Hoffmann-Ostenhof, Maria; Hoffmann-Ostenhof, Thomas Local properties of solutions of Schrödinger equations. (English) Zbl 0783.35054 Commun. Partial Differ. Equations 17, No. 3-4, 491-522 (1992). This paper is concerned with the local behaviour of distributional solutions \(u\) to \[ -\Delta u+Vu=0 \tag{1} \] in some open set \(\Omega \subset \mathbb{R}^ n\). For the potential, \(V \in K^{n,\delta}(\Omega)\) is assumed with some \(\delta>0\). The class \(V \in K^{n,\delta}(\Omega)\) consists of those potentials \(V\) for which \[ \lim_{\varepsilon \downarrow 0}\sup_{x\in\Omega} \int_{\{y \in \Omega:| x-y |<\varepsilon\}}V(y) | x-y|^{-n-\delta+2}dy=0 \] holds. The authors show that near some point in \(\Omega\), say 0, any solution \(u \not\equiv 0\) of (1) may be represented as \[ u=P_ M+\Phi \tag{2} \] where \(P_ M\) is a nonzero homogeneous harmonic polynomial of some degree \(M \in \mathbb{N}\), and \(\Phi(x)=O\bigl( | x |^{M+\min\{1;\delta'\}}\bigr)\) as \(x\to 0\) for any \(\delta'<\delta\). Conversely for given \(P_ M\), equation (1) has a solution of the form (2) in some sufficiently small ball. In the same spirit, a representation result is proved for certain inhomogeneous equations. Reviewer: K.J.Witsch (Essen) Cited in 12 Documents MSC: 35Q40 PDEs in connection with quantum mechanics 35B40 Asymptotic behavior of solutions to PDEs 35J10 Schrödinger operator, Schrödinger equation Keywords:asymptotics of solutions at some point; distributional solutions; potentials PDFBibTeX XMLCite \textit{M. Hoffmann-Ostenhof} and \textit{T. Hoffmann-Ostenhof}, Commun. Partial Differ. Equations 17, No. 3--4, 491--522 (1992; Zbl 0783.35054) Full Text: DOI References: [1] DOI: 10.1016/0022-0396(90)90078-4 · Zbl 0778.35109 · doi:10.1016/0022-0396(90)90078-4 [2] DOI: 10.1002/cpa.3160350206 · Zbl 0459.60069 · doi:10.1002/cpa.3160350206 [3] DOI: 10.1080/03605308808820567 · Zbl 0649.35040 · doi:10.1080/03605308808820567 [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/BF01393691 · Zbl 0659.58047 · doi:10.1007/BF01393691 [7] DOI: 10.1016/0022-1236(90)90125-5 · Zbl 0703.35029 · doi:10.1016/0022-1236(90)90125-5 [8] Gilberg D., Elliptic partial differential equations (1983) [9] DOI: 10.1515/crll.1986.370.83 · Zbl 0594.35008 · doi:10.1515/crll.1986.370.83 [10] Hinz A. M., J. reine angw. Math. 404 pp 118– (1990) [11] DOI: 10.1080/03605309908820692 · Zbl 0725.35005 · doi:10.1080/03605309908820692 [12] \(au: \)fn:M. \(sn:Hoffmann--Ostenhof \)eau: \(au: \)fn:T. \(sn:Hoffmann--Ostenhof \)eau: \(title:Generalized cusp conditions for Coulombic systems (in preparation) \)vol:71 [13] Hoffmann–Ostenhof M., Phys. Rev. 16 pp 1782– (1977) · doi:10.1103/PhysRevA.16.1782 [14] Hoffmann–Ostenhof M., Phys. Rev. 23 pp 21– (1981) · doi:10.1103/PhysRevA.23.21 [15] Hardt R., J. Diff. Geometry 30 pp 505– (1989) [16] DOI: 10.1002/cpa.3160100201 · Zbl 0077.20904 · doi:10.1002/cpa.3160100201 [17] DOI: 10.1007/BF02760233 · Zbl 0246.35025 · doi:10.1007/BF02760233 [18] Kenig C., Restriction theorems, Carleman estimates, uniform Sobolev inequalities and unique continution, Lecture Notes in Mathematics 1384 pp 69– (1989) · Zbl 0685.35003 [19] Kinderlehrer D., An introduction to variational inequalities and their application (1980) · Zbl 0457.35001 [20] DOI: 10.1080/03605308708820513 · Zbl 0654.35036 · doi:10.1080/03605308708820513 [21] Rabbiano L., 2 114, in: Bull. Sc. math. pp 329– (1990) [22] Reed M., Methods of modern mathematical physics II, Fourier analysis, self-adjointness (1975) [23] DOI: 10.1090/S0273-0979-1982-15041-8 · Zbl 0524.35002 · doi:10.1090/S0273-0979-1982-15041-8 [24] Sawyer E., Ann. Inst. Fourier (Grenobles) 33 pp 189– (1984) · Zbl 0535.35007 · doi:10.5802/aif.982 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.