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On the local behaviour of nodes of solutions of Schrödinger equations in dimensions \(\geq 3\). (English) Zbl 0725.35005
In 1955 L. Beers showed that any solution \(\psi\) of Schrödinger’s equation \([-\nabla^ 2+V)\psi =0\), which tends to zero at infinity with finite order, must have the same behaviour at infinity as a harmonic polynomial \(P_ M\). In the present paper this result is exploited in order to investigate the nodes of \(\psi\) in a neighborhood of the origin. Suppose the nodal set is known, that is the set of points for which \(\psi =0\) around the origin. Does this set locally coincide with the nodal set of a harmonic polynomial of M-degree? The answer is affirmative in the sense that the difference between the measures of the two nodal sets, intersected with the (n-1)-dimensional sphere of radius r, tends to zero with n. If, moreover, \(n=3\), then the eigenfunctions of Schrödinger’s equation converge to the eigenfunctions of spherical harmonics on the two-dimensional sphere of radius r.
Reviewer: P.Villaggio (Pisa)

35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
Full Text: DOI
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