Communications in Analysis and Geometry

Volume 30 (2022)

Number 4

Eigenvalues upper bounds for the magnetic Schrödinger operator

Pages: 779 – 814

DOI: https://dx.doi.org/10.4310/CAG.2022.v30.n4.a3

Authors

Bruno Colbois (Institut de Mathématiques, Université de Neuchâtel, Switzerland)

Ahmad El Soufi (Laboratoire de Mathématiques et Physique Théorique, Université François Rabelais de Tours, France)

Saïd Ilias (Laboratoire de Mathématiques et Physique Théorique, Université François Rabelais de Tours, France)

Alessandro Savo (Dipartimento SBAI, Sezione di Matematica, Sapienza Università di Roma, Italy)

Abstract

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