Asymptotics of the Schr\"odinger operator levels for a crystal film with a nonlocal potential

M. S. Smetanina

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 28 (2018) no. 4, pp. 462-473

Résumé

We consider a three-dimensional Schrödinger operator for a crystal film with a nonlocal potential, which is a sum of an operator of multiplication by a function, and an operator of rank two (“separable potential”) of the form $V=W (x) +\lambda _1(\cdot,\phi _1)\phi _1+\lambda _2(\cdot,\phi _2)\phi _2 $. Here the function $W(x)$ decreases exponentially in the variable $x_3$, the functions $\phi _1(x)$, $\phi _2(x)$ are linearly independent, of Bloch type in the variables $x_1,\,x_2$ and exponentially decreasing in the variable $x_3$. Potentials of this type appear in the pseudopotential theory. A level of the Schrödinger operator is its eigenvalue or resonance. The existence and uniqueness of the level of this operator near zero is proved, and its asymptotics is obtained.

Détail

Keywords: Schrödinger equation, nonlocal potential, eigenvalues, resonances, asymptotics.