frac{1}{a} + \frac{1}{a} \sum_{m=1}^{\infty} \frac{A \rho(0)}{E - \pi^2 m^2/a^2} \cos \left( \frac{2\pi m x}{a} \right). \label{EqRHOX}
\end{eqnarray}
From this expression, we can solve for $\rho(0)$. Setting $x = 0$ and using the identity
\begin{eqnarray}
    \cot z = \frac{1}{z} + \sum_{n=1}^{\infty} \frac{2z}{z^2 - n^2 \pi^2},
\end{eqnarray}
we obtain
\begin{eqnarray}
    \rho(0) = \frac{2}{A \left[ \frac{2a}{A} + \frac{1}{E} - \frac{a}{\sqrt{E}} \cot \left( a \sqrt{E} \right) \right]}. \label{EqRHOZERO}
\end{eqnarray}
We plot this solution in Figure\begin{figure}[t!]

\includegraphics[clip,width=12cm]{ERHOZEROANALYTIC.png}
\centering

\caption{Analytic solution for $E$ as a function of $\rho(0)$ for $a = A = 2$.}
\label{fig:rhozeroanaly}
\end{figure} \ref{fig:rhozeroanaly} for $a = A = 2$. Observe that the function centres on $\rho(0) = 1/a$, and that negative values of $\rho(0)$ are allowed by this expression. This latter fact appears unphysical, and we will return to this subtlety shortly.
\medskip

As noted before, $\rho(0)= 0$ when $E = m^2 \pi^2/a^2$ for $m \in \mathbb{Z}/\lbrace 0 \rbrace$. The poles physically correspond to the limit $a \rightarrow 0$ where it is impossible to define a wavefunction (as the potential is infinite everywhere). Having computed $\rho(0)$ as a function of $E$, equation (\ref{EqTN}) now only depends upon $E$, so numerically, our bootstrapping procedure needs only to consider one parameter.
\section{Numeric implementation of the bootstrap}
\label{SecNumericImplementation}
Having constructed (and in our case, solved) a recursion relation between the moments of the probability distribution, the other key element of the bootstrapping procedure is to restrict the values of those moments via a positivity constraint. In particular, consider a list of linearly independent operators $\mathcal{O}_i$. Then some linear combination $\mathcal{O} = \sum_i a_i O_i$ must obey the constraint \cite{lin_bootstraps_2020,han_bootstrapping_2020}
\begin{eqnarray}
    \left \langle \psi \left| \mathcal{O}^{\dagger} O \right| \psi \right\rangle \geq 0, \label{EqPosConstraint}
\end{eqnarray}
where $\psi$ is an energy eigenstate as before. This constraint of positive definiteness can now be checked numerically, given the initial data from the recursion.
\medskip

As mentioned earlier, previous work suggests using an operator basis with commensurate symmetries to increase the convergence speed of the bootstrap.
Thus, given the operators we have encountered in our periodic potential, we are motivated to consider operators of the form
\begin{eqnarray}
    \hat{O} = \sum_{n=0}^K \sum_{s=0}^L a_{n,s} \hat{p}^s t_n,
\end{eqnarray}
where $a_{n,s}$ is set of $(K+1) \times (L+1)$ arbitrary complex coefficients \cite{tchoumakov_bootstrapping_2021}. In the case, expanding equation (\ref{EqPosConstraint}) yields
\begin{eqnarray}
    \sum_{n,n'=0}^K\sum_{s,s'=0}^L a_{n,s}^* a_{n',s'} \left \langle t_{-n} \hat{p}^{s+s'} t_{n'} \right \rangle \geq 0,
\end{eqnarray}
which is equivalent to the requirement that the matrix $\mathcal{M}_{n\sigma,m\tau} = \langle t_{-m}p^{\sigma+\tau}t_n \rangle$ be positive semi-definite. If we fix $\sigma + \tau$, we have a $(K+1) \times (K+1)$ matrix. From this matrix, we numerically obtain allowed energies by scanning over the parameter space (in our case, the energy $E$) and ruling out energies which do not correspond to a positive definite matrix. The philosophy of the bootstrapping process is that as $K$ increases, the allowed energies should converge to those obtained from explicitly solving the Schr\"{o}dinger equation. In the case $\sigma+\tau = 0$, the requirement for $\mathcal{M}_{n\sigma,m\tau}$ to be positive semi-definite is equivalent to the solution to the classical trigonometric moment problem (as proven in \cite{caratheodory_uber_1911}) and the matrix is referred to as the Toeplitz matrix.
\medskip

Note that when calculating equation (\ref{EqRHOZERO}), we used the fact that the sum in equation (\ref{EqRHOX}) is over infinite $\langle t_n \rangle$. As $\mathcal{M}_{n\sigma,m\tau}$ is computed for a finite number of modes, in what follows, we numerically compute $\rho(0)$ using the finite sum
\begin{eqnarray}
    \rho(0) = \left( a - A \sum_{m=1}^K \frac{1}{E - \pi^2 m^2/a^2} \right)^{-1},
\end{eqnarray}
which is equivalent to equation (\ref{EqRHOZERO}) in the $K \rightarrow \infty$ limit.
\subsection{Obtaining band gaps from the Toeplitz matrix}
We begin by considering the $\sigma + \tau = 0$ case, fixing $a = A = 2$\footnote{The same patterns would occur for $a \neq A$, we simply study this example for aesthetic reasons.}, and then scan over different values of energy $E$. We plot the result of this process for different values of $K$ in Figure\begin{figure}[t!]

