Spectral shift and multiplicity of the first eigenvalue of the magnetic Schrödinger operator in two dimensions

Erdős, László

doi:10.5802/aif.1936

Erdős, László ¹

¹Georgia Institute of Technology, School of Mathematics, Atlanta GA 30332 (USA)

Annales de l'Institut Fourier, Volume 52 (2002) no. 6, pp. 1833-1874

Abstract (Original language)
Résumé

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 $\int_{M} | B |$ 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 $\int_{M} | B |$ même dans le cas du fibré trivial.

Article information
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Zbl MR

DOI: 10.5802/aif.1936

Classification: 35P15, 58J35, 81Q10
Keywords: magnetic laplacian, multiplicity of the ground state, Riemann surface
Mots-clés : laplacien magnétique, multiplicité de l'état fondamental, surface riemannienne

Author's affiliations:

Erdős, László ¹

¹ Georgia Institute of Technology, School of Mathematics, Atlanta GA 30332 (USA)

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

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References
Cited by

[A] T. Aubin Some Nonlinear Problems in Riemannian Geometry, Springer-Verlag, 1998 | Zbl | MR

[AC] Y. Aharonov; A. Casher Ground state of spin $- \frac{1}{2}$ charged particle in a two-dimensional magnetic field., Phys. Rev., Volume A19 (1979), pp. 2461-2462 | MR

[BCC] G. Besson; B. Colbois; G. Courtois Sur la multiplicité de la première valeur propre de l'opérateur de Schrödinger avec champ magnétique sur la sphère $S^{2}$ , Trans. Amer. Math. Soc., Volume 350 (1998) no. 1, pp. 331-345 | DOI | Zbl | MR

[Be] G. Besson Sur la multiplicité de la première valeur propre des surfaces riemanniennes, Ann. Inst. Fourier, Volume 30 (1980) no. 1, pp. 109-128 | DOI | Zbl | MR | Numdam

[C] S.Y. Cheng Eigenfunctions and nodal sets, Comment. Math. Helv., Volume 51 (1976), pp. 43-55 | DOI | Zbl | MR

[CdV86] Y. Colin de Verdière Sur la multiplicité de la première valeur propre non nulle du laplacien, Comment. Math. Helv., Volume 61 (1986), pp. 254-270 | DOI | Zbl | MR

[CdV87] Y. Colin de Verdière Construction de laplaciens dont une partie finie du spectre est donnée, Ann. Sci. Éc. Norm. Sup., 4e série, Volume 20 (1987), pp. 599-615 | Zbl | MR | Numdam

[CdV88] Y. Colin de Verdière Sur une hypothèse de transversalité d'Arnold, Comment. Math. Helv., Volume 63 (1988), pp. 184-193 | DOI | Zbl | MR

[CdV97] Y. Colin de Verdière Spectre d'opérateurs différentiels sur les graphes to appear in the Proceedings of `Random walks and discrete potential theory' (Cortona, 1997) | Zbl

[CdVT] Y. Colin de Verdière; N. Torki Opérateurs Schrödinger avec champs magnétique, Séminaire de théorie spectrale et géométrie (Grenoble), Volume 11 (1992-1993), pp. 9-18 | Numdam | Zbl | MR | EuDML

[CFKS] H.L. Cycon; R.G. Froese; W. Kirsch; B. Simon Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Springer-Verlag, 1987 | Zbl | MR

[Ch] I. Chavel Riemannian Geometry: A Modern Introduction, Cambridge University Press, 1993 | Zbl | MR

[CL] S.Y. Cheng; P. Li Heat kernel estimates and lower bound of eigenvalues, Comment. Math. Helv., Volume 56 (1981) no. 3, pp. 327-338 | DOI | Zbl | MR | EuDML

[ES] L. Erdős; J.P. Solovej The kernel of the Dirac operator, Rev. Math. Phys., Volume 13 (2001) no. 10, pp. 1247-1280 | DOI | Zbl | MR

[EV] L. Erdős; V. Vougalter Pauli operator and Aharonov-Casher theorem for measure valued magnetic fields, Commun. Math. Phys., Volume 225 (2002), pp. 399-421 | DOI | Zbl | MR

[HH] R. Hempel; I. Herbst Strong magnetic fields, Dirichlet boundaries, and spectral gaps, Commun. Math. Phys., Volume 169 (1995) no. 2, pp. 237-259 | DOI | Zbl | MR

[HLMW] T. Hupfer; H. Leschke; P. Müller; S. Warzel The absolute continuity of the integrated density of states for magnetic Schrödinger operators with certain unbounded random potentials, Comm. Math. Phys., Volume 221 (2001) no. 2, pp. 229-254 | DOI | Zbl | MR

[HM] B. Hellfer; A. Morame Magnetic bottles in connection with superconductivity, J. Funct. Anal., Volume 185 (2001) no. 2, pp. 604-680 | DOI | Zbl | MR

[HON] M. Hoffmann-Ostenhof; T. Hoffmann-Ostenhof; N. Nadirashvili On the multiplicity of eigenvalues of the Laplacian on surfaces, Ann. Global Anal. Geom., Volume 17 (1999) no. 1, pp. 43-48 | DOI | Zbl | MR

[J] J. Jost Riemannian Geometry and Geometric Analysis, Springer-Verlag, 1998 | Zbl | MR

[LW] A. Laptev; T. Weidl Hardy inequalities for magnetic Dirichlet forms, Operator Theory: Advances and Applications, Volume 108 (1999), pp. 299-305 | Zbl | MR

[R] S. Rosenberg Semigroup domination and vanishing theorems., Geometry of random motion (Ithaca, N.Y. 1987) (Contemporary Math.), Volume 73 (1988), pp. 287-302 | Zbl

[Si] B. Simon Maximal and minimal Schrödinger forms, J. Operator Theory., Volume 1 (1979), pp. 37-47 | Zbl | MR

[St] Stein E. Harmonic Analysis, Princeton University Press, 1993 | Zbl | MR

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