## On the location of resonances for Schrödinger operators in the semiclassical limit. I: Resonances free domains.(English)Zbl 0629.47043

One shows that in the semi-classical regime and under suitable non- trapping conditions on the potential V at energy E of Virial relation type: $$2(V-E)+x\nabla V<0$$ the Schrödinger operator $$H-\hslash^ 2\Delta +V$$ has no resonances in some complex neighbourhood of E.
Some critical values of E like thresholds or barrier tops are also discussed.

### MSC:

 47F05 General theory of partial differential operators 47A10 Spectrum, resolvent 35J10 Schrödinger operator, Schrödinger equation
Full Text:

### References:

 [1] Aguilar, J; Combes, J.M, A class of analytic perturbations for one body Schrödinger Hamiltonians, Comm. math. phys., 22, 269-279, (1971) · Zbl 0219.47011 [2] Balslev, E; Combes, J.M, Spectral properties of many-body Schrödinger operators with dilation analytic potentiels, Comm. math. phys., 22, 280-294, (1971) · Zbl 0219.47005 [3] {\scP. Briet, J. M. Combes, and P. Duclos}, On the location of resonances for Schrödinger operators in the classical limit II, in preparation. · Zbl 0622.47047 [4] Combes, J.M; Duclos, P; Seiler, R, Krein’s formula and one dimensional multiple well, J. funct. anal., 52, 257-301, (1983) · Zbl 0562.47002 [5] Combes, J.M; Duclos, P; Seiler, R, On the shape resonance, (), 64 · Zbl 0629.47044 [6] Combes, J.M; Duclos, P; Seiler, R, Resonances and scattering in the classical limit, () · Zbl 0562.47002 [7] {\scJ. M. Combes, P. Duclos, M. Klein, and R. Seiler}, The spahe resonance, CPT 85/1797, Marseille, preprint. · Zbl 0629.47044 [8] Erdmann, C; Cycon, H.L, (), preprint [9] {\scS. Graffi and K. Yajima}, Exterior scaling and the AC-Stark effect in a · Zbl 0522.35085 [10] Helffer, B; Sjöstrand, J, Effet tunnel pour l’opérateur de Schrödinger semiclassique, II, resonances, (), to appear · Zbl 0607.35027 [11] Helffer, B; Sjöstrand, J, Resonances en limite semi classique, (1985), prepublication de l’université de Nantes · Zbl 0595.35031 [12] Harrel, E.M; Svirsky, R, Potentials producing maximaly sharp resonances, (1984), preprint [13] Ikawa, M, On the poles of the scattering matrix, () · Zbl 0746.35027 [14] Kato, T, Pertubation theory for linear operators, (1966), Springer-Verlag Berlin/Heidelberg/New York [15] Klein, M, On the absence of resonances for Schrödinger operators with non trapping potentials in the classical limit, Cpt-85/p, (1798), Marseille, preprint [16] Mourre, E, Absence of singular continuous spectrum for certain selfadjoint operators, Comm. math. phys., 78, 391-408, (1981) · Zbl 0489.47010 [17] Nussenszveig, H.M, Causality and dispersion relation, (1972), Academic Press New York [18] Reed, M; Simon, B, Methods of modern mathematical physics. III. scattering theory, (1979), Academic Press New York · Zbl 0405.47007 [19] Reed, M; Simon, B, Methods of modern mathematical physics. IV. analysis of operators, (1978), Academic Press New York · Zbl 0401.47001 [20] Robert, D; Tamura, H, Semiclassical bounds for resolvents of Schrödinger operators and asymptotics of scattering phase, Comm. partial differential equations, 9, 1017, (1984) · Zbl 0561.35021 [21] Simon, B, The definition of molecular resonances curves by the methode of exterior complex scaling, Phys. lett. A, 71, 211-214, (1979) [22] Simon, B, Semi classical analysis of low lying eigenvalues. I. nondegenerate minima: asymptotic expensions, Ann. inst. H. Poincaré, 38, 295-307, (1983)
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.