\includegraphics[clip,width=12cm]{Toeplitz1.png}
\centering
\caption{Bootstrap for the Kronig-Penney model at various $K$ and $a = A = 2$. Parallel vertical lines mark the exact energy bands.}
\label{fig:Toeplitz}
\end{figure} \ref{fig:Toeplitz}. The horizontal axis is the energy, and the vertical lines mark the boundaries of the exact energy spectrum from the equation (\ref{EqKPDispersion}). For $K=2$, only the first two band gaps are noticeable, and the numerical band gaps are smaller than the exact value. As $K$ increases, all band gaps become evident and converge to the exact values. We quantify this in Table\begin{table}[t!]
\centering
\begin{tabular}{cc|cccccc|c}
\cline{3-8}
\multicolumn{1}{l}{}                           & \multicolumn{1}{l|}{} & \multicolumn{6}{c|}{K}                                                                                & \multicolumn{1}{l}{}          \\ \cline{3-9}
                                               &                       & \multicolumn{1}{c|}{2}    & \multicolumn{1}{c|}{5}    & \multicolumn{1}{c|}{10}   & \multicolumn{1}{c|}{20}   & \multicolumn{1}{c|}{50}   & 100  & \multicolumn{1}{c|}{Analytic} \\ \hline
\multicolumn{1}{|c|}{\multirow{3}{*}{$\Delta E$}} & 1st Gap               & \multicolumn{1}{c|}{1.09} & \multicolumn{1}{c|}{1.45} & \multicolumn{1}{c|}{1.58} & \multicolumn{1}{c|}{1.63} & \multicolumn{1}{c|}{1.64} & 1.66 & \multicolumn{1}{c|}{1.66}     \\ \cline{2-9}
\multicolumn{1}{|c|}{}                         & 2nd Gap               & \multicolumn{1}{c|}{0.90} & \multicolumn{1}{c|}{1.35} & \multicolumn{1}{c|}{1.65} & \multicolumn{1}{c|}{1.80} & \multicolumn{1}{c|}{1.85} & 1.87 & \multicolumn{1}{c|}{1.87}     \\ \cline{2-9}
\multicolumn{1}{|c|}{}                         & 3rd Gap               & \multicolumn{1}{c|}{0.09} & \multicolumn{1}{c|}{1.03} & \multicolumn{1}{c|}{1.57} & \multicolumn{1}{c|}{1.79} & \multicolumn{1}{c|}{1.91} & 1.93 & \multicolumn{1}{c|}{1.94}     \\ \hline
\end{tabular}
\centering
\caption{First three energy gaps computed numerically at various $K$ and $a = A = 2$. Values are quoted to a numeric error of $\pm 0.01$.}
\label{TabDeltaE}
\end{table} \ref{TabDeltaE}, up to a numeric error of $\pm 0.01$. The band gaps between higher energies converge less rapidly than those between lower energies.
\medskip

We also plot the values of $\rho(0)$ corresponding to allowed values for $E$ in Figure\begin{figure}[t!]

\includegraphics[clip,width=\columnwidth]{EvsRZeroNumeric.png}

\caption{Numeric solution for $E$ as a function of $\rho(0)$ for $a = A = 2$ for some values of $K$. Observe that for $K=100$ the negative $\rho(0)$ region stops being allowed, in line with physical intuition.}
\label{fig:EvRZnumer}
\end{figure} \ref{fig:EvRZnumer} for $K =  5$, $10$ and $100$. Given the finite sum, the curves qualitatively differ from the analytic solution shown in Figure \ref{fig:rhozeroanaly} for small $K$. As $K$ increases, the range of allowed $\rho(0)$ decreases; in particular, only positive values of $\rho(0)$ are allowed. This indicates that requiring $\rho(0) \geq 0$ could improve the effectiveness of the bootstrap for small $K$.
\medskip

However, there is an obvious problem: as is evident in Figure \ref{fig:Toeplitz}, requiring the positivity of the Toeplitz matrix does not constrain the minimum energy. Regardless of the value of $K$, all energies below $\pi^2/a^2$ (including negative energies) are allowed by the bootstrap. This indicates that the positivity of the Toeplitz matrix is not a sufficient constraint to find the energy spectrum of the model and that it is necessary to add another tool to our bootstrapping philosophy.
\subsection{Further matrices and constraints for the Kronig-Penney model}
We now move on to the higher constraints, i.e. consider the consequence of enforcing the positivity of the matrix $\mathcal{M}_{n \sigma, m \tau}$ for $\sigma + \tau \geq 1$. We start by constructing elements of the form $\langle t_{-m} p^s t_n \rangle$ from $\langle t_m p^s \rangle$ and the commutation relation
\begin{eqnarray}
    \left[ p, t_m \right] = \frac{2m \pi}{a} t_m. \label{EqCommut}
\end{eqnarray}
From equations (\ref{EqTP1}) and (\ref{EqTP2}) we immediately have
\numparts
\begin{eqnarray}
    \langle t_n p \rangle = - n \frac{\pi}{a} \langle t_n \rangle, \\
    \langle t_n p^2 \rangle = E \langle t_n \rangle - A \rho(0),
\end{eqnarray}
\endnumparts
and we construct $\langle t_m p^s \rangle$ from equation (\ref{EqFCon2}) by choosing $f = t_m p^s$. In particular, for $s \geq 1$ we obtain (after simplifying with equation (\ref{EqCommut})),
\begin{eqnarray}
    \fl \langle t_n p^{s+2} \rangle = \left( E - 4n^2 \left( \frac{\pi}{a} \right)^2 \right) \langle t_n p^s \rangle - 4n \frac{\pi}{a} \langle t_n p^{s+1} \rangle - \frac{A}{a} \sum_{m=-\infty}^{\infty} \langle t_{m+n} p^s \rangle. \label{EqTNPRecursion}
\end{eqnarray}
A similar double recursion relation was obtained in \cite{tchoumakov_bootstrapping_2021}; however, in that work, the higher order moments (specifically for $n = 0$) were found to be numerically unstable. Analytically computing the next two terms in the recursion, we can easily see the reason:
\begin{eqnarray}
    \langle t_n p^3 \rangle = - n \frac{\pi}{a} \left( E \langle t_n \rangle - 2 A \rho(0) \right), \\
    \langle t_n p^4 \rangle = E^2 \langle t_n \rangle - A \rho(0) \left[ 2E + 4 n^2 (\pi/a)^2 - \frac{A}{a} \sum_{m=-\infty}^{\infty} \right], \label{EqTNP4}
\end{eqnarray}
or to say specifically, the recursion leads to a non-convergent infinite sum.\footnote{The symbol $\sum_{m=-\infty}^{\infty}$ implies an infinite series of $...+1+1+1+...$ for our case.}

