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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)

MSC:
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
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[1] DOI: 10.1007/BF00251498 · Zbl 0664.35016 · doi:10.1007/BF00251498
[2] Aronszajn N., J. Math. Pures Appl. 36 pp 235– (1957)
[3] Berard P., Annal. Sci. Ec. Norm. Super. 15 pp 513– (1982)
[4] DOI: 10.1002/cpa.3160080404 · Zbl 0066.08101 · doi:10.1002/cpa.3160080404
[5] DOI: 10.1016/0022-0396(85)90133-0 · Zbl 0593.35047 · doi:10.1016/0022-0396(85)90133-0
[6] DOI: 10.1007/BF02568142 · Zbl 0334.35022 · doi:10.1007/BF02568142
[7] DOI: 10.1007/BF01393691 · Zbl 0659.58047 · doi:10.1007/BF01393691
[8] Donnelly H., Nodal sets of eigenfunctions on Riemannian manifolds with boundar 93 (1988)
[9] Hardt R., Nodal sets for solutions of elliptic equations 93 (1988) · Zbl 0692.35005
[10] DOI: 10.1007/BF01228411 · Zbl 0658.35021 · doi:10.1007/BF01228411
[11] DOI: 10.1007/BF01163288 · Zbl 0627.35024 · doi:10.1007/BF01163288
[12] Hoffmann–ostenhof M., Proceedings of the Congerence on Partia Differential Equations in Holzhau (GDR) Teubner Texte zur mathematik, in press 198 (1988)
[13] DOI: 10.1080/03605307708820059 · Zbl 0377.31008 · doi:10.1080/03605307708820059
[14] DOI: 10.1080/03605308708820513 · Zbl 0654.35036 · doi:10.1080/03605308708820513
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