Abstract
We study the spectral properties of multiple well Schrödinger operators on ℝn. We give in particular upper bounds on energy shifts due to tunnel effect and localization properties of wave packets. Our methods are based on Agmon type estimates for resolvents in classically forbidden regions and geometric perturbation theory. Our results are valid also for an infinite number of wells, arbitrary spectral type and in non-semi-classical regimes.
Similar content being viewed by others
References
[Ad] Adams, R.A.: Sobolev spaces. New York: Academic Press 1975
[Ag] Agmon, S.: Lectures on exponential decay of solutions of second order elliptic operators. Princeton, NJ: Princeton University Press 1982
[AsHa] Asbaugh, M., Harrell, E.: Perturbation theory for shape resonances and large barrier potentials. Commun. Math. Phys.83, 151 (1982)
[BCD1] Briet, Ph., Combes, J.M., Duclos, P.: Spectral properties of Schrödinger operators in the semi-classical limit. Proc. Int. Conf. on Diff. Eqn. Math. Phys., Knowles, I., Saito, Y. (eds.). Lecture Notes in Math. Berlin, Heidelberg, New York: Springer 1987
[BCD2] Briet, Ph., Combes, J.M., Duclos, P.: On the location of resonances in the semi-classical limit III: Shape resonances (to appear)
[BCD3] Briet, Ph., Combes, J.M., Duclos, P.: Spectral stability under tunneling for Schrödinger operators, to appear in the Proc. of the Conf. on Partial Differential Equations, Holzhau (DDR) April 1988. Teubner-Text zur Mathematik
[CDS1] Combes, J.M., Duclos, P., Seiler, R.: Convergent expansions for tunneling. Commun. Math. Phys.92, 229 (1983)
[CDS2] Combes, J.M., Duclos, P., Seiler, R.: On the shape resonances. Lecture Notes in Physics, vol. 211, p. 64. Berlin, Heidelberg, New York: Springer 1984
[CDKS] Combes, J.M., Duclos, P., Klein, M., Seiler, R.: The shape resonance. Commun. Math. Phys.110, 215 (1987)
[GGJ] Graffi, S., Grecchi, V., Jona-Lasinio, G.: Tunneling instability via perturbation theory. J. Phys. A17, 2935 (1984)
[Ha1] Harrell, E.: Double Wells. Commun. Math. Phys.119, 351 (1979)
[Ha2] Harrell, E.: The band structure of a one-dimensional periodic system in a scaling limit. Ann. Phys.75, 239 (1980)
[HSj1] Helffer, B., Sjöstrand, J.: Multiple Wells in the semi-classical limit. Commun. Part. Diff. Eq.9, 337 (1984)
[HSj2] Helffer, B., Sjöstrand, J.: Puits multiples. II. interaction moléculaire, symetries, perturbation. Ann. Inst. H. Poincaré2, 127 (1985)
[HSj3] Helffer, B., Sjöstrand, J.: Interaction through non-resonant Wells. Mathematishe Nachriten (1985)
[HSj4] Helffer, B., Sjöstrand, J.: Resonances en limite semi-classique, [Suppl.] Bulletin S.M.F.114 (1986)
[HiSig] Hislop, P., Sigal, I.M.: Semi-classical theory of shape resonances in quantum mechanics. Mem. Am. Math. Soc.78 (399), (1989)
[JMSc1] Jona-Lasinio, G., Martinelli, F., Scoppola, E.: New approach in the semi-classical limit of quantum mechanics. I. Multiple tunnelings in one dimension. Commun. Math. Phys.80, 223 (1981)
[JMSc2] Jona-Lasinio, G., Martinelli, F., Scoppola, E.: Multiple tunneling ind-dimensions: a quantum particle in a hierarchical potential. Ann. Inst. H. Poincaré43, 2 (1985)
[JMSc3] Jona-Lasinio, G., Martinelli, F., Scoppola, E.: A quantum particle in a hierarchical potential with tunneling over arbitrary large scales. J. Phys. A17 (1984)
[JMSc4] Jona-Lasinio, G., Martinelli, F., Scoppola, E.: Tunneling in one dimension: general theory, instabilities, rules of calculation, applications. Mathematics and physics, lecture on recent results, Vol. 11, Streit, L. (ed.). World Scientific: Singapore 1986
[K] Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1966
[MaR] Martinez, A., Rouleux, M.: Effet tunnel entre puits dégénérés. Comm. P.D.E. (1988)13, 1157
[O] Outassourt, A.: Analyse semi-classique pour l'opérateur de Schrödinger à Potentiel Périodique. J. Funct. Anal.72, 1 (1987)
[RS] Reed, M., Simon, B.: Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. New York, San Francisco, London: Academic Press 1975
[Sig1] Sigal, I.M.: Geometric Parametrices and the Many-Body Birman-Schwinger Principle. Duke Math. J.50, 517 (1983)
[Sig2] Sigal, I.M.: Sharp Expnential bounds on resonance states and width of resonances. Adv. Appl. Math.9, 127 (1988)
[Sim1] Simon, B.: Semi-classical analysis of low-lying eigenvalues. II. Tunneling. Ann. Math.120, 89 (1984)
[Sim2] Simon, B.: Semi-classical analysis of low-lying eigenvalues. III. Width of the ground-state band in strongly coupled solids. Ann. Phys.158, 415 (1984)
[Sim3] Simon, B.: Semi-classical analysis of low-lying eigenvalues. IV. The flea on the elephant. J. Funct. Anal.63, 123 (1985)
Author information
Authors and Affiliations
Additional information
Communicated by B. Simon
Laboratoire Propre, Centre National de la Recherche Scientifique
Phymat, Université de Toulon et du Var
Rights and permissions
About this article
Cite this article
Briet, P., Combes, J.M. & Duclos, P. Spectral stability under tunneling. Commun.Math. Phys. 126, 133–156 (1989). https://doi.org/10.1007/BF02124334
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02124334