We will be discussing a way of analytically treating this sum in Section \ref{SecDispersionRelation}. However, for numerics, here we treat it as a finite size correction by summing over the (finite) number of modes determined by the bootstrap matrix of dimension $K$. With that in mind, in what follows, we consider only the cases where $\sigma + \tau \leq 4$, where the infinite sum first appears, as the first investigation into these finite size effects. To do so, we can analytically construct the following matrices

\begin{eqnarray}
   \fl \langle t_{-m} p t_n \rangle = \frac{\pi(n+m)}{a}\langle t_{n-m} \rangle, \\
    \fl\langle t_{-m} p^2 t_n \rangle = \left( \left( \frac{2\pi}{a} \right)^2 nm + E \right) \langle t_{n-m} \rangle - A \rho(0), \\
    \fl \langle t_{-m} p^3 t_n \rangle = \frac{\pi}{a} \left(n+m\right) \left[ \left[ 3 E  - 2 \left( m^2 - 4mn + n^2 \right) \left( \frac{\pi}{a} \right)^2 \right] \langle t_{n-m} \rangle - 3 A \rho(0) \right], \\
    \fl \langle t_{-m} p^4 t_n \rangle = E^2 \langle t_{n-m} \rangle + E 8n \left( \frac{\pi}{a} \right)^2 \left( n+ 2m \right) \langle t_{n-m} \rangle \\
    - 8n \left[ m^3 -3m^2 n - mn^2 + n^3 \right] (\pi/a)^4 \langle t_{n-m} \rangle \nonumber \\
    - A \rho(0) \left[ 2E + 4(m+n)(m+2n) (\pi/a)^2 - \frac{A}{a} \sum_{m=-\infty}^{\infty} \right]\,. \nonumber
\end{eqnarray}

Using the expressions of  $\langle t_n p^s \rangle$. It is evident that as $\sigma + \tau$ increases, the matrices become more unwieldy.
\subsection{Constraining the minimum energy}
As discussed earlier, the problem with using the Toeplitz matrix constraint was that it did not constrain the lowest energy value and allowed an infinite sea of negative values. To demonstrate a potential way of improving this using higher constraints, we start with $K = 5$, and fixing $a = A = 2$, we scan over different values of energy $E$ for different $\sigma + \tau$. We plot the result of this process for each of the matrices with $0 \leq \sigma + \tau \leq 4$ in Figure \ref{fig:ComparingPA}.
\begin{figure}[t!]

\subfloat[][$K = 5$]{%
  \includegraphics[clip,width=0.51
 \columnwidth]{ComparingP1K5.png} \label{fig:ComparingPA}
}
\subfloat[][$K = 100$]{%
  \includegraphics[clip,width=0.51 \columnwidth]{ComparingP1.png} \label{fig:ComparingPb}
}
\caption{Bootstrap for the Kronig-Penney model at $a = A = 2$ for $0 \leq \sigma + \tau \leq 4$ for a) $K = 5$ and b) $K = 100$. Parallel vertical lines mark the exact energy bands. The minimum energy is more constrained for even $\sigma + \tau$ values. }
\label{fig:ComparingPK5K100}
\end{figure}
\medskip

Remember, we are now interested in constraining both the band gaps and the minimum allowed energy. Observe that as $\sigma + \tau$ increases for $K=5$, the band gaps are narrower, i.e. less accurate. This difference becomes negligible as $K$ increases: we plot the result of the bootstrapping process for $K = 100$ in Figure \ref{fig:ComparingPb}, and the exact band gaps are obtained for all $\sigma + \tau$ up to the numerical error of $0.3 \%$. The bootstrapping procedure is most efficient at obtaining the exact band gap structure for smaller $\sigma + \tau$.
\medskip

On the other hand, now consider the behaviour of the minimum energy. For $\sigma + \tau = 1$ a near-continuum spectrum of negative energies is still allowed, but for $\sigma + \tau > 1$, i.e. increasing the power of momentum leads us to a finite minimum bound on the energy. There appears to be an even-odd asymmetry, as evident in Figure \ref{fig:ComparingPA} and \ref{fig:ComparingPb}, where the even powers of momentum in the calculated moment lead to a more accurate constraint than the odd powers. One can interpret this as an artefact of the analytic dispersion relation (equation (\ref{EqKPDispersion})) being only dependent on even powers of Bloch momentum, which is a consequence of the spatial symmetry of the system. We will return to this discussion in Section \ref{SecDispersionRelation}.
\medskip

The best approximation to the minimum energy is achieved for $\sigma + \tau = 4$. This matrix contains a non-convergent infinite sum as it depends on $\langle t_n p^4 \rangle$. In Figure\begin{figure}[t!]
\subfloat[]{
\includegraphics[clip,width=0.52\columnwidth]{FSP1.png}\label{fig:FSPa}}
\subfloat[]{
\includegraphics[clip,width=0.52\columnwidth]{FSC1.png}\label{fig:FSCb}}

\caption{ Bootstrap for the Kronig-Penney model at $\sigma + \tau = 4$ and $a = A = 2$ at various $K$, where we zoom into the structure of the first band. The exact minimum energy is marked in a) by a vertical line. In b), we plot the minimum energy obtained numerically as a function of $K$. }
\label{fig:FSC}
\end{figure} \ref{fig:FSPa} we plot the allowed energy levels in the $\sigma + \tau = 4$ case for different values of $K$. We plot the minimum energy as a function of $K$ in Figure \ref{fig:FSCb}. As $K$ increases, the minimum energy increases and converges towards a constant value, which is a lower bound on the minimum energy. This demonstrates that, numerically, the infinite sum contributes a finite size correction, which converges in the limit of a large number of modes.
\medskip

