Abstract
Resonances which appear as perturbed bound states are discussed in the framework of Balslev-Combes theory. The corresponding metastable states are constructed using the formal perturbation expansion to orderN−1 for the (nonexistent) perturbed bound states. They are shown to have exponential decay in time governed by the complex resonance energies, up to a background of order 2N in the perturbation parameter. The results apply in lowest orderN=1 to the perturbation of bound states embedded in the continuum and in arbitrary order to cases like the Stark effect.
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References
Balslev, E., Combes, J.M.: Spectral properties of Schrödinger operators with dilation analytic interactions. Commun. Math. Phys22, 280–294 (1971)
Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger operators. Berlin, Heidelberg, New York: Springer 1987
Cycon, H.L.: Resonances defined by modified dilations. Helv. Phys. Acta58, 969–981 (1985)
Herbst, I.: Exponential decay in the Stark effect. Commun. Math. Phys.87, 429–447 (1982/83)
Herbst, I., Simon, B.: Dilation analyticity in constant electric field. II. The N-body problem, Borel summability. Commun. Math. Phys.80, 181–216 (1981)
Hunziker, W., Pillet, C.-A.: Degenerate asymptotic perturbation theory. Commun. Math. Phys.90, 219–233 (1983)
Hunziker, W.: Distortion analyticity and molecular resonance curves. Ann. Inst. Henri Poincaré45, 339–358 (1986)
Hunziker, W.: Notes on asymptotic perturbation theory for Schrödinger eigenvalue problems. Helv. Phys. Acta61, 257–304 (1988)
Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1966
Orth, A.: Quantum mechanical resonance and limiting absorption: the many body problem. Commun. Math. Phys.126, 559–573 (1990)
Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. New York: Academic Press 1978
Simon, B.: Resonances inN-body quantum systems with dilation analytic potentials and the foundations of time-dependent perturbation theory. Ann. Math.97, 247–274 (1973)
Simon, B.: Resonances and complex scaling: a rigorous overview. Int. J. Quant. Chem.14, 529–542 (1978)
Simon, B.: The definition of molecular resonance curves by the method of exterior complex scaling. Phys. Lett.71A, 211–214 (1979)
Sigal, I.M.: Complex transformation method and resonances in one-body quantum systems. Ann. Inst. Henri Poincaré41, 103–114 (1984)
Skibsted, E.: Truncated Gamov functions, α-decay and the exponential law Commun. Math. Phys.104, 591–604 (1986)
Skibsted, E.: On the evolution of resonance states. J. Math. Anal. Appl. (to appear)
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Communicated by A. Jaffe
Dedicated to Res Jost and Arthur Wightman
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Hunziker, W. Resonances, metastable states and exponential decay laws in perturbation theory. Commun.Math. Phys. 132, 177–188 (1990). https://doi.org/10.1007/BF02278006
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DOI: https://doi.org/10.1007/BF02278006