Semiclassical and weak-magnetic-field eigenvalue asymptotics for the Schrödinger operator with electromagnetic potential. (English) Zbl 0812.35113
Summary: We consider the discrete spectrum of the Schrödinger operator \({\mathfrak H}_{h,\mu} :=( ih\nabla+ \mu A)^ 2-V\) where \(A\) is the magnetic potential, \(-V\) is the electric potential, \(h\) is the Planck constant, and \(\mu\) is the magnetic-field coupling constant. We study the asymptotic behaviour of the number of the eigenvalues of \({\mathfrak H}_{h,\mu}\) smaller than \(\lambda\leq 0\) as \(h\downarrow 0\), \(\mu>0\) being fixed, or \(\mu\downarrow 0\), \(h>0\) being fixed.
MSC:
| 35Q40 | PDEs in connection with quantum mechanics |
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 35J10 | Schrödinger operator, Schrödinger equation |
Keywords:
discrete spectrum of the Schrödinger operator; asymptotic behaviour of the number of the eigenvaluesReferences:
| [1] | A.B. Alekseev , On the spectral asymptotics of differential operators with polynomial occurrence of a small parameter , Probl. Mat. Fiz. , Vol. 8 , 1976 , pp. 3 - 15 (in Russian). MR 510062 |
| [2] | J. Avron , I. Herbst and B. Simon , Schrödinger operators with magnetic fields. I. General interactions , Duke Math. J. , Vol. 45 , 1978 , pp. 847 - 883 . Article | MR 518109 | Zbl 0399.35029 · Zbl 0399.35029 · doi:10.1215/S0012-7094-78-04540-4 |
| [3] | J. Avron , I. Herbst and B. Simon , Schrödinger operators with magnetic fields. III. Atoms in homogeneous magnetic field , Commun. Math. Phys. , Vol. 79 , 1981 , pp. 529 - 572 . Article | MR 623966 | Zbl 0464.35086 · Zbl 0464.35086 · doi:10.1007/BF01209311 |
| [4] | M.S. Birman , On the spectrum of singular boundary value problems , AMS Translations , ( 2 ), Vol. 53 , 1966 , pp. 23 - 80 . Zbl 0174.42502 · Zbl 0174.42502 |
| [5] | M.S. Birman and M.Z. Solomjak , Estimates of the singular numbers of integral operators , Russian Math. Surveys , Vol. 32 , 1977 , pp. 15 - 89 . MR 438186 | Zbl 0376.47023 · Zbl 0376.47023 · doi:10.1070/RM1977v032n01ABEH001592 |
| [6] | M.S. Birman and M.Z. Solomjak , Quantitative analysis in Sobolev imbedding theorems and applications to spectral theory , AMS Translations ( 2 ), Vol. 114 , 1980 . · Zbl 0426.46020 |
| [7] | M.S. Birman and M.Z. Solomjak , Spectral Theory of Selfadjoint Operators in Hilbert Space , D. REIDEL , Dordrecht , 1987 . · Zbl 0744.47017 |
| [8] | Y. Colin De Verdière , L’asymptotique de Weyl pour les bouteilles magnétiques , Comm. Math. Phys. , Vol. 105 , 1986 , pp. 327 - 335 . Article | MR 849211 | Zbl 0612.35102 · Zbl 0612.35102 · doi:10.1007/BF01211105 |
| [9] | J.M. Combes , R. Schrader and R. Seiler , Classical bounds and limits for energy distributions of Hamiltonian operators in electromagnetic fields , Ann. Physics , Vol. 111 , 1978 , pp. 1 - 18 . MR 489509 |
| [10] | H.L. Cycon , R.G. Froese , W. Kirsch and B. Simon , Schrödinger operators with Application to Quantum Mechanics and Global Geometry . Springer , Berlin , 1987 . MR 883643 | Zbl 0619.47005 · Zbl 0619.47005 |
| [11] | M. Dauge and D. Robert , Weyl’s formula for a class of pseudodifferential operators of negative order on L2 (Rn) , Lect. Notes Mah. , Vol. 1256 , 1987 , pp. 91 - 122 . MR 897775 | Zbl 0629.35098 · Zbl 0629.35098 |
| [12] | U. Grenander and G. Szegö , Toeplitz Forms and Their Applications , University of California Press , Berkeley - Los Angeles , 1958 . MR 94840 | Zbl 0080.09501 · Zbl 0080.09501 |