We conclude from Figure \ref{fig:ComparingPK5K100} that the bootstrapping philosophy needs to be updated for periodic potentials. Not only does increasing the value of $K$ increase the accuracy of the energy spectrum obtained, but it is also necessary to explore multiple powers of momentum $p$ (higher values of $\sigma + \tau$). Lower powers of momentum converge more quickly to the desired band gaps, whilst higher powers of momentum converge to provide a better constraint on the minimum energy. This is because, by exploring more powers of momentum, more moments to constrain the dispersion relation are being accessed.
\section{Towards constructing the dispersion relation}
\label{SecDispersionRelation}
As discussed in the introduction, constraining the dispersion relation for periodic systems using the moment recursion relation is much more involved. We sketch here a resolution of the issue using quantities computed in the previous sections. The reader is reminded that a wavefunction of the Bloch form satisfies $\psi(x) = \exp(ikx) u(x)$, applying the exponential of the momentum operator yields $e^{i\hat{p}a}| \psi \rangle = e^{ika} | \psi \rangle$, so the Bloch wavevector $k$ is extracted via $\langle e^{i\hat{p}a} \rangle = e^{ika}$. The dispersion relation is thus extracted from
\begin{eqnarray}
    \cos \left( ka \right) = \sum_{n=0}^{\infty} \left(- 1 \right)^n \frac{a^{2n}}{(2n)!} \langle \hat{p}^{2n} \rangle, \label{EqCOSKADefin}
\end{eqnarray}
where $p^{2n}$ is extracted from the $n = 0$ instance of equation (\ref{EqTNPRecursion}), which reads,
\begin{eqnarray}
    \langle p^{s+2}\rangle = E \langle p^s \rangle- \frac{A}{a} \sum_{m=-\infty}^{\infty} \langle t_m p^s \rangle,
\end{eqnarray}
and $\langle t_m p^s \rangle$ are themselves computed from equation (\ref{EqTNPRecursion}). It was found in \cite{tchoumakov_bootstrapping_2021} that following such a procedure proved numerically unstable; we propose that this is because this leads to larger and larger sums, even for potential with a finite number of Fourier modes $ \lbrace V_m \rbrace$. We instead deal with the infinite sums using two known values of the Riemann zeta function
\begin{eqnarray}\label{EqZeta2n}
    \zeta \left( 0 \right) = -\frac{1}{2}, ~~~~~~~~~~    \zeta \left( - 2n \right) = 0 \qquad& n \in \mathbb{N}.
\end{eqnarray}
With that, the infinite sum in equation (\ref{EqTNP4}) simplifies, and we find via induction
\begin{eqnarray}
   \fl \langle t_n p^{2s} \rangle = E^s \langle t_n \rangle - A \rho(0) \left( E^{s-1} + \sum_{m=1}^{s-1} c^{(1)}_{m,n} E^{s-1-m} \left( n^2 \pi^2 / a ^2 \right)^m \right), \\
     \fl\langle t_n p^{2s+1} \rangle = - n\frac{\pi}{a} \left[ E^s \langle t_n \rangle - A \rho(0) \left( E^{s-1} + \sum_{m=1}^{s-1} c^{(2)}_{m,n} E^{s-1-m} \left( n^2 \pi^2 / a ^2 \right)^m \right) \right],
\end{eqnarray}

where the $c^{(i)}_{m,n}$ for $i \in \lbrace 1,2 \rbrace$ are numeric factors without an apparent closed form, but those will not prove relevant to our interests. Using equation (\ref{EqZeta2n}), we readily obtain:
\numparts
\begin{eqnarray}
    \langle p^{2s} \rangle = E^s \left( 1 - \frac{s A \rho(0) }{ E} \right), \label{EqP2SRHOZERO} \\
    \langle p^{2s+1} \rangle = E^s \langle p \rangle,
\end{eqnarray}
\endnumparts
where $\langle p \rangle = - a/2 j \left( a/2 \right)$ as computed in terms of the particle current $j$ in \ref{AppComputingExpP} using a proper calculation of the anomaly $\mathcal{A}_x$. We do not solve a recursion relation for the particle current in this work, but by considering different operator forms of $\mathcal{O}$, one could do so.
\medskip

Substituting the solution from equation (\ref{EqP2SRHOZERO}) back into equation (\ref{EqCOSKADefin}) and using equation (\ref{EqRHOZERO}) for $\rho(0)$ does not help us a lot.
However, the dispersion relation is a semi-classical quantity. Keeping in with that, in \cite{tchoumakov_bootstrapping_2021} it was noted that the dispersion relation for Bloch momenta could be obtained only by taking the average of a statistical ensemble. In accordance with that, we then focus on the expectation value of $\rho(x)$ instead, which is simply
\begin{eqnarray}
    \langle \rho(x) \rangle = 1/a,
\end{eqnarray}
and then trivially $\langle \rho(0) \rangle = 1/a$. Now, using this expectation value in (\ref{EqP2SRHOZERO}), we get,
\begin{eqnarray}
    \langle p^{2s} \rangle = E^s\left(1 - s \frac{A}{aE}\right),
\end{eqnarray}
which we can then substitute the above into equation (\ref{EqCOSKADefin}) to obtain
\begin{eqnarray}
    \cos ka = \langle \cos pa \rangle = \cos \left( a \sqrt{E} \right) + \frac{A}{2\sqrt{E}} \sin \left( a \sqrt{E} \right),
\end{eqnarray}
exactly the relation in equation (\ref{EqKPDispersion}). We have thus proposed an analytic approach to deal with the unstable summations present in the recursion for $p^s$ observed in \cite{tchoumakov_bootstrapping_2021} and demonstrated its effectiveness for the Kronig-Penney model. It remains a question for future investigations whether (and how) this trick can be implemented numerically.
\section{Conclusions}
\subsection*{Summary}
In this work, motivated by recent advancements, we explore the effectiveness of the bootstrap method for the Dirac comb model of a 1d lattice system (the Kronig-Penney model). This problem generally can be easily solved analytically, but it is very important to pass it through this algorithm for the sake of further development. We successfully achieved some progress in terms of developing the bootstrap as a tool for solving generic periodic systems using this model as an example. We derived a general dispersion relation for a Bloch periodic problem, considering anomalies generated by the boundary conditions. We also demonstrated the benefit of explicitly reconstructing probability density functions to have analytical handling of the band structure associated with the problem.
\medskip

Numerically, we proposed novel positivity constraints involving a spatially periodic operator basis to zero in on the allowed energy eigenvalues of the system. An intriguing question associated with the Quantum Mechanics bootstrap program is finding constraints which are enough, with a sufficiently sized Hankel matrix, to converge on the exact spectrum of the system. For our case, it turned out that not only a larger size of the constraint matrix but also the power of the momentum used to change the allowed energies pretty drastically, with higher powers generating the band structure better. We augmented our discussion by providing a roadmap toward reconstructing the Kronig-Penney dispersion relation from our analytic considerations.
\subsection*{Comments about the minimum energy}
One intriguing feature of our numerical calculation is the presence of the seemingly unconstrained minimal energy, which we could minimize by using higher powers of momentum. However, there lingers a question about how to constrain this value properly. In a recent work by Morita, \cite{morita_bootstraping_2022}, it was shown that the positive definiteness of the Hamburg matrix for $K = 2$ is equivalent to the Heisenberg uncertainty relation. Computing $\langle x^2 \rangle$ explicitly from equation (\ref{EqRHOX}), requiring $\langle x^2 \rangle \langle p^2 \rangle \geq 1/4$ can be used numerically to obtain finite minimum energy for our model.
\medskip

A bit of tinkering reveals, for $a = A = 2$, the Heisenberg constraint obtains $E_{min} = 0.245$, as compared to the numerical value $E_{min} = -0.435$ obtained via the $\sigma + \tau = 4$ matrix in the $K = 400$ case, and the analytic value $E_{min} = 0.741$. Whilst the Heisenberg constraint provides a more effective constraint on the minimal value; it allows all energies $E > E_{min}$; that is, it cannot obtain the rest of the energy spectrum. That observation feeds into the narrative, which is apparent from our results; the Toeplitz matrix converges quickly to the band gaps for $E > \pi^2/a^2$, but isn't useful for constraining the minimum possible energy.
\medskip

