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Distortion analyticity for two-body Schrödinger operators. (English) Zbl 0736.35026

A complex distortion which is local in momentum is studied for a Schrödinger operator \(-\Delta+V\). A class of distortion analytic \(V\)’s is given which includes analytic potentials and exponentially decaying potentials.
The method enables to study semiclassical resonances for \(-\hbar^ 2\Delta+V\), \(V\) exponentially decaying [see the author, Commun. Partial Differ. Equations 14, 1385-1419 (1989; Zbl 0699.35204)].
Reviewer: J.Asch (Berlin)

MSC:

35J10 Schrödinger operator, Schrödinger equation
35Q40 PDEs in connection with quantum mechanics

Citations:

Zbl 0699.35204
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References:

[1] F. Aguilar and J.M. Combes , On a Class of Analytic Perturbations of One-Body Schrödinger Operators , Commun. Math. Phys. , 1971 , Vol. 22 , pp. 269 - 279 . Article | MR 345551 | Zbl 0219.47011 · Zbl 0219.47011 · doi:10.1007/BF01877510
[2] D. Babbit and E. Balslev , Local Distortion Techniques and Unitarity of the S-Matrix for 2-Body Problem , J. Funct. Anal. , 1975 , Vol. 18 , pp. 1 - 14 . MR 413860
[3] E. Balslev , Analyticity Properties of Eigenfunctions and Scattering Matrix , Commun. Math. Phys. , 1988 , Vol. 114 , pp. 599 - 612 . Article | MR 929132 | Zbl 0662.35079 · Zbl 0662.35079 · doi:10.1007/BF01229457
[4] E. Balslev and J.M. Combes , Spectral Properties of Many-Body Schrödinger Operators with Dilation-Analytic Interactions , Commun. Math. Phys. , 1971 , Vol. 22 , pp. 280 - 299 . Article | MR 345552 | Zbl 0219.47005 · Zbl 0219.47005 · doi:10.1007/BF01877511
[5] E. Balslev and E. Skibsted , Resonance Theory for Two-Body Schrödinger Operators (to appear). Numdam | MR 1033614 · Zbl 0714.35063
[6] J.M. Combes , P. Duclos , M. Klein and R. Seiler , The Shape Resonance , Commun. Math. Phys. , 1987 , Vol. 110 , pp. 215 - 236 . Article | MR 887996 | Zbl 0629.47044 · Zbl 0629.47044 · doi:10.1007/BF01207364
[7] H.L. Cycon , Resonances Defined by Modified Dilations , Helv. Phys. Acta , 1989 , Vol. 58 , pp. 969 - 987 . MR 821113
[8] H.L. Cycon , R.G. Froese , W. Kirsch and B. Simon , Schrödinger Operators, with Applications to Quantum Mechanics and Global Geometry , Berlin , Springer-Verlag , 1987 . MR 883643 | Zbl 0619.47005 · Zbl 0619.47005
[9] B. Helffer and J. Sjöstrand , Resonances en limite semi-classique , Suppl. Bull. Soc. Math. France , 1986 , Vol. 114 . Numdam | Zbl 0631.35075 · Zbl 0631.35075
[10] W. Hunziker , Distortion Analyticity and Molecular Resonance Curves , Ann. Inst. Henri Poincaré , 1986 , Vol. 45 , pp. 339 - 358 . Numdam | MR 880742 | Zbl 0619.46068 · Zbl 0619.46068
[11] K. Jorgens , Linear Integral Operators , London , Pitman , 1982 . MR 647629 | Zbl 0499.47029 · Zbl 0499.47029
[12] S. Nakamura , Shape Resonances for Distortion Analytic Schrödinger Operators, Commun. P.D.E. , Vol. 14 , 1989 , pp. 1385 - 1419 . MR 1022991 | Zbl 0699.35204 · Zbl 0699.35204 · doi:10.1080/03605308908820659
[13] M. Reed and B. Simon , Methods of Modern Mathematical Physics , Vols. I - IV , New York , Academic Press , 1972 - 1979 . MR 493419 · Zbl 0242.46001
[14] I.M. Sigal , Complex Transformation Method and Resonances in One-Body Quantum Systems , Ann. Inst. Henri Poincaré , 1984 , Vol. 41 , pp. 103 - 114 . Numdam | MR 760129 | Zbl 0568.47008 · Zbl 0568.47008
[15] B. Simon , The Definition of Molecular Resonance Curves by the Method of Exterior Complex Scalong , Phys. Lett. , 1979 , Vol. 71A , pp. 211 - 214 .
[16] M. Taylor , Pseudodifferential Operators , New Jersey , Princeton Univ. Press , 1981 . Zbl 0453.47026 · Zbl 0453.47026
[17] P. Hislop and I.M. Sigal , Shape Resonances in Quantum Mechanics, Differential Equation and Mathematical Physics , I. KNOWLES and Y. SAITO Eds., Lect. Notes Math. , 1987 , Vol. 1285 . MR 921268 | Zbl 0653.46074 · Zbl 0653.46074
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