## 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

### Citations:

Zbl 0002.25902; Zbl 0066.345; Zbl 0229.35008
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### References:

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