Schrödinger equation with time-dependent boundary conditions

Abstract

A Schrödinger equation for a well potential with varying width is studied. Generalized canonical transformations are shown to transform the problem into a time-dependent harmonic oscillator problem submitted to fixed boundary conditions. This transformed problem is solved by a perturbation technique and gives the evolution of the average energy of the system according to the motion of the well. Motions corresponding to a renormalization or compaction group are shown to be solvable by separation of variables.

On the linearization of the generalized Ermakov systems

“…The time dependence in (22)(23), is thus, in a sense, spurious. Consequently, there remains, in fact, only four fundamental arbitrary functions in the linearizable generalized Ermakov system (22)(23), that is, F may be incorporated in C and ρ may be eliminated by the quasi-invariance transformation. It must be stressed, however, that the quasi-invariance transformation is not an essential step in the linearization procedure.…”

“…This is manifest in equation (22) where we see, by inspection, that B can account for F . The second and perhaps more interesting remark is that the linearizable generalized Ermakov system (22-23) can be expressed in autonomous form, by means of a quasi-invariance [15,22] transformation…”

“…While equation (32) represents the orbits for the nonlinear generalized Ermakov system, equations (30) and (33) give the time evolution of the dynamic variables, in terms of the general solution for the linearized system. A particular case in the class of generalized linearizable Ermakov systems (22)(23), are the usual Ermakov systems, with frequency depending only on time, ω ≡ ω(t). Let A ≡ B ≡ C ≡ 0 in the definition (18) of generalized frequencies, which impliesρ…”

Time-dependent quantum graph

“…Treatment of such system requires solving the Schrödinger equation with time-dependent boundary conditions. Earlier, the problem of time-dependent boundary conditions in the Schrödinger equation has attracted much attention in the context of quantum Fermi acceleration [12][13][14], although different aspects of the problem were treated by many authors [16][17][18][19][20][21][22][23][24][25][26][27]. Detailed study of the problem can be found in a series of papers by Makowski and co-authors [21][22][23].…”

Quantum metastability in a class of moving potentials

“…However, for the scaling form of V (x, t) in Eq. (1) and the linear form of L(t), separation of variables can be achieved through a series of transformations introduced in [11,12] (see also: [9,10]). One first transforms the coordinate frame into a rescaled frame with a rescaled coordinatex defined bȳ…”