On the positive spectrum of Schrödinger operators with long range potentials

Khosrovshahi, G. B.; Levine, H. A.; Payne, L. E.

doi:10.1090/S0002-9947-1979-0536943-1

On the positive spectrum of Schrödinger operators with long range potentials
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by G. B. Khosrovshahi, H. A. Levine and L. E. Payne PDF

Trans. Amer. Math. Soc. 253 (1979), 211-228 Request permission

Abstract:

In this paper we are concerned with solutions of the equation $\Delta u + p(x)u = 0$ in an unbounded domain $\Omega$ in ${R^n}$, $\Omega \supset \{ x| \left \| x \right \| \geqslant {R_0}\}$. The main result is a determination of conditions on the asymptotic behavior of $p(x)$ sufficient to guarantee that no nontrivial ${L_2}$ solution exists. Our results contain those of previous authors as special cases. The principal application is to the determination of upper bounds for positive eigenvalues of Schrödinger operators.

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