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Isolated singularities of the solutions of the Schrödinger equation with a radial potential. (English) Zbl 0645.35017

Let \(B_ R\) be the ball with centre 0 and radius \(R\geq 0\) in \({\mathbb{R}}^ N\), \(N\geq 2\), and \(E_ N\) the singularity function of the Laplacian. The following theorem on harmonic functions u in \(\dot B_ R:=B_ R\setminus \{0\}\) goes back to Picard. If \(u+dE_ N\) is bounded from below for some \(d\geq 0\) then there exist a number of \(c\in {\mathbb{R}}\) and a harmonic function h such that \(u=cE_ N+h\) [cf. W. Stożek, Ann. Soc. Pol. Math. 4, 52-58 (1925)]. Results of this genre are given for \(C^ 2(\dot B_ R)\)-solutions of (*) \((-\Delta +V(| \cdot |)u=0\), when \(V\in C^ 0((0,R))\) is a real-valued function. Theorem A assumes in addition that \(VE_ N\in L^ 1(B_ R)\). Theorem C relaxes this to \(V_-E_ N\in L^ 1(B_ R)\) but presupposes that (*) has a singular spherically symmetric solution that is of power-growth. Theorem B characterizes those \(C^ 2(\dot B_ R)\)-solutions of (*) that satisfy \(Vu\in L^ 1_{loc}(B_ R)\) when \(V_+E_ N\not\in L^ 1(B_ R)\) and \(V_-E_ N\in L^ 1(B_ R)\). The main result of the paper is
Theorem D. Suppose \(N\geq 3\) and \(V_-E_ N\in L^ 1({\mathbb{R}}^ N)\). Let \(U_ 1\), \(U_ 2\) be a nonprincipal and a principal solution at zero of the radial equation corresponding to (*). If \(u\in C^ 2({\mathbb{R}}^ N\setminus \{0\})\) is a solution of (*) with \[ u(x)=o(U_ 1(| x|))\text{ as } | x| \to 0 \] and \[ u(x)=o(U_ 2(| x|))\text{ as } | x| \to \infty \] then u is spherically symmetric.
{Reviewer’s remarks. Picard’s theorem was first generalized to solutions of \((-\Delta +V)u=0\) with \(V\geq 0\) though not necessarily spherically symmetric in M. Brelot’s thesis [Ann. Sci. Éc. Norm. Supér., III. Sér. 48, 153-246 (1931; Zbl 0002.25902)]. A result in the vein of but less precise than Theorem A can be found on p. 233. The case \(V=V(r)\) is treated on pp. 203-207. - A paper by P. Hartman and A. Wintner [Rend. Circ. Mat. Palermo, II. Ser. 4, 237-255 (1955; Zbl 0066.345)] is devoted to the case that V behaves like \(r^{-2}\) without necessarily being spherically symmetric. - The authors’ Propositions 2.1 and 4.3 should be compared with Theorems 4 and 5 in G. M. Verzhbinskij and V. G. Maz’ya [Sib. Math. J. 12, 874-899 (1972); translation from Sib. Mat. Zh. 12, 1217-1249 (1971; Zbl 0229.35008)].}
Reviewer: H.Kalf

MSC:

35J10 Schrödinger operator, Schrödinger equation
35A20 Analyticity in context of PDEs
35A30 Geometric theory, characteristics, transformations in context of PDEs
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