Exactly solvable models with time-dependent boundary conditions

Absorbing boundary conditions for Schrödinger equation in a time-dependent interval

“…Earlier, the Schrödinger equation with time-dependent boundary conditions given in a one-dimensional box has been studied by many authors (see, e.g., [12,17] and references therein) The Schrödinger equation with timedependent boundary conditions is an ill posed problem unless the moving boundaries are not mapped onto fixed ones [12,17]. This can be done by introducing a new coordinate as [12,17]:…”

“…Strict mathematical treatment of ABC for different wave equations, including quantum mechanical Schrödinger equation can be found in the Refs. [11][12][13][14][15][16]. For both processes the boundary conditions can be derived by factorization of the differential operator corresponding to a wave equation which in general lead to complicated equations for the boundary conditions The explicit form of such boundary conditions is much more complicated than those of Dirichlet, Neumann and Robin conditions.…”

“…In particular, using the approach developed in the Ref. [12], we reduce time-dependent Schrödinger equation with moving boundary conditions into that of fixed ones. Imposing the ABC for the problem with moving boundaries, we derive the corresponding boundary conditions for the Schrödinger equation on a finite interval with unit length which is then solved numerically.…”

Time dependent quantum graph with loop

“…A detailed study of the problem can be found in [11]. In particular, it was pointed out that the problem of 1D box with a moving wall can be mapped onto that of an harmonic oscillator with time-dependent frequency confined inside the static box.…”

Quantum metastability in a class of moving potentials

“…The second class of potentials involves time-dependent boundaries. Unlike the first class, this class of potentials attracts much less attention, and almost all previous works in this area concerned only the simplest of all cases, namely, an infinite potential well with a moving wall [9,10]. The last class is the combination of the previous two classes.…”