Two-dimensional Time-dependent Point Interactions

Abstract

We study the time-evolution of a quantum particle subjected to time-dependent zero-range forces in two dimensions. After establishing a conceivable ansatz for the solution to the Schrödinger equation, we prove that the wave packet time-evolution is completely specified by the solutions of a system of Volterra-type equations -- the {\it charge equations} -- involving the coefficients of the singular part of the wave function, thus extending to the two-dimensional case known results in one and three dimensions.

17 pages, AMS-LaTex; presentation of the model changed, small changes to Lemma 2.1 and Proposition 2.1

References in corpus (3)

• The point-like limit for a NLS equation with concentrated nonlinearity in dimension three
• Energy lower bound for the unitary N+1 fermionic model
• Decay of a Bound State under a Time-Periodic Perturbation: a Toy Case

Cited by in corpus (9)

• Ground states for the planar NLSE with a point defect as minimizers of the constrained energy
• The point-like limit for a NLS equation with concentrated nonlinearity in dimension three
• Blow-up for the pointwise NLS in dimension two: absence of critical power
• The action of Volterra integral operators with highly singular kernels on Hölder continuous, Lebesgue and Sobolev functions
• Nonlinear singular perturbations of the fractional Schrödinger equation in dimension one
• Complete ionization for a non-autonomous point interaction model in d = 2
• Microscopic Derivation of Time-dependent Point Interactions
• Time dependent delta-prime interactions in dimension one
• An introduction to the two-dimensional Schrödinger equation with nonlinear point interactions