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
HTML articles powered by AMS MathViewer

by G. B. Khosrovshahi, H. A. Levine and L. E. Payne

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

DOI: https://doi.org/10.1090/S0002-9947-1979-0536943-1

PDF | 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.

References

Shmuel Agmon, Lower bounds for solutions of Schrödinger equations, J. Analyse Math. 23 (1970), 1–25. MR 276624, DOI 10.1007/BF02795485
Shmuel Agmon, Lower bounds for solutions of Schrödinger-type equations in unbounded domains, Proc. Internat. Conf. on Functional Analysis and Related Topics (Tokyo, 1969) Univ. Tokyo Press, Tokyo, 1970, pp. 216–224. MR 0264242

Linear partial differential operators

Tosio Kato, Growth properties of solutions of the reduced wave equation with a variable coefficient, Comm. Pure Appl. Math. 12 (1959), 403–425. MR 108633, DOI 10.1002/cpa.3160120302
Tosio Kato, Some mathematical problems in quantum mechanics, Progr. Theoret. Phys. Suppl. 40 (1967), 3–19. MR 0218080, DOI 10.1143/ptps.40.3
G. B. Khosrovshahi, Nonexistence of nontrivial solutions of Schrödinger type systems, SIAM J. Math. Anal. 8 (1977), no. 6, 998–1013. MR 454314, DOI 10.1137/0508077
Reiji Konno, Non-existence of positive eigenvalues of Schrödinger operators in infinite domains, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 19 (1972), 393–402. MR 336046
Richard Lavine, Absolute continuity of positive spectrum for Schrödinger operators with long-range potentials, J. Functional Analysis 12 (1973), 30–54. MR 0342880, DOI 10.1016/0022-1236(73)90088-8
F. Odeh, Note on differential operators with a purely continuous spectrum, Proc. Amer. Math. Soc. 16 (1965), 363–366. MR 177193, DOI 10.1090/S0002-9939-1965-0177193-6
Barry Simon, On positive eigenvalues of one-body Schrödinger operators, Comm. Pure Appl. Math. 22 (1969), 531–538. MR 247300, DOI 10.1002/cpa.3160220405

Über merkwürdige diskrete Eigenwerte

50

Joachim Weidmann, On the continuous spectrum of Schrödinger operators, Comm. Pure Appl. Math. 19 (1966), 107–110. MR 192186, DOI 10.1002/cpa.3160190108
Joachim Weidmann, The virial theorem and its application to the spectral theory of Schrödinger operators, Bull. Amer. Math. Soc. 73 (1967), 452–456. MR 208197, DOI 10.1090/S0002-9904-1967-11781-6