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Asymptotic behavior of solutions of \(-\Delta v+qv=\lambda v\) and the distance of \(\lambda\) to the essential spectrum. (English) Zbl 0622.35053

The main results of the paper are upper bounds for the distance of \(\lambda\in {\mathbb{R}}\) to the essential spectrum of an operator \(\overline{-\Delta +q},\) depending on growth or decay properties of distributional solutions v for the equation \(-\Delta v+qv=\lambda v\). From these bounds it is derived that if \(q(x) \geq -o(| x|^ 2)\) outside a ball, then the existence of a bounded solution guarantees \(\lambda \in \sigma (\overline{-\Delta +q}),\) and every eigenfunction to a discrete eigenvalue decays exponentially. Furthermore, an example of an eigenvalue in the essential spectrum and lower bounds for \(non\)-L\({}_ 2\)-solutions are given. \(q(x) \sim -O(| x|^ 2)\) appears as a borderline case, and the question is raised if, in that case, there may be a \(\lambda\) outside the spectrum of \(\overline{-\Delta +q}\) with a bounded solution v of \(-\Delta v+qv=\lambda v.\)

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
35B40 Asymptotic behavior of solutions to PDEs
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References:

[1] Adams, R.A.: Sobolev spaces. New York-San Francisco-London: Academic Press 1975 · Zbl 0314.46030
[2] Eastham, M.S.P., Kalf, H.: Schrödinger-type operators with continuous spectra. Boston-London-Melbourne: Pitman 1982 · Zbl 0491.35003
[3] Glazman, I.M.: Direct methods of qualitative spectral analysis of singular differential operators. Jerusalem: Israel Program for Scientific Translations 1965 · Zbl 0143.36505
[4] Hinz, A.M.: Obere Schranken für Eigenfunktionen eines Operators ??+q. Math. Z.185, 291-304 (1984) · Zbl 0528.35009 · doi:10.1007/BF01181700
[5] Hinz, A.M.: Pointwise bounds for solutions of the equation ??v+pv=0. J. Reine Angew. Math.370, 83-100 (1986) · Zbl 0594.35008 · doi:10.1515/crll.1986.370.83
[6] Simader, C.G.: Bemerkungen über Schrödinger-Operatoren mit stark singulären Potentialen. Math. Z.138, 53-70 (1974) · Zbl 0317.35028 · doi:10.1007/BF01221884
[7] Simon, B.: Spectrum and continuum eigenfunctions of Schrödinger operators. J. Funct. Anal.42, 347-355 (1981) · Zbl 0471.47028 · doi:10.1016/0022-1236(81)90094-X
[8] Sleeman, B.D., Michael, I.M. (ed.): Ordinary and partial differential equations. Lecture Notes in Mathematics415. Berlin, Heidelberg, New York: Springer 1974
[9] Weidmann, J.: Linear operators in Hilbert spaces. Berlin, Heidelberg, New York: Springer 1980 · Zbl 0434.47001
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