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The shape resonance. (English) Zbl 0629.47044

For a class of Schrödinger operators \(H-h^ 2\Delta +V\) on \(L^ 2({\mathbb{R}}^ n)\), with potentials having minima embedded in the continuum of the spectrum and non-trapping tails, the authors show the existence of shape resonances exponentially close to the real axis as \(h\to 0\). Resonances are defined by exterior complex scaling. The perturbation of introducing a Dirichlet condition on some boundary sphere is estimated via Krein’s formula. Application of Brillouin-Wigner perturbation theory then leads to a convergent expansion of the resonance energy exhibiting the expected exponentially small behavior for tunneling.

MSC:

47F05 General theory of partial differential operators
81Q15 Perturbation theories for operators and differential equations in quantum theory
47A10 Spectrum, resolvent
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[1] Aguilar, J., Combes, J.M.: A class of analytic perturbations for one-body Schrödinger Hamiltonians. Commun. Math. Phys.22, 269-279 (1971) · Zbl 0219.47011
[2] Agmon, S.: Lectures on exponential decay of second-order elliptic equations: Bounds on eigenfunctions ofN-body Schrödinger operators. Princeton Math. notes29 (1982) · Zbl 0503.35001
[3] Ashbaugh, M.S., Harrell, E.M.: Perturbation theory for shape resonances and high barrier potentials Commun. Math. Phys.83, 151-170 (1982) · Zbl 0494.34044
[4] Baumgartel, H.: A decoupling approach to resonances forN-particle scattering systems. ZIMM preprint, Berlin
[5] Briet, P., Combes, J.M., Duclos, P.: On the location of resonances for Schrödinger operators in the classical limit I: Resonance free domains. Preprint CPT-85/1829. To appear in J. Math. Anal. Appl. · Zbl 0629.47043
[6] 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
[7] Combes, J.M., Duclos, P., Seiler, R.: Convergent expansions for tunneling. Commun. Math. Phys.92, 229-245 (1983) · Zbl 0579.47050
[8] Combes, J.M., Duclos, P., Seiler, R.: On the shape resonance. Lecture Notes in Physics, Vol. 211, pp. 64-77. Berlin, Heidelberg, New York: Springer 1984
[9] Resonances and scattering in the classical limit. Proceedings of ?Methodes semiclassique en mecanique quantique? Luminy 1984, Publ. Univ. Nantes
[10] Shape Resonances at threshold in one dimension (in preparation)
[11] Cycon, H.L.: Resonances defined by modified dilations. Helv. Phys. Act.58, 968-981 (1985)
[12] Dieudonné, J.: Calcul infinitesimal. Paris: Hermann 1968 · Zbl 0155.10001
[13] Faris, W.G.: Selfadjoint operators. Lecture Notes in Mathematics, Vol. 233. Berlin, Heidelberg, New York: Springer 1975 · Zbl 0317.47016
[14] Gamov, G.: Zur Quantentheorie der Atomkerne. Z. Phys.51, 204-212 (1928)
[15] Zur Quantentheorie der Atomzertrümmerung. Z. Phys.52, 510-515 (1929) · JFM 54.0969.05
[16] Gurney, R.W., Condon, E.U.: Quantum mechanics and radioactive disintegration. Phys. Rev.33, 127-132 (1929) · JFM 55.0550.05
[17] Nature122, 439 (1928) · JFM 54.1006.07
[18] Hunziker, W.: Distortion analyticity and molecular resonance curves. To appear in Ann. Inst. Henri Poincaré · Zbl 0619.46068
[19] Helffer, B., Sjöstrand, J.: Multiple wells in the semiclassical limit. I. Commun. P.D.E.9 (4), 337-369 (1984) · Zbl 0546.35053
[20] Helffer, B., Sjöstrand, J.: Puits multiples en limite semi-classique. II. Ann. Inst. Henri Poincaré42, 127-212 (1985) · Zbl 0595.35031