In this work, we found that the matrices $\mathcal{M}_{n \sigma,m\tau}$ for $\sigma + \tau > 0$ provide one means to get a finite lower bound on the minimum energy (whilst at the same time constraining the band gaps). The Heisenberg relation simply samples another constraint from the set of possible quadratic constraints. It remains an open question of which constraint is most effective for finding the exact minimum energy.
\subsection*{ Comments about a quasi-periodic case}

In this work, we have considered periodic potentials and focused on the Kronig-Penney model as a particular example. A closely related class of systems are those with quasi-periodic potentials, which have emerged in recent years as important classes of models in condensed matter physics, like the celebrated Aubry-André-Harper (AAH) model \cite{harper_single_1955,aubry_analyticity_1980}. The simplest example of such a quasi-periodic potential in the continuous regime is:
\begin{eqnarray}
    V(x) = \cos \left( \frac{2\pi}{a} x \right) + \cos \left( \alpha \frac{2\pi}{a} x + \beta \right),
\end{eqnarray}
where $\alpha$ is irrational (so that there is no periodic solution) and $\beta$ contributes a phase. We can consider the machinery developed in this work to try and construct a recursion relation for moments for such a potential (although those results were derived for the explicitly periodic case). Computing the Fourier modes $V_m$ to substitute into equation (\ref{EqRecurrenceFourier}), we obtain
\begin{eqnarray}
    \fl V_m = \frac{1}{2} \left[ \delta_{m,1} + \delta_{m,-1} \right]  + \frac{1}{\pi (\alpha^2-m^2)} \left( -1 \right)^m \sin \left( \pi \alpha \right) \left[ m \cos \beta - i \alpha \sin \beta \right],
\end{eqnarray}
so we can see the set $\lbrace V_m \rbrace$ is infinite in size. This would require an infinite sum in the recursion relation, meaning that it is not numerically solvable. Alternatively, one may explicitly compute
\begin{eqnarray}
    \langle t_n V \rangle = \frac{1}{2} \left \langle t_{1-n} + t_{-1-n} + e^{-i\beta} e^{i \frac{2\pi}{a} \left( \alpha - n \right)} + e^{i\beta} e^{i \frac{2\pi}{a} \left( - \alpha - n \right)} \right \rangle.
\end{eqnarray}
Observe that, as $\alpha$ is irrational, the terms dependent on it are outside the sequence of $t_n$, and it is not possible to rescale the operator basis to include them in the sequence. Furthermore, to conclude the vanishing of anomalies, we assumed an exactly periodic (Bloch) wavefunction density, which is no longer necessarily valid. A probable way out is to consider an explicitly quasi-periodic basis of operators and rework the whole algorithm. This is beyond the scope of this work and will be discussed in future communication.
\subsection*{Future directions}
The bootstrap technique has already proven to be of much interest and seemingly very powerful in solving Quantum Mechanical problems. Periodic problems, especially ones with Bloch symmetry, are very new to this growing list of explorations. We considered one of the simpler examples in this work, and extending our results to other tight-binding models makes perfect sense.
\medskip

Of course, other tractable Quantum Mechanical problems are open to exploration using similar methods. Recently \cite{cho_bootstrapping_2022} has initiated an intriguing study of infinite lattice Ising models in different dimensions using similar bootstrap techniques. One could think of these as a viable alternative to the conformal bootstrap procedure, and this leaves ample scope for improvements in the formalism employed, given the success of the conformal bootstrap program. We hope to contribute to these refinements in future work.

\section*{Acknowledgements}
The authors would like to thank Jyotirmoy Bhattacharya and Diptarka Das for their comments on the draft. MJB was supported by an internship at the Quantum Gravity Unit of Okinawa Institute of Science and Technology Graduate University (OIST) during the course of this project. MJB would like to thank Yasha Neiman for discussions. AB is supported by Start-Up Research Grant (SRG/2020/001380), Mathematical Research Impact Centric Support Grant (MTR/2021/000490) by the Department of Science and Technology Science and Engineering Research Board (India) and Relevant Research Project grant (202011BRE03RP06633-BRNS) by the Board Of Research In Nuclear Sciences (BRNS), Department of atomic Energy, India. The work of ArB is supported by the Quantum Gravity Unit of the OIST. ArB would like to thank TU Wien and Ecole Polytechnique, Paris for kind hospitality during the course of this project.

redpajama_set_name: RedPajamaArXiv


Situated on the prestigious Oak Circle Drive in the highly desirable Sparr Heights neighborhood is this gorgeous newly renovated charmer. Enter the bright living room with lots of windows, new engineered hardwood floors, a warm fireplace and views of the mountains. A brand new decorator kitchen with stunning quartz countertops, beautiful glass backsplash and new stainless steel appliances is ready for the chef to entertain in style. There are three bedrooms and 2 remodeled bathrooms. Enjoy the lovely and serene backyard which is perfect for gatherings with family and friends. Expand your options with the large multi-purpose room off the backyard that could be used as a game room, office or fulfill other possible uses. Delight in all this Sparr Heights home has to offer with a location convenient to restaurants, shopping, farmer's market and all Montrose has to offer.


Because it is Friday and because we collect quotes, here is one on statistics being the best and worst of disciplines. Which one of the two views are closest to your opinion?
If mathematics is the handmaiden of science, statistics is the whore: all that scientists are looking for is a quick fix without the encumbrance of a meaningful relationship. Statisticians are second-class mathematicians, third-rate scientists and fourth-rate thinkers. They are the hyenas, jackals and vultures of the scientific ecology: picking over the bones and carcasses of the game that the big cats, the biologists, the physicists and the chemists, have brought down.
Statistics is a wonderful discipline. It has it all: mathematics and philosophy, analysis and empiricism, as well as applicability, relevance and the fascination of data. It demands clear thinking, good judgement and flair. Statisticians are engaged in an exhausting but exhilarating struggle with the biggest challenge that philosophy makes to science: how do we translate information into knowledge?


