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Eigenfunctions and Hardy inequalities for a magnetic Schrödinger operator in ℝ2

✍ Scribed by Bénédicte Alziary; Jacqueline Fleckinger-Pellé; Peter Takáč

Publisher
John Wiley and Sons
Year
2003
Tongue
English
Weight
349 KB
Volume
26
Category
Article
ISSN
0170-4214

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✦ Synopsis

Abstract

The zero set {z∈ℝ^2^:ψ(z)=0} of an eigenfunction ψ of the Schrödinger operator ℒ︁~V~=(i∇+A)^2^+V on L^2^(ℝ^2^) with an Aharonov–Bohm‐type magnetic potential is investigated. It is shown that, for the first eigenvalue λ~1~ (the ground state energy), the following two statements are equivalent: (I) the magnetic flux through each singular point of the magnetic potential A is a half‐integer; and (II) a suitable eigenfunction ψ associated with λ~1~ (a ground state) may be chosen in such a way that the zero set of ψ is the union of a finite number of nodal lines (curves of class C^2^) which emanate from the singular points of the magnetic potential A and slit the two‐dimensional plane ℝ^2^. As an auxiliary result, a Hardy‐type inequality near the singular points of A is proved. The C^2^ differentiability of nodal lines is obtained from an asymptotic analysis combined with the implicit function theorem. Copyright © 2003 John Wiley & Sons, Ltd.

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