Asymptotic behavior of eigenvalues of Schr\"odinger type operators with degenerate kinetic energy

TL;DR

This paper investigates how the eigenvalues of Schrödinger type operators behave asymptotically as the coupling constant approaches zero, especially when the kinetic energy degenerates on a manifold of codimension one.

Contribution

It provides new insights into the asymptotic behavior of eigenvalues for operators with degenerate kinetic energy on a manifold, extending previous understanding to this specific degeneracy case.

Findings

Eigenvalues exhibit specific asymptotic behavior in the small coupling limit.

Degeneracy of the kinetic energy significantly influences eigenvalue distribution.

Results apply to Schrödinger operators with non-trivial degeneracy manifolds.

Abstract

We study the eigenvalues of Schr\"odinger type operators $T + \lambda V$ and their asymptotic behavior in the small coupling limit $\lambda \to 0$, in the case where the symbol of the kinetic energy, $T(p)$, strongly degenerates on a non-trivial manifold of codimension one.