| [13] | B. Helffer and J. Sjöstrand , Équation de Schrödinger avec champ magnétique et équation de Harper, II. Approximation champ magnétique faible: substitution de Peierls . In: Schrödinger operators , Lect. Notes Phys. , Vol. 345 , 1988 , pp. 138 - 191 . · Zbl 0699.35189 |
| [14] | B. Helffer and J. Sjöstrand , On the link between the Schrödinger operator with magnetic field and Harper’s equation , In: Proceedings of Symp. Part. Diff. Equ., HOLZHAU, 1988 . Teubner-Texte zur Mathematik , Vol. 112 , 1989 , pp. 139 - 156 . Zbl 0684.35074 · Zbl 0684.35074 |
| [15] | V.Ya. Ivrii , Estimations pour le nombre de valeurs propres négatives de l’opérateur de Schrödinger avec potentiels singuliers , C. R. Acad. Sci. Paris , Sér. I, Vol. 302 , 1986 , pp. 467 - 470 . MR 838401 | Zbl 0611.35064 · Zbl 0611.35064 |
| [16] | V.Ya. Ivrii , Estimates of the number of negative eigenvalues of the Schrödinger operator with a strong magnetic field , Soviet Math. Dokl. , Vol. 36 , 1988 , pp. 561 - 564 . MR 936071 | Zbl 0661.35066 · Zbl 0661.35066 |
| [17] | V.Ya. Ivrii , Sharp spectral asymptotics for the two-dimensional Schrödinger operator with a strong magnetic field , Soviet Math. Dokl. , Vol. 39 , 1989 , pp. 437 - 441 . MR 1006532 | Zbl 0714.35060 · Zbl 0714.35060 |
| [18] | V.Ya. Ivrii , Semiclassical microlocal analysis and precise spectral asymptotics , Preprints 1-9, École Polytechnique , 1990 - 1992 . |
| [19] | H. Leinfelder , Gauge invariance of Schrödinger operators and related spectral properties , J. Opt. Theory , Vol. 9 , 1983 , pp. 163 - 179 . MR 695945 | Zbl 0528.35024 · Zbl 0528.35024 |
| [20] | G.D. Raikov , Spectral asymptotics for the Schrödinger operator with potential which steadies at infinity , Comm. Math. Phys. , Vol. 124 , 1989 , pp. 665 - 685 . Article | MR 1014119 | Zbl 0704.35110 · Zbl 0704.35110 · doi:10.1007/BF01218455 |
| [21] | G.D. Raikov , Eigenvalue asymptotics for the Schrödinger operator with homogeneous magnetic potential and decreasing electric potential. I. Behaviour near the essential spectrum tips , Comm. P.D.E. , Vol. 15 , 1990 , pp. 407 - 434 . MR 1044429 | Zbl 0739.35055 · Zbl 0739.35055 · doi:10.1080/03605309908820690 |
| [22] | G.D. Raikov , Strong electric field eigenvalue asymptotics for the Schrödinger operator with electromagnetic potential , Letters Math. Phys. , Vol. 21 , 1991 , pp. 41 - 49 . MR 1088409 | Zbl 0727.35102 · Zbl 0727.35102 · doi:10.1007/BF00414634 |
| [23] | G.D. Raikov , Semiclassical and weak-magnetic-field eigenvalue asymptotics for the Schrödinger operator with electromagnetic potential , C. R. Acad. Sci. Bulg. , Vol. 44 , 1991 , No. 1 , pp. 15 - 18 . MR 1117317 | Zbl 0739.35050 · Zbl 0739.35050 |
| [24] | M. Reed and B. Simon , Methods of Modem Mathematical Physics. IV. Analysis of Operators , Academic Press , London , 1978 . Zbl 0401.47001 · Zbl 0401.47001 |
| [25] | G.V. Rozenbljum , An asymptotic of the negative discrete spectrum of the Schrödinger operator , Math. Notes U.S.S.R. , Vol. 21 , 1977 , pp. 222 - 227 . MR 447838 | Zbl 0399.35083 · Zbl 0399.35083 · doi:10.1007/BF01106748 |
| [26] | B. Simon , Functional Integration and Quantum Physics , Academic Press , London , 1979 . MR 544188 | Zbl 0434.28013 · Zbl 0434.28013 |
| [27] | H. Tamura , Asymptotic distribution of eigenvalues for Schrödinger operators with homogeneous magnetic fields , Osaka J. Math. , Vol. 25 , 1988 , pp. 633 - 647 . MR 969024 | Zbl 0731.35073 · Zbl 0731.35073 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
