NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2016, 7 (2), P. 303–314
Time dependent delta-prime interactions in dimension one
C. Cacciapuoti – DiSAT, Sezione di Matematica, Università dell’Insubria, via Valleggio 11, 22100 Como, Italy; claudio.cacciapuoti@uninsubria.it
A. Mantile – Laboratoire de Mathématiques, Université de Reims FR3399 CNRS, Moulin de la Housse BP 1039, 51687 Reims, France; andrea.mantile@univreims.fr
A. Posilicano – DiSAT, Sezione di Matematica, Università dell’Insubria, via Valleggio 11, 22100 Como, Italy; andrea.posilicano@uninsubria.it
We solve the Cauchy problem for the Schrödinger equation corresponding to the family of Hamiltonians Hγ(t) in L2(R) which describes a δ’-interaction with time-dependent strength 1/γ(t). We prove that the strong solution of such a Cauchy problem exists whenever the map t –> γ(t) belongs to the fractional Sobolev space H3/4(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.
Keywords: time dependent point interactions, delta-prime interaction, non-autonomous Hamiltonians.
PACS 02.30.Jr, 03.65.Db, 02.30.Rz
DOI 10.17586/2220-8054-2016-7-2-303-314