Time dependent delta-prime interactions in dimension one

Abstract

We solve the Cauchy problem for the Schrödinger equation corresponding to the family of Hamiltonians $H_{γ(t)}$ in $L^{2}(\mathbb{R})$ which describes a $δ'$-interaction with time-dependent strength $1/γ(t)$. We prove that the strong solution of such a Cauchy problem exits whenever the map $t\mapstoγ(t)$ 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.

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