We show that the lowest eigenvalue of the magnetic Schrödinger operator on a line bundle over a compact Riemann surface is bounded by the -norm of the magnetic field . This implies a similar bound on the multiplicity of the ground state. An example shows that this degeneracy can indeed be comparable with 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 est majorée par la norme du champ magnétique . On en déduit une borne analogue pour la multiplicité de l’état fondamental. Un exemple démontre que cette multiplicité peut être comparable avec même dans le cas du fibré trivial.
Keywords: magnetic laplacian, multiplicity of the ground state, Riemann surface
Mot clés : laplacien magnétique, multiplicité de l'état fondamental, surface riemannienne
Erdős, László 1
@article{AIF_2002__52_6_1833_0, author = {Erd\H{o}s, L\'aszl\'o}, title = {Spectral shift and multiplicity of the first eigenvalue of the magnetic {Schr\"odinger} operator in two dimensions}, journal = {Annales de l'Institut Fourier}, pages = {1833--1874}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {52}, number = {6}, year = {2002}, doi = {10.5802/aif.1936}, zbl = {1106.35039}, mrnumber = {1954326}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1936/} }
TY - JOUR AU - Erdős, László TI - Spectral shift and multiplicity of the first eigenvalue of the magnetic Schrödinger operator in two dimensions JO - Annales de l'Institut Fourier PY - 2002 SP - 1833 EP - 1874 VL - 52 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1936/ DO - 10.5802/aif.1936 LA - en ID - AIF_2002__52_6_1833_0 ER -
%0 Journal Article %A Erdős, László %T Spectral shift and multiplicity of the first eigenvalue of the magnetic Schrödinger operator in two dimensions %J Annales de l'Institut Fourier %D 2002 %P 1833-1874 %V 52 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1936/ %R 10.5802/aif.1936 %G en %F AIF_2002__52_6_1833_0
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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