×

A note on the absence of resonances for Schrödinger operators. (English) Zbl 0688.35073

It is well-known that the Schrödinger operator \(H:=-\Delta +V\) in \(L^ 2({\mathbb{R}}^ n)\) has no eigenvalues larger than a given number \(\Lambda\) if \[ (*)\quad 2(V(x)-\Lambda)+x\cdot (\nabla V)(x)\leq 0\quad (x\in {\mathbb{R}}^ n) \] [cf., for example, M. S. P. Eastham and the reviewer, Schrödinger-type operators with continuous spectra (1982; Zbl 0491.35003)], and this condition admits of an interpretation in terms of notions from classical mechanics. M. Klein [Commun. Math. Phys. 106, 485-494 (1986; Zbl 0651.47007)] showed that an assumption similar to (*) can be used to rule out resonances of H in a certain energy interval. In the present paper Klein’s result is generalized by introducing a further modification of (*). To obtain the required resolvent estimate, a special case of Hörmander’s construction of a parametrix for elliptic pseudodifferential operators is used. As an additional reference to the circle of questions involved we mention a paper by H. L. Cycon and C. Erdmann [Rep. Math. Phys. 23, 169-176 (1986; Zbl 0646.35017)].
Reviewer: H.Kalf

MSC:

35P99 Spectral theory and eigenvalue problems for partial differential equations
35J10 Schrödinger operator, Schrödinger equation
47B25 Linear symmetric and selfadjoint operators (unbounded)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] AsadaK. and FujiwaraD., On some oscillatory integral transformations in L 2(R n ), Japan J. Math. 4, 299-361 (1978).
[2] AvronJ. E. and HerbstI., Spectral and scattering theory of Schrödinger operators related to the Stark effect, Commun. Math. Phys. 52, 239-254 (1977). · Zbl 0351.47007
[3] BrietPh., CombesJ. M., DuclosP., On the location of resonances for Schrödinger operators in the semiclassical limit. I. Resonance free domains, J. Math. Anal. Appl. 126, 90-99 (1098). · Zbl 0629.47043
[4] Briet, Ph., Combes, J. M., and Duclos, P., Spectral properties of Schrödinger operators with trapping potentials in the semi-classical limit, in L. W. Knowles and Y. Saito (eds.), Differential Equations and Mathematical Physics, Springer Lecture Notes in Math. 1285, 55-72 (1987). · Zbl 0654.47032
[5] CombesJ. M., DuclosP., KleinM., and SeilerR., The shape resonance, Commun. Math. Phys. 110, 215-236 (1987). · Zbl 0629.47044
[6] Helffer, B. and Martinez, A., Comparaison entre les diverses notions de resonances, preprint (1987).
[7] HelfferB. and RobertD., Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles, J. Funct. Anal. 53, 246-268 (1983). · Zbl 0524.35103
[8] HelfferB. and SjöstrandJ., Résonances en limite semi-classique, Bull. Société Math. France. Mémoire 24/25, 1-228 (1986).
[9] HerbstI., Exponential decay in the Stark effect, Commun. Math. Phys. 75, 197-205 (1980). · Zbl 0482.35034
[10] Hörmander, L., The Analysis of Partial Differential Operators, Vol. 3, Springer-Verlag, 1985. · Zbl 0601.35001
[11] HunzikerW., Distortion analyticity and molecular resonance curves, Ann. Inst. Henri Poincaré 45, 339-358 (1986).
[12] KleinM., On the absence of resonances for Schrödinger operators in the classical limit, Commun. Math. Phys. 106, 485-494 (1986). · Zbl 0651.47007
[13] Nakamura, S., Scattering theory for the shape resonance model, preprint (1987).
[14] RobertD. and TamuraH., Semi-classical bounds for the resolvents of Schrödinger operators and asymptotics for scattering phases, Commun. Partial Differential Equations 9, 1017-1058 (1984). · Zbl 0561.35021
[15] SimonB., The definition of molecular resonance curves by the method of exterior complex scaling, Phys. Lett. A71, 211-214 (1979).
[16] Taylor, M., Pseudo-differential Operators, Princeton Univ. Press, 1981. · Zbl 0482.34021
[17] Hislop, P. D. and Sigal, I. M., Shape resonance in quantum mechanics, in I. W. Knowles and Y. Saito (eds.), Differential Equations and Mathematical Physics, Springer Lecture Notes in Math. 1285, 180-196 (1987). · Zbl 0653.46074
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.