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Eigenfunctions and Hardy inequalities for a magnetic Schrödinger operator in ℝ2

The zero set {z∈ℝ2:ψ(z)=0} of an eigenfunction ψ of the Schrödinger operator ℒ︁V=(i∇+A)2+V on L2(ℝ2) with an Aharonov–Bohm‐type magnetic potential is investigated. It is shown that, for the first eigenvalue λ1 (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 λ1 (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 C2) which emanate from the singular points of the magnetic potential A and slit the two‐dimensional plane ℝ2. As an auxiliary result, a Hardy‐type inequality near the singular points of A is proved. The C2 differentiability of nodal lines is obtained from an asymptotic analysis combined with the implicit function theorem. Copyright © 2003 John Wiley & Sons, Ltd.