Eigenvalues upper bounds for the magnetic Schrödinger operator

We study the eigenvalues $\lambda_k (H_{A,q})$ of the magnetic Schrödinger operator $ H_{A,q}$ associated with a magnetic potential $A$ and a scalar potential $q$, on a compact Riemannian manifold $M$, with Neumann boundary conditions if $\partial M \neq \emptyset$. We obtain various bounds on $\lambda_1 (H_{A,q}),\lambda_2 (H_{A,q})$ and, more generally, on $\lambda_k (H_{A,q})$. Some of them are sharp. Besides the dimension and the volume of the manifold, the geometric quantities which plays an important role in these estimates are: the first eigenvalue $\lambda^{\prime\prime}_{1,1} (M)$ of the Hodge–de Rham Laplacian acting on co-exact $1$-forms, the mean value of the scalar potential $q$, the $L^2$-norm of the magnetic field $B = dA$, and the distance, taken in $L^2$, between the harmonic component of $A$ and the subspace of all closed $1$-forms whose cohomology class is integral (that is, having integral flux around any loop). In particular, this distance is zero when the first cohomology group $H^1 (M, \mathbf{R})$ is trivial. Many other important estimates are obtained in terms of the conformal volume, the mean curvature and the genus (in dimension $2$). Finally, we also obtain estimates for sum of eigenvalues (in the spirit of Kröger estimates) and for the trace of the heat kernel.

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Authors’ note: Our colleague and friend Ahmad El Soufi passed away on December 29, 2016.

Received 16 June 2017

Accepted 19 September 2019

Published 30 January 2023