×

Weyl asymptotics for magnetic Schrödinger operators and de Gennes’ boundary condition. (English) Zbl 1167.82024

The paper is devoted to consideration of the semi-classical Schrödinger operator with constant magnetic field at the given smooth function \(\gamma\) and number \(\alpha\). This operator arises from the analysis of the onset superconductivity for a superconductor placed adjacent to another material. The usual Neumann condition corresponds to the case of \(\gamma = 0\). The bottom of the essential spectrum of the Schrodinger operator is above \(h\) and the operator has a discrete spectrum below \(h\). The main goal of the paper is to study the asymptotic behavior of the number of eigenvalues of the operator. It is shown, in particular that depending on the type of the boundary condition one can produce many additional eigenvalues below the essential spectrum.
The main results are connected with the proof of the set of Theorems on asymptotic behavior of the above number of eigenvalues for various cases of \(\gamma\) and \(\alpha\). First, the model operator is analyzed in a half-cylinder when \(\gamma\) is constant. With this aim are present the main results concerning a family of differential operators with Robin boundary conditions and an important consequence of the standard Sturm-Liouville theory. Then, the functions on the domain of the model operator are considered to satisfy the periodic conditions and the de Gennes boundary condition at \(t=0\). Further, the model operator is treated on a Dirichlet strip, i.e., the functions on the domain of the model operator satisfy the de Gennes condition at \(t=0\) and Dirichlet condition on the other sides of the boundary. Then, the results obtained for the case in a half-cylinder are extended to a general domain by which the first Theorem is proved by using investigation of the correspondent quadratic forms. A finer approximation of the quadratic forms leads to the analysis of a family of ordinary different operators on a weighted \(L\) in the two power space that is connected with the magnetic Sobolev space and takes into account the curvature effects of the boundary.
With the aim to account of the curvature effects, first the main results for the lowest eigenvalue problem concerning this family of operators are present. Then the consideration of the spectral function for the model operator on the half-cylinder and on the Dirichlet strip allows one to estimate the spectral counting function of the operator in general domains and as a result to prove the remaining theorems.

MSC:

82D55 Statistical mechanics of superconductors
35J10 Schrödinger operator, Schrödinger equation
35Q40 PDEs in connection with quantum mechanics
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35P15 Estimates of eigenvalues in context of PDEs
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bonnaillie V., Asymptot. Anal. 41 pp 215–
[2] DOI: 10.1007/BF01211105 · Zbl 0612.35102 · doi:10.1007/BF01211105
[3] DOI: 10.1006/jdeq.1993.1071 · Zbl 0784.34021 · doi:10.1006/jdeq.1993.1071
[4] DOI: 10.5802/aif.2171 · Zbl 1097.47020 · doi:10.5802/aif.2171
[5] DOI: 10.1112/plms/pdl024 · Zbl 1131.35076 · doi:10.1112/plms/pdl024
[6] de Gennes P. G., Superconductivity of Metals and Alloys (1966) · Zbl 0138.22801
[7] Helffer B., J. Funct. Anal. 181 pp 604–
[8] DOI: 10.1007/978-3-662-12496-3 · doi:10.1007/978-3-662-12496-3
[9] Kachmar A., C. R. Math. Acad. Sci. Paris 332 pp 701–
[10] DOI: 10.1063/1.2218980 · Zbl 1112.81035 · doi:10.1063/1.2218980
[11] Kachmar A., Asymptot. Anal. 55 pp 145–
[12] Kachmar A., Asymptot. Anal. 54 pp 125–
[13] DOI: 10.1016/j.crma.2008.01.018 · Zbl 1138.35087 · doi:10.1016/j.crma.2008.01.018
[14] Kato T., Perturbation Theory for Linear Operators (1995) · Zbl 0836.47009 · doi:10.1007/978-3-642-66282-9
[15] Persson A., Math. Scand. 8 pp 143– · Zbl 0145.14901 · doi:10.7146/math.scand.a-10602
[16] Reed M., Methods of Modern Mathematical Physics VI: Analysis of Operators (1979)
[17] Tamura H., Nagoya Math. J. 105 pp 49– · Zbl 0623.35048 · doi:10.1017/S002776300000074X
[18] Truc F., Asymptot. Anal. 15 pp 385–
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.