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


35J10 Schrödinger operator, Schrödinger equation
35A20 Analyticity in context of PDEs
35A30 Geometric theory, characteristics, transformations in context of PDEs
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[1] M. Aizenman & B. Simon, Brownian Motion and Harnack Inequality for Schrödinger Operators, Comm. Pure Applied Math. 35 (1982), 209–273. · Zbl 0459.60069
[2] P. Baras & J. A. Goldstein, Remarks on the inverse square potential in Quantum Mechanics, in Differential Equations, I. W. Knowles & R. T. Lewis eds., North-Holland, 1984, 31–35. · Zbl 0566.35035
[3] R. Bellman, Stability Theory of Differential Equations, Dover Publications Inc., New York, 1953. · Zbl 0053.24705
[4] H. Brezis & P. L. Lions, A note on isolated singularities for Linear elliptic equations, Mathematical Analysis and Applications, A. L. Nachbin ed. Academic Press, New York, 1981, 263–266.
[5] H. Brezis & L. Véron, Removable Singularities of some Nonlinear Elliptic Equations, Arch. Rational Mech. Anal. 75 (1980), 1–6. · Zbl 0459.35032
[6] L. A. Caffarelli & W. Littman, Representation formulas for solutions to {\(\Delta\)}u-u = 0 in \(\mathbb{R}\)n, Studies in Partial Differential Equations, Math. Assoc. America Studies in Maths. # 23 (1982), 249–263.
[7] N. Garofalo & F. H. Lin, Monotonicity properties of
[8] D. Gilbarg & J. Serrin, On Isolated Singularities of Solutions of Second Order Elliptic Differential Equations, J. d’Analyse Math. 4 (1955–56), 309–340. · Zbl 0071.09701
[9] D. Gilbarg& N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1977. · Zbl 0361.35003
[10] T. Kato, Schrödinger Operators with Singular Potentials, Israel J. Math. 13 (1972), 135–148. · Zbl 0246.35025
[11] D. Jerison & C. E. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Annals of Mathematics 121 (1985), 483–494. · Zbl 0593.35119
[12] G. Dal Maso & U. Mosco, The Wiener modulus of a radial measure, IMA Preprint Series No. 194, University of Minnesota.
[13] M. Murata, Positive solutions of Schrödinger equations, IMA Preprint Series # 124, University of Minnesota.
[14] M. Naito, Asymptotic behavior of solutions of second order differential equations with integrable coefficients, Transactions Amer. Math. Soc. 282 (1984), 577–588. · Zbl 0556.34055
[15] M. Reed & B. Simon, Methods of Modern Mathematical Physics vol. II Fourier Analysis Self-Adjointness, and vol. IV, Analysis of Operators, Ac. Press, New York, 1975 and 1978. · Zbl 0308.47002
[16] J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Mathematica 111 (1964), 247–302. · Zbl 0128.09101
[17] J. Serrin, Isolated singularities of solutions of elliptic equations, Acta Mathematica 113 (1965), 219–240. · Zbl 0173.39202
[18] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 447–522. · Zbl 0524.35002
[19] E. M. Stein, Appendix to ”Unique continuation” ([JK] above). Annals of Mathematics 121 (1985), 489–494.
[20] J. Serrin & H. F. Weinberger, Isolated singularities of solutions of linear equations, Amer. J. math. 88 (1966), 258–272. · Zbl 0137.07001
[21] J. L. Vázquez & L. Véron, Isolated singularities of some semilinear elliptic equations, J. Diff. Equations, 60 (1985), 301–321. · Zbl 0549.35043
[22] J. L. Vázquez & C. Yarur, Singularités isolées et comportement à l’infini des solutions de l’équation de Schrödinger stationnaire, C. Rendus Acad. Sci. Paris I, 300 (1985), 105–108.
[23] L. Véron, Singular Solutions of Some Nonlinear Elliptic Equations, Nonlinear Analysis 5 (1981), 225–242. · Zbl 0457.35031
[24] D. Willet, Classification of Second Order Linear Differential Equations with Respect to Oscillations, Advances in Mathematics 3 (1969), 594–623. · Zbl 0188.40101
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