GLYPH<Minus> 1 1 2 3 4 GLYPH<Minus> 5 5 10 15 t$_{1}$ B$_{eff}$ Stringy Stringy = B$_{eff}$ 7 must project out the nonphysical Q -degrees of freedom. When using the standard formula 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$_{+}$ 〉 , where the non-normalizible Q -system states | Ψ$_{±}$ 〉 are chosen as FIG. 4: Expectation values of the position and momentum as functions of time for the Gaussian wave packet, drawn for the stringy and effective Bateman propagators (17) and (22), respectively. The initial wave packet characteristics and the meaning of solid and dashed color lines are the same as in Figure 3. A trial B ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ ) = GLYPH<2> dQA$_{B}$ ( q$_{1}$, Q, t$_{1}$ | q$_{0}$, Q, t$_{0}$ ) , A eff B ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ ) := Amp $^{(}$| q$_{0}$ 〉| Ψ$_{−}$ 〉 → | q$_{1}$ 〉| Ψ$_{+}$ 〉 ) = = GLYPH<2> dQdQ ′ Ψ$_{+}$( Q $^{′}$) A$_{B}$ ( q$_{1}$, Q $^{′}$, t$_{1}$ | q$_{0}$, Q, t$_{0}$ ) Ψ$_{−}$( Q ) , (21) Ψ$_{±}$( Q ) = √ i cosh κ $_{2}$T$^{√}$ ∓ 2 iα π$_{ℏ}$ exp $^{{}$± i ℏ αQ 2 $^{}}$. i$_{ℏ}$∂$_{t}$ Ψ( q, t ) = $^{[}$− ℏ 2 2 d 2 dq 2 + U ( q ) + $_{ℏ}$K [Ψ] $^{]}$Ψ( q, t ) , (23) A eff B ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ ) := √ κ 4 πi$_{ℏ}$ tanh κ $_{2}$T × × exp { i ℏ α 4$^{[}$$^{(}$ q$_{1}$ − q$_{0}$β$_{−}$ ) 2 + $^{(}$q$_{0}$ − q$_{1}$β$_{+}$ ) 2 $^{]}}$. (22) K [Ψ] = i κ 2$^{[}$ ln Ψ( q, t ) Ψ( q, t ) − GLYPH<2> dq ′ Ψ( q $^{′}$, t ) ln Ψ( q $^{′}$, t ) Ψ( q $^{′}$, t ) Ψ( q $^{′}$, t ) $^{]}$. equivalent description of the system by the Schrödinger equation [4]: Substituting (20) into (21) we obtain the following effec- tive propagator for the damped (sub)system: where K [Ψ] is the Ψ -dependent Kostin potential defined as The form of the auxiliary states | Ψ$_{+}$ 〉 and | Ψ$_{−}$ 〉 guar- antees that the effective Bateman propagator for the damped (sub)system satisfies the normalization condi- tion (2). This is known as Kostin-Schrödinger(-Langevin) equa- tion. In the Kostin’s paper a complete set of solu- tions of this equation in the special case of free particle ( U ( q ) = 0 ) is presented. The set consists of the states Ψ$_{p}$$_{0}$( q, t ) labeled by the continuous quantum number p$_{0}$ , the initial momentum of the particle. The state Ψ$_{p}$$_{0}$( q, t ) starts as a momentum eigenstate with the momentum p$_{0}$ , and remains the momentum eigenstate also later, but with the decreasing momentum, The Bateman effective propagator (22) depends only on the time difference T , which favors it in compari- son with the CK-propagator (18). On the other hand, the artificial “repairing procedure” involving nonphysical states discredits the Bateman approach with respect to the stringy one. The Gaussian wave packet character- istics obtained from (17) and (22) are drawn in Figure 4. As seen from the figure, the stringy quantization pos- sesses again better qualitative features than the approach to which we have compared it. From the ensemble of non-localized Kostin’s states one would like to form normalized wave packet solutions, localized in configuration as well as momentum space. This is, however, impossible since the Kostin-Schrödinger equation is nonlinear; thus, the Kostin approach belongs to nonlinear quantum mechanics and lacks superposition principle. In Figure 5 we plot the expectation value of the momentum for the Kostin state Ψ$_{p}$$_{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. 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 ˆ p Ψ$_{p}$$_{0}$( q, t ) = p$_{0}$ e − κ ( t − t$_{0}$ $^{)}$Ψ$_{p}$$_{0}$ ( q, t ) . ˙ ˆ q = ˆ p , ˙ ˆ p = F (ˆ q, ˆ p ) , C. Stringy versus Kostin approach