…Yes, The Boat's Fixed; And It's Ferry Nice.
Bert was loaded and off we sailed for the one hour crossing to the small island of Ko Yao Noi. It was pleasant. Well, after suffering crossings on the long-tail boat, rowing a kayak would be pleasant. The car ferry was sheer bliss by comparison. Although, I must admit, it does roll a lot. I think it must have a flat bottom. After an hour on the wooden seats, it wasn't the only one. There was one tall truck on board which I thought was going to fall over.
And since this article is about the ferry, I'll save the vacation stories for the next article or three.
So, yes, I also came back. Yesterday, in fact. I was supposed to be at the jetty at the uncivilized hour of 7am, but my friends' restaurant, Ciao Bella Italian Restaurant & Pizzeria is only five minutes up the road and round the bend by the mosque, so I turned up at 7:40 and guess what. No ferry.
Last time, I wrote "Although I wasn't happy at missing out on a vacation, I'll still go, when the captain decides the boat has been fixed. Perhaps next time he can arrange for it to break down for a lengthy period while I'm on the island. I like the sound of that" and I thought my prayers had been answered. But it turned out, it was everyone else's prayers that were being answered.
Being the first day of Eid al'Fitr (your spelling may vary) everyone on Ko Yao Yai, the ferry's starting point, had to go to the mosque, which delayed the sailing by … well, no one seemed to know. So it was back to the jetty at 9am and 9:30am until a friend volunteered to go at 10am on her motorbike and then called me in a panic to say "the boat is here." At this point, I was getting ready to go back to the hotel to unpack, but instead, Bert was hastily reversed onto the boat – a few centimeters closer to the front than I would have liked. "We'll teach this obnoxious farang a lesson" they were thinking, in Thai of course.
Yes, that's as far as the ramp gets raised!


text:
Constitutively Activated G Protein-Coupled Receptors: A Novel Approach to CNS Drug Discovery
Author(s): Frederique D.P. Menzaghi, Dominic P. Behan, Derek T. Chalmers	Arena Pharmaceuticals,Inc, 6166 Nancy Ridge Drive, San Diego, CA 92131, USA
Journal Name: Current Drug Targets - CNS & Neurological Disorders
Continued as CNS & Neurological Disorders - Drug Targets
DOI : 10.2174/1568007023339508
G protein-coupled receptors (GPCRs) represent a major class of signal transduction proteins that modulate various biological functions. GPCRs are one of the most common targets for drug development-currently, 39 of the top 100 marketed drugs in use act directly or indirectly through activation or blockade of GPCR-mediated receptors. Nearly 160 GPCRs have been identified based on their gene sequence and their ability to interact with known endogenous ligands. However, an estimated 500-800 additional GPCRs have been classified as "orphan" receptors (oGPCRs) because their endogenous ligands have not yet been identified. Given that known GPCRs have proven to be such clinically useful drug targets, these oGPCRs represent a rich group of receptor targets for the development of novel and improved medicines. To develop ligands for these potential drug targets requires the ability to identify groups or pools of GPCRs that are likely to be involved in a specific disease process (obesity, schizophrenia, depression, etc.) and to dissect out the pharmacological and signal transduction differences between these GPCR subtypes. It also requires the development of assays to detect ligands of GPCRs even when the endogenous ligands are unidentified. This paper will review novel strategies to identify clinically interesting oGPCRs and to screen for small molecules that act as ligands without prior knowledge of endogenous ligands. This involves the use of constitutively activated GPCRs, a technology that provides a unique opportunity to identify several classes of pharmacological agents, including agonists, inverse agonists and allosteric modulators.
Keywords: Activated G Protein-Coupled Receptors, G protein-coupled receptors (GPCRs), Orphan receptors
Current Drug Targets - CNS & Neurological Disorders
Title: Constitutively Activated G Protein-Coupled Receptors: A Novel Approach to CNS Drug Discovery
Author(s):Frederique D.P. Menzaghi, Dominic P. Behan and Derek T. Chalmers
Affiliation:Arena Pharmaceuticals,Inc, 6166 Nancy Ridge Drive, San Diego, CA 92131, USA
Keywords:Activated G Protein-Coupled Receptors, G protein-coupled receptors (GPCRs), Orphan receptors
Abstract: G protein-coupled receptors (GPCRs) represent a major class of signal transduction proteins that modulate various biological functions. GPCRs are one of the most common targets for drug development-currently, 39 of the top 100 marketed drugs in use act directly or indirectly through activation or blockade of GPCR-mediated receptors. Nearly 160 GPCRs have been identified based on their gene sequence and their ability to interact with known endogenous ligands. However, an estimated 500-800 additional GPCRs have been classified as "orphan" receptors (oGPCRs) because their endogenous ligands have not yet been identified. Given that known GPCRs have proven to be such clinically useful drug targets, these oGPCRs represent a rich group of receptor targets for the development of novel and improved medicines. To develop ligands for these potential drug targets requires the ability to identify groups or pools of GPCRs that are likely to be involved in a specific disease process (obesity, schizophrenia, depression, etc.) and to dissect out the pharmacological and signal transduction differences between these GPCR subtypes. It also requires the development of assays to detect ligands of GPCRs even when the endogenous ligands are unidentified. This paper will review novel strategies to identify clinically interesting oGPCRs and to screen for small molecules that act as ligands without prior knowledge of endogenous ligands. This involves the use of constitutively activated GPCRs, a technology that provides a unique opportunity to identify several classes of pharmacological agents, including agonists, inverse agonists and allosteric modulators.
Frederique D.P. Menzaghi, Dominic P. Behan and Derek T. Chalmers, " Constitutively Activated G Protein-Coupled Receptors: A Novel Approach to CNS Drug Discovery", Current Drug Targets - CNS & Neurological Disorders (2002) 1: 105. https://doi.org/10.2174/1568007023339508
https://doi.org/10.2174/1568007023339508 Print ISSN
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redpajama_set_name: RedPajamaCommonCrawl