[21] Helffer, B., Sjöstrand, J.: Effet tunnel pour l’operateur de Schrödinger; semiclassique. II. Resonances. To appear in the Proceedings of the Nato Inst. on Micro-Analysis at ?Il Ciorco?, Sept. 1985 (Dortrecht: Reidel)
[22] Resonances en limite semi classique. Prepublication de l’Universite de Nantes. To appear in the Suppl. Bull. Soc. Math. France
[23] Jona-Lasinio, G., Martinelli, F., Scoppola, E.: Decaying quantum-mechanical states: an informal discussion within stochastic mechanics. Lett. Nuovo Cim.34, 13-17 (1982)
[24] Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1966 · Zbl 0148.12601
[25] Klein, M.: On the absence of resonances for Schrödinger operators with non-trapping potentials in the classical limit. Commun. Math. Phys.106, 485-494 (1986) · Zbl 0651.47007
[26] Klein, M.: On the mathematical theory of predissociation. TUB-Preprint Nr. 144 Berlin (1985)
[27] Krein, M.: Über Resolventen hermitescher Operatoren mit Defektindex (m, m). Dokl. Akad. Nauk. SSSR52, 657-660 (1946)
[28] Lavine, R.: Spectral density and sojourn times. Atomic and scattering theory. Ed. J. Nuttall. Univ. of West. Ontario (1978)
[29] Lions, J.L., Magenes, E.: Problemes aux limites non homogenes et applications. Paris: Dunod 1968 · Zbl 0165.10801
[30] Mourre, E.: Absence of singular continuous spectrum for certain selfadjoint operators. Commun. Math. Phys.78, 391-408 (1981) · Zbl 0489.47010
[31] Mourre, E.: Operateurs conjugués et proprieté de propagation. Commun. Math. Phys.91, 279-300 (1983) · Zbl 0543.47041
[32] Majda, A., Ralston, J.: An analogue of Weyl’s theorem for unbounded domains. I. Duke Math. J.45, 183-196 (1978) · Zbl 0408.35069
[33] An analogue of Weil’s theorem for unbounded domains. II. Duke Math. J.45, 513-536 (1978) · Zbl 0416.35058
[34] An analogue of Weil’s theorem for unbounded domains. III. An epilogue. Duke Math. J.46, 725-731 (1979) · Zbl 0433.35055
[35] Orth, A.: Die mathematische Beschreibung von Resonanzen im Vielteilchen-Quantumsystem. Thesis Frankfurt (1985) · Zbl 0628.46072
[36] Reed, M., Simon, B.: Methods of modern mathematical physics. I. New York: Academic Press 1980 · Zbl 0459.46001
[37] Robert, D., Tamura, H.: Semiclassical bounds for resolvents of Schrödinger operators and asymptotics of scattering phase. Commun. P.D.E.9, 1017 (1984) · Zbl 0561.35021
[38] Simon, B.: The definition of molecular resonance curves by the method of exterior complex scaling. Phys. Lett.71 A, 211-214 (1979)
[39] Simon, B.: Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions. Ann. Inst. Henri Poincaré38, 295-307 (1983)
[40] Simon, B.: Semiclassical analysis of low lying eigenvalues. II. Tunneling. Ann. Math.120, 89-118 (1984) · Zbl 0626.35070
[41] Sigal, I.: Complex transformation method and resonances in one-body quantum system. Ann. Inst. Henri Poincaré41, 103-114 (1984) · Zbl 0568.47008
[42] Siedentop, H.K.H.: Bound on resonance eigenvalue of Schrödinger operators-local Birman Schwinger bound. Phys. Lett.99 A, 65 (1983)
[43] Sjöstrand, J.: Tunnel effect for semiclassical Schrödinger operators. Proceedings of the Workshop and Symposium on ?Hyperbolic equations and related topics?. Katada and Kyoto 1984 · Zbl 0669.35083
[44] Neuman, J. von, Wigner, E.P.: Über das Verhalten von Eigenwerten bei adiabatischen Prozessen. Phys. Z.30, 467-470 (1929) · JFM 55.0520.05
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