Tunneling effect in two dimensions with vanishing magnetic fields

In this paper, we consider the semiclassical 2D magnetic Schrödinger operator in the case where the magnetic field vanishes along a smooth closed curve. Assuming that this curve has an axis of symmetry, we prove that semiclassical tunneling occurs. The main result is an expression of the splitting o...

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Bibliographic Details
Published inJournal of spectral theory Vol. 14; no. 3; pp. 837 - 889
Main Author Abou Alfa, Khaled
Format Journal Article
LanguageEnglish
Published European Mathematical Society 14.08.2024
Subjects
Mathematics
Agmon estimates
eikonal equation
magnetic Laplacian
pseudo-differential operator
tunnel effect
Schrödinger equation
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ISSN1664-039X
1664-0403
DOI10.4171/jst/516

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Summary:In this paper, we consider the semiclassical 2D magnetic Schrödinger operator in the case where the magnetic field vanishes along a smooth closed curve. Assuming that this curve has an axis of symmetry, we prove that semiclassical tunneling occurs. The main result is an expression of the splitting of the first two eigenvalues and an explicit tunneling formula.
ISSN:1664-039X
1664-0403
DOI:10.4171/jst/516