Bernard Helffer, Yuri A. Kordyukov
Semiclassical Spectral Asymptotics for a Two-Dimensional Magnetic Schrödinger Operator: The Case of Discrete Wells
Preprint series: ESI preprints

MSC:

35P20 Asymptotic distribution of eigenvalues and eigenfunctions for PDO

35J10 Schrodinger operator, See also {35Pxx}

47A75 Eigenvalue problems, See also {49Rxx}

58G25 Spectral problems; spectral geometry; scattering theory, See also {35Pxx}

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis

Abstract: We consider a magnetic Schr\"odinger operator $H^h$, depending on
the semiclassical parameter $h>0$, on a two-dimensional Riemannian
manifold. We assume that there is no electric field. We suppose that
the minimal value $b_0$ of the magnetic field $b$ is strictly
positive, and there exists a unique minimum point of $b$, which is
non-degenerate. The main result of the paper is a complete
asymptotic expansion for the low-lying eigenvalues of the operator
$H^h$ in the semiclassical limit. We also apply these results to
prove the existence of an arbitrary large number of spectral gaps in
the semiclassical limit in the corresponding periodic setting.


Keywords: Spectral theory, Schrodinger operators, magnetic fields, eigenvalue asymptotics