content_image/414113.jpg content_image/414114.jpg content_image/414115.jpg content_image/414117.jpg content_image/414129.jpg content_image/414107.jpg content_image/414116.jpg content_image/414118.jpg content_image/414128.jpg
must project out the nonphysical Q -degrees of freedom. When using the standard formula 

<img src='content_image/414113.jpg'>

we do not reproduce the free particle propagator for κ = 0 as desired. To overcome this trouble the following “repair- ing prescription” is introduced (for more detail see [25]): the amplitude for the damped (sub)system to pass from | q$_{0}$ 〉 to | q$_{1}$ 〉 within the time T is equal to the amplitude for the whole system to pass from the nonphysical state | q$_{0}$ 〉| Ψ$_{−}$ 〉 to the nonphysical state | q$_{1}$ 〉| Ψ$_{+}$ 〉 , 

<img src='content_image/414114.jpg'>

where the non-normalizible Q -system states | Ψ$_{±}$ 〉 are chosen as 

<img src='content_image/414115.jpg'>

Substituting (20) into (21) we obtain the following effec- tive propagator for the damped (sub)system: 

<img src='content_image/414117.jpg'>

The form of the auxiliary states | Ψ$_{+}$ 〉 and | Ψ$_{−}$ 〉 guar- antees that the effective Bateman propagator for the damped (sub)system satisfies the normalization condi- tion (2). 

The Bateman effective propagator (22) depends only on the time difference T , which favors it in compari- son with the CK-propagator (18). On the other hand, the artificial “repairing procedure” involving nonphysical states discredits the Bateman approach with respect to the stringy one. The Gaussian wave packet character- istics obtained from (17) and (22) are drawn in Figure 4. As seen from the figure, the stringy quantization pos- sesses again better qualitative features than the approach to which we have compared it. 

A possible incorporation of the classical dynamics into the quantum one can be obtained by using Heisenberg equations for the position and momentum operators [34]: 

where ˆ q and ˆ p obey $^{[}$ˆ q, ˆ p ] = i$_{ℏ}$ for all t . In the special case when F (ˆ q, ˆ p ) = − dU (ˆ q ) /dq − κ ˆ p , Kostin found an 

## C. Stringy versus Kostin approach 

<img src='content_image/414129.jpg'>

<img src='content_image/414107.jpg'>

equivalent description of the system by the Schrödinger equation [4]: 

<img src='content_image/414116.jpg'>

where K [Ψ] is the Ψ -dependent Kostin potential defined as 

<img src='content_image/414118.jpg'>

This is known as Kostin-Schrödinger(-Langevin) equa- tion. In the Kostin’s paper a complete set of solu- tions of this equation in the special case of free particle ( U ( q ) = 0 ) is presented. The set consists of the states Ψ$_{p}$$_{0}$( q, t ) labeled by the continuous quantum number p$_{0}$ , the initial momentum of the particle. The state Ψ$_{p}$$_{0}$( q, t ) starts as a momentum eigenstate with the momentum p$_{0}$ , and remains the momentum eigenstate also later, but with the decreasing momentum, 

From the ensemble of non-localized Kostin’s states one would like to form normalized wave packet solutions, localized in configuration as well as momentum space. This is, however, impossible since the Kostin-Schrödinger equation is nonlinear; thus, the Kostin approach belongs to nonlinear quantum mechanics and lacks superposition principle. In Figure 5 we plot the expectation value of the momentum for the Kostin state Ψ$_{p}$$_{0}$( q, t$_{1}$ ) and the stringy evolved state Ψ( q, t$_{1}$ ) = GLYPH<1> dq $^{′}$A$_{1}$$_{str}$ ( q, t$_{1}$ | q $^{′}$, t$_{0}$ )Ψ$_{p}$$_{0}$( q $^{′}$, t$_{0}$ ) for various values of κ [35]. As κ tends to zero, the stringy and Kostin results match each other. 

<img src='content_image/414128.jpg'>

