an:04196504
Zbl 0725.35005
Hoffman-Ostenhof, M.; Hoffman-Ostenhof, T.
On the local behaviour of nodes of solutions of Schr??dinger equations in dimensions \(\geq 3\)
EN
Commun. Partial Differ. Equations 15, No. 4, 435-451 (1990).
00175152
1990
j
35B05 35J10
nodal set; harmonic polynomial
In 1955 L. Beers showed that any solution \(\psi\) of Schr??dinger'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 Schr??dinger's equation converge to the eigenfunctions of spherical harmonics on the two-dimensional sphere of radius r.
P.Villaggio (Pisa)