Counting function of the embedded eigenvalues for some manifold with cusps, and magnetic Laplacian

TL;DR

This paper studies the spectral properties of magnetic Laplacians on certain non-compact manifolds with cusps, establishing Weyl asymptotics and bounds for embedded eigenvalues, advancing understanding of spectral geometry in these settings.

Contribution

It proves Weyl asymptotics with sharp remainder for magnetic Laplacians on manifolds with cusps and provides bounds for embedded eigenvalues of the Laplace-Beltrami operator.

Findings

Weyl asymptotic formula with sharp remainder established for magnetic Laplacian.

Upper bounds derived for the counting function of embedded eigenvalues.

Analysis applies to manifolds with cusps and non-exact boundary forms.

Abstract

We consider a non compact, complete manifold {\bf{M}} of finite area with cuspidal ends. The generic cusp is isomorphic to ${\bf{X}}\times ]1,+\infty [$ with metric $ds^2=(h+dy^2)/y^{2\delta}.$ {\bf{X}} is a compact manifold with nonzero first Betti number equipped with the metric $h.$ For a one-form $A$ on {\bf{M}} such that in each cusp $A$ is a non exact one-form on the boundary at infinity, we prove that the magnetic Laplacian $-\Delta_A=(id+A)^\star (id+A)$ satisfies the Weyl asymptotic formula with sharp remainder. We deduce an upper bound for the counting function of the embedded eigenvalues of the Laplace-Beltrami operator $-\Delta =-\Delta_0.$