On Scr\"odinger equation with non-local potential

M. S. Smetanina

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 26 (2002) no. 3, pp. 99-114

Résumé

We consider the Schrödinger operator of the form $H=-d^2/dx^2+V$ acting in $L^2({\mathbf R})$ where $V=\varepsilon W(x)+\lambda(\cdot,\phi _0)\phi_0$ is non-local potential. It is proved, that the unique level (i. e. eigenvalue or resonance of the operator $H$) exists for $V=\lambda(\cdot,\phi_0)\phi_0$ for all sufficiently small $\lambda$. We investigate the asymptotic behaviour of level for a small $\lambda$. We prove that there are no levels for $V=\varepsilon W(x)+\lambda(\cdot,\phi_0)\phi_0$ for all sufficiently small $\varepsilon$, if $\lambda \not=0$.