I say "in your particular case" because your reason may be different than someone else's reason. That's why there are so many different natural treatments, and that's why only some will work for you.
Join the Natural Solutions for PCOS Online Conference to hear Lara's interview!
Don't be too quick to accept a PCOS diagnosis. Proper diagnosis requires a blood test that shows high testosterone or other male hormones like androstenedione or DHEA. Never accept a diagnosis that was made by ultrasound alone.
This is the classic type of PCOS and by far the most common. High insulin and leptin impede ovulation and stimulate the ovaries to make testosterone. Insulin resistance is caused by sugar, smoking, trans fat, and environmental toxins.
Is this you? You have been told that you are borderline diabetic, or you had an abnormal glucose tolerance test. You probably have elevated insulin, and you may also have elevated LH (luteinizing hormone). You are probably overweight, although you may be normal weight. Normal-weight insulin resistance can occur in the years following dieting or eating disorder.
Treatment ideas: The first step is to quit sugar. A small amount of fructose is healthy but a large amount is a major contributor to insulin resistance. Please also consider gentle intermittent fasting, which works well to improve insulin sensitivity. Best supplements for insulin resistance are magnesium, lipoic acid, inositol, and berberine. The Pill is not a treatment for this type (or any type) of PCOS because it impairs insulin sensitivity. Improvement for Type 1 PCOS is slow and gradual over 6-9 months.
Hormonal birth control suppresses ovulation. For most women, it's a temporary effect, and ovulation will usually resume fairly soon after the Pill is stopped. But for some women, ovulation-suppression can persist for months or even years. During that time, it is not unusual to be given the diagnosis of PCOS. Some experts deny the existence of Pill-induced PCOS, but it is very real. It is the second most common type of PCOS that I see. I have spoken to many other clinicians who are seeing the same thing. We desperately need more research into it.
Is this you? You had regular periods before starting the Pill, although you may have had acne. You probably now have elevated LH on a blood test, although you may have normal LH and high-normal prolactin.
Treatment ideas: If your LH is elevated, the best herbal treatment is Peony & Licoricecombination. If your prolactin is high-normal, then the best herbal treatment is Vitex (also called chaste tree or chaste berry). Do not use Vitex if your LH is elevated. Vitex stimulates LH so it will make things worse. Indeed, many of my PCOS patients report feeling worse on Vitex.
Both Peony and Vitex work on your pituitary-ovarian axis, and they are powerful herbs. I recommend you do not use them too soon or for too long. Do not take them if you are a teenager, or if you have just come off the Pill. Give yourself at least 3-4 months off the Pill. Do not use Peony or Vitex for more than 10 months in a row. They should not need to be used that way. If they are the right herbs, they will work fairly quickly (within 3-4 months). And then, your periods should stay regular after you stop the herbs. Do not take licorice if you have high blood pressure. Please seek professional advice.
Is this you? You have other symptoms of immune dysfunction such as recurring infections, headaches, joint pain or skin conditions. Your blood test shows inflammatory biomarkers such as vitamin D deficiency, abnormal blood count, elevated C-RP, thyroid antibodies, or gluten antibodies. You may have elevated DHEA or androstenedione, and a positive urine test for intestinal permeability.
Treatment ideas. Reduce stress and exposure to environmental toxins like pesticides and plastics. Eliminate inflammatory foods such as wheat, dairy, and sugar. Treat intestinal permeability with zinc, berberine, and probiotics. Supplement magnesium because it is anti-inflammatory and normalizes adrenal hormones (HPA axis). Improvement is slow and gradual over 6-9 months.
Iodine deficiency, because your ovaries need iodine. Please be careful with iodine supplementation.
Artificial sweeteners, because they impair insulin and leptin signaling.
Know when it's time to let go of your PCOS Diagnosis. In most cases, PCOS is not a permanent condition. With the right diagnosis and the right treatment, PCOS can become a thing of the past.
I am the first to acknowledge that there is some overlap between these 4 types. For example, inflammation is a major cause of both insulin resistance and thyroid disease (Types 1, 3, and 4). My intention was not to over-simplify, but rather to alert you to some treatment ideas that you have not yet considered.
Join the Natural Solutions for PCOS Online Conference right now, where you can hear Lara's interview!


KUALA LUMPUR: Foreign investors have been making a return to the local market since the start of the new year, with a net inflow of about RM3.1 billion in the first 18 trading days.
Local institutions, however, were making use of higher share prices to lock in profits. Local fund managers were the net sellers, with about RM3.2 billion in the same period.
According to Rakuten Trade Sdn Bhd vice-president of research Vincent Lau, the selling seen among the local institutions was mainly to rejig their investment portfolios. "Most of them are gradually taking profit as they relook at their portfolios. It makes sense for some of them to lock in their profit.
"Some of them are switching in their investments from exporters to banking, while the OPR (overnight policy rate) hike has also led to the 'underweight' of the REIT (real estate investment trust) players," Lau told The Edge Financial Daily. Lau sees the selling as an anticipation of a slight pullback on Bursa Malaysia. But, he believes that the local institutions would eventually re-enter the local market.
A fund manager with a local asset management firm, who requests to remain anonymous, commented that the timing of the local funds to re-enter the market is also dependent on the date of the upcoming 14th general election (GE14). Since the pre-election rally appears to have taken a breather after a strong run at the beginning of the year, it is likely to see local institutions time their entry when the Parliament is dissolved, he said.
"When Parliament is dissolved, it'll create some volatility in the stock market and I think that's when the local institutions will deploy their money into the market. Right now, they will probably take some of the money off the table ahead of GE14 and re-strategise," he said.
Affin Hwang Asset Management Bhd (Affin Hwang AM) managing director Teng Chee Wei concurs that local institutions will turn net buyers sooner or later. He noted that Malaysia has a big pool of local investment funds. This local liquidity will come in handy to lend support to share prices on Bursa, for instance when foreign investors leave the domestic market in a big way.
Teng believes with the anticipated rising cash positions, local institutions have to remain invested to generate yield. "If you look at institutions, the local institutions at some point will have to invest. Where else do they want to invest when the required return on their asset is a lot more higher than deposit rates?" Teng told The Edge Financial Daily.
Given the rising cash position, Teng said the local fund managers will have to remain invested to generate sufficient returns of about 5% to 6%, or even higher, for their contributors or unitholders. Teng concluded that under such expectations, local institutions will have to eventually invest selectively and time their entry at some point into Bursa.
On the difference in sentiment between local and foreign investors, Teng explained that it is due mainly to the influence of local noises. "For local institutions, there is an influence of the local noises. Foreigners do not stay here. Their lives are not affected day by day. However, as a local, you are impacted daily. There are very different perspectives all together … My conclusion is that when you talk to locals, they are very biased against the country. They are inherently biased from what they [have] experienced on a day-to-day basis. From a foreign perspective, it's always on a big macro perspective and they invest using the macro approach," he said.
For Affin Hwang AM, Teng pointed that the firm has not sold on the recent rally and believed that there is still upside in the local bourse. He noted that it will be difficult to time the market but it could be too early to head for the exit now.


text:
Q: ng-model values not being captured when generated by php function So I have a function that generates honeypot form fields in Laravel.
The issue is when I submit the form with Angular, the javascript object doesn't contain the my_time or my_name fields.
public function generate(array $honey_name, array $honey_time)
{
    $honey_time_encrypted = parent::getEncryptedTime();
    // Encrypt the current time

