Spectral shift and multiplicity of the first eigenvalue of the magnetic Schrödinger operator in two dimensions
Annales de l'Institut Fourier, Volume 52 (2002) no. 6, pp. 1833-1874.

We show that the lowest eigenvalue of the magnetic Schrödinger operator on a line bundle over a compact Riemann surface M is bounded by the L 1 -norm of the magnetic field B. This implies a similar bound on the multiplicity of the ground state. An example shows that this degeneracy can indeed be comparable with M |B| even in case of the trivial bundle.

On démontre que la première valeur propre de l’opérateur de Schrödinger avec champ magnétique sur un fibré en droites au-dessus d’une surface riemannienne compacte M est majorée par la norme L 1 du champ magnétique B. On en déduit une borne analogue pour la multiplicité de l’état fondamental. Un exemple démontre que cette multiplicité peut être comparable avec M |B| même dans le cas du fibré trivial.

DOI: 10.5802/aif.1936
Classification: 35P15, 58J35, 81Q10
Keywords: magnetic laplacian, multiplicity of the ground state, Riemann surface
Mot clés : laplacien magnétique, multiplicité de l'état fondamental, surface riemannienne

Erdős, László 1

1 Georgia Institute of Technology, School of Mathematics, Atlanta GA 30332 (USA)
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Erdős, László. Spectral shift and multiplicity of the first eigenvalue of the magnetic Schrödinger operator in two dimensions. Annales de l'Institut Fourier, Volume 52 (2002) no. 6, pp. 1833-1874. doi : 10.5802/aif.1936. https://aif.centre-mersenne.org/articles/10.5802/aif.1936/

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