Time dependent delta-prime interactions in dimension one

Abstract

We solve the Cauchy problem for the Schrödinger equation corresponding to the family of Hamiltonians $H_{\gamma (t)}$ in $L^2(\mathbb{R})$ which describes a ${\delta }^{\prime }$-interaction with time-dependent strength $1/\gamma (t)$. We prove that the strong solution of such a Cauchy problem exits whenever the map $t\mapsto \gamma (t)$ belongs to the fractional Sobolev space $H^{3/4}(\mathbb{R})$, thus weakening the hypotheses which would be required by the known general abstract results. The solution is expressed in terms of the free evolution and the solution of a Volterra integral equation.

minor changes, 10 pages

References in corpus (4)

• The point-like limit for a NLS equation with concentrated nonlinearity in dimension three
• Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity I: Basic theory
• Decay of a Bound State under a Time-Periodic Perturbation: a Toy Case
• Two-dimensional Time-dependent Point Interactions

Cited by in corpus (1)

• Application of a quantum wave impedance method for zero-range singular potentials