Spectral asymptotics for magnetic Schrödinger operators with rapidly decreasing electric potentials

Raikov, Georgi D.; Warzel, Simone

Comptes Rendus. Mathématique, Tome 335 (2002) no. 8, pp. 683-688.

On considère l'opérateur de Schrödinger H(V) agissant dans ${\mathrm{L}}^2\left({ℝ}^2\right)$ ou ${\mathrm{L}}^2\left({ℝ}^3\right)$ avec un champ magnétique constant et un potentiel électrique V qui génériquement décroı̂t à l'infini exponentiellement vite ou est à un support compact. On étudie le comportement asymptotique du spectre discret de H(V) en voisinage des points de la frontière de son spectre essentiel. Si la décroissance de V est gaussienne ou plus rapide ce comportement ne se décrit pas par les formules semi-classiques connues dans le cas où V décroı̂t comme une puissance.

We consider the Schrödinger operator H(V) on ${\mathrm{L}}^2\left({ℝ}^2\right)$ or ${\mathrm{L}}^2\left({ℝ}^3\right)$ with constant magnetic field, and a class of electric potentials V which typically decay at infinity exponentially fast or have a compact support. We investigate the asymptotic behaviour of the discrete spectrum of H(V) near the boundary points of its essential spectrum. If V decays like a Gaussian or faster, this behaviour is non-classical in the sense that it is not described by the quasi-classical formulas known for the case where V admits a power-like decay.

[1] Avron, J.; Herbst, I.; Simon, B. Schrödinger operators with magnetic fields. I. General interactions, Duke Math. J., Volume 45 (1978), pp. 847-883

[2] Birman, M.S̆. On the spectrum of singular boundary value problems, Amer. Math. Soc. Transl., Volume 53 (1966), pp. 23-80

[3] Blankenbecler, R.; Goldberger, M.L.; Simon, B. The bound states of weakly coupled long-range one-dimensional quantum Hamiltonians, Ann. Phys. (N.Y.), Volume 108 (1977), pp. 69-78

[4] Hardy, G.H.; Wright, E.M. An Introduction to the Theory of Numbers, Clarendon Press, Oxford, 1954

[5] Hupfer, T.; Leschke, H.; Warzel, S. Upper bounds on the density of states of single Landau levels broadened by Gaussian random potentials, J. Math. Phys., Volume 42 (2001), pp. 5626-5641

[6] Ivrii, V.Ya. Microlocal Analysis and Precise Spectral Asymptotics, Springer, Berlin, 1998

[7] M. Melgaard, G. Rozenblum, Eigenvalue asymptotics for even-dimensional perturbed Dirac and Schrödinger operators with constant magnetic fields, Preprint mp_arc 02-140, March 2002

[8] Raikov, G.D. Eigenvalue asymptotics for the Schrödinger operator with homogeneous magnetic potential and decreasing electric potential. I. Behaviour near the essential spectrum tips, Comm. Partial Differential Equations, Volume 15 (1990), pp. 407-434

[9] Raikov, G.D. Border-line eigenvalue asymptotics for the Schrödinger operator with electromagnetic potential, Integral Equations Operator Theory, Volume 14 (1991), pp. 875-888

[10] G.D. Raikov, S. Warzel, Quasi-classical versus non-classical spectral asymptotics for magnetic Schrödinger operators with decreasing electric potentials, Preprint , January 2002 | arXiv

[11] Sobolev, A.V. Asymptotic behavior of the energy levels of a quantum particle in a homogeneous magnetic field, perturbed by a decreasing electric field. I, J. Soviet Math., Volume 35 (1986), pp. 2201-2212

[12] Solnyshkin, S.N. Asymptotics of the energy of bound states of the Schrödinger operator in the presence of electric and homogeneous magnetic fields, Sel. Math. Soviet., Volume 5 (1986), pp. 297-306

[13] Tamura, H. Asymptotic distribution of eigenvalues for Schrödinger operators with homogeneous magnetic fields, Osaka J. Math., Volume 25 (1988), pp. 633-647