Heat kernel asymptotics for magnetic Schrödinger operators. (English) Zbl 1284.81106
Summary: We explicitly construct parametrices for magnetic Schrödinger operators on \({\mathbb{R}}^d\) and prove that they provide a complete small-\(t\) expansion for the corresponding heat kernel, both on and off the diagonal.{
©2013 American Institute of Physics}
©2013 American Institute of Physics}
MSC:
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 35K08 | Heat kernel |
| 35J10 | Schrödinger operator, Schrödinger equation |
References:
| [1] | Minakshisundaram, S.; Pleijel, Å., Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Can. J. Math., 1, 242-256 (1949) · Zbl 0041.42701 · doi:10.4153/CJM-1949-021-5 |
| [2] | Minakshisundaram, S., Eigenfunctions on Riemannian manifolds, J. Indian Math. Soc., New Ser., 17, 159-165 (1953) · Zbl 0055.08702 |
| [3] | Berger, M.; Gauduchon, P.; Mazet, E., Le spectre d’une variété riemannienne, 194, vii+251 (1971) · Zbl 0223.53034 |
| [4] | Berline, N.; Getzler, E.; Vergne, M., Heat Kernels and Dirac Operators, 298, viii+369 (1992) · Zbl 0744.58001 |
| [5] | Gilkey, P. B., Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, x+516 (1995) · Zbl 0856.58001 |
| [6] | Kirsten, K., Spectral Functions in Mathematics and Physics, 400 (2001) |
| [7] | Atiyah, M.; Bott, R.; Patodi, V. K., On the heat equation and the index theorem, Invent. Math., 19, 279-330 (1973) · Zbl 0257.58008 · doi:10.1007/BF01425417 |
| [8] | Loss, M.; Thaller, B., Optimal heat kernel estimates for Schrödinger operators with magnetic fields in two dimensions, Commun. Math. Phys., 186, 95-107 (1997) · Zbl 0873.35078 · doi:10.1007/BF02885674 |
| [9] | Erdős, L., Dia- and paramagnetism for nonhomogeneous magnetic fields, J. Math. Phys., 38, 1289-1317 (1997) · Zbl 0875.81047 · doi:10.1063/1.531909 |
| [10] | Colin de Verdière, Y., Une formule de traces pour l”opérateur de Schrödinger dans \(\textbf{R}^3\), Ann. Sci. Éc. Normale Super. (4), 14, 27-39 (1981) · Zbl 0482.35068 |
| [11] | Hitrik, M., Existence of resonances in magnetic scattering, J. Comput. Appl. Math., 148, 91-97 (2002) · Zbl 1010.81084 · doi:10.1016/S0377-0427(02)00575-7 |
| [12] | Hitrik, M.; Polterovich, I., Regularized traces and Taylor expansions for the heat semigroup, J. Lond. Math. Soc., 68, 2, 402-418 (2003) · Zbl 1167.35335 · doi:10.1112/S0024610703004538 |
| [13] | Korotyaev, E.; Pushnitski, A., On the high-energy asymptotics of the integrated density of states, Bull. London Math. Soc., 35, 770-776 (2003) · Zbl 1075.35542 · doi:10.1112/S0024609303002467 |
| [14] | Simon, B., Schrödinger operators with singular magnetic vector potentials, Math. Z., 131, 361-370 (1973) · Zbl 0277.47006 · doi:10.1007/BF01174911 |
| [15] | Leinfelder, H.; Simader, C. G., Schrödinger operators with singular magnetic vector potentials, Math. Z., 176, 1-19 (1981) · Zbl 0468.35038 · doi:10.1007/BF01258900 |
| [16] | Simon, B., Functional Integration and Quantum Physics, xiv+306 (2005) · Zbl 1061.28010 |
| [17] | Kac, M., Can one hear the shape of a drum?, Am. Math. Monthly, 73, 1-23 (1966) · Zbl 0139.05603 · doi:10.2307/2313748 |
| [18] | McKean, H. P. Jr.; Singer, I. M., Curvature and the eigenvalues of the Laplacian, J. Diff. Geom., 1, 43-69 (1967) · Zbl 0198.44301 |
| [19] | Yosida, K., On the fundamental solution of the parabolic equation in a Riemannian space, Osaka Math. J., 5, 65-74 (1953) · Zbl 0052.32702 |
| [20] | Grieser, D., “Notes on heat kernel asymptotics” (2004), see . |
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