    $html = '
' .
                '<input type="text" ' . $this->attributes($honey_name) . ' id="' . $honey_name['name'] . '" />' .
                '<input type="text" ' . $this->attributes($honey_time) . ' value="' . $honey_time_encrypted . '" />' .
            '
';

    echo $html;
}

Any thoughts why?
UPDATE:
Blade Template
<form ng-submit="submitForm(formData)">

        {!! Form::text('name', null, ['class' => 'form-control', 'placeholder' => 'Name', 'ng-model' => 'formData.name']) !!}
        {!! Honeypot::generate(['name' => 'my_name', 'ng-model' => 'formData.my_name'], ['name' => 'my_time', 'ng-model' => 'formData.my_time']) !!}

        {!! Form::email('email', null, ['class' => 'form-control', 'placeholder' => 'Email', 'ng-model' => 'formData.email']) !!}

        {!! Form::text('subject', null, ['class' => 'form-control', 'placeholder' => 'Subject','ng-model' => 'formData.subject']) !!}

        {!! Form::textarea('message', null, ['class' => 'form-control', 'rows' => '5', 'placeholder' => 'Message', 'ng-model' => 'formData.message']) !!}

        {!! Form::submit('Send', ['class' => 'btn btn-default center-block']) !!}

</form>

Submitted form data:
$scope.submitForm = function(formData) {
    console.log(formData);
    Contact.submit(formData).then(function(results) {
        SweetAlert.swal({
            title: "Success!",
            text: "Thanks for getting in touch! I usually respond within 12 hours.",
            type: "success",
        });
    });
};

// Output of console.log();
Object {name: "Daniel", email: "abc@123.com", subject: "asdsadsafas fasfaf", message: "asjfasf aalhfsfa flaskfh aklsflakf a"}

So it's missing the my_time and my_name models, which causes validation to fail.

A: You don't have the <form> element.
Try it this way:
$html = '
' .
'<form method="post" action="some action here">' .
'<input type="text" ' . $this->attributes($honey_name) . ' id="' . $honey_name['name'] . '" />' .
'<input type="text" ' . $this->attributes($honey_time) . ' value="' . $honey_time_encrypted . '" />' .
'</form>
';

A: Okay, I figured it out. Angular doesn't post empty form fields, so that was part of the problem.
The second issue had to do with formData.my_time. If you plan to set the input value server side, then you must use ng-init="foo = 'bar'" ng-model="foo".
So once I added ng-init="formData.my_time = \'' . $honey_time_encrypted . '\'" I was good to go.

redpajama_set_name: RedPajamaStackExchange


January 17 2017, 9:49 p.m.
The Intercept revealed last week that Donald Trump's nominee to lead the the Department of Homeland Security, retired Marine Corps Gen. John F. Kelly, did not disclose on his ethics form that he was listed as a vice chairman at the Spectrum Group, a defense contractor lobbying firm.
Now, it appears that Kelly failed to list two other positions as well: board memberships with Michael Baker International and Sallyport Global, two defense contractors that do business with the U.S. government.
Federal law requires Senate-confirmed nominees to list all positions at outside organizations on ethics forms that are vetted by the Office of Government Ethics and released to both the public and lawmakers. But the board memberships, like the Spectrum Group vice chairmanship, were not listed on Kelly's federal ethics disclosure form nor his letter to the Homeland Security Department's designated ethics official.
Kelly's professional biography has been wiped from the Michael Baker International website. But Brian Peiritsch, the firm's director of corporate communications, told The Intercept that Kelly joined the board, "I would say 2016, that's about how specific I can be right now." Peiritsch declined to comment further.
Screenshot from the leadership page of Sallyport Global.
Kelly's disclosure currently shows that he received $37,500 from DC Capital Partners, a position he has held since April 2016. The payments are described as "fees, member of board of directors" on the form.
DC Capital Partners is the controlling shareholder of the Spectrum Group, and lists Michael Baker International and Sallyport Global as portfolio companies. DC Capital Partners, using a subsidiary, took control of Sallyport Global in 2011, and purchased Michael Baker International in 2013. A press release notes that Sallyport Global is now a subsidiary of Michael Baker International.
T. Gail Dady, a partner at DC Capital Partners, confirmed that Kelly joined the board of Michael Baker International. Dady did not respond to multiple requests for further comment. Online records appear to show that Kelly was added to Michael Baker International's website in July 2016.
Michael Baker International provides a range of government clients with managing consulting. The firm announced last year that it won a $500 million contract to provide environmental consulting for the Air Force and a contract with the Department of Homeland Security to develop guidelines for first responders.
Sallyport Global, which directed questions about the firm to DC Capital Partners, has a long, scandal-plagued history as a contractor to the military. Thomas Charron, Sallyport Global's founder, launched the company then partnered with John DeBlasio, a former adviser to the Coalition Provisional Authority. The company was awarded lucrative contracts, transporting Iraqi police trainees to Jordan, building secure compounds, and doing other security work in Iraq and Afghanistan.
In 2006, insurgents boarded a bus and killed 17 Iraqi-born translators working for Sallyport in Basrah, then scattered their bodies around the city.
Sallyport was providing security for General Dynamics when three of that company's contractors were kidnapped in Baghdad a year ago and held for 31 days.
Despite questions about their security record, the firm continued to win contracts. In January 2016, the firm was awarded a $271 million contract to manage upgrades at the Balad Air Base in Iraq.
As The Intercept previously reported, the Spectrum Group widely advertised Kelly's role with the company, claiming that he worked on multiple client teams, that he was among the people at the firm who could help guide clients with information about the Trump transition, and that he specializes in helping clients with border and infrastructure security issues.
The same Trump transition official had previously acknowledged that Kelly maintained a relationship with Spectrum Group, however; last week, he told The Intercept that Kelly had mentioned his relationship with Spectrum to Senate staffers during a meeting.
Ethics experts say the failure to disclose Kelly's listed board memberships violate ethics guidelines. "One of the purposes of the financial disclosure requirement is for OGE and the public to identify an official's potential conflicts of interest," said Brendan Fischer, an attorney with the Campaign Legal Center.
"If Gen. Kelly only identifies his relationship with the parent private equity firm, and not his relationship with specific subsidiaries engaged in lobbying or other activities, OGE and the public are left guessing as to which clients of which subsidiaries might pose a potential conflict," Fischer added, noting that OGE guidelines do require even non-compensated positions to be disclosed.