Abstract
This paper is devoted to an asymptotic formula for splitting of the lowest eigenvalues of a two-dimensional Schrödinger operator with a potential having two symmetric wells. We rigorously prove the corresponding formula, obtained earlier in a paper by J. Brüning, S. Yu. Dobrokhotov, and E. S. Semenov [“Unstable Closed Trajectories, Librations and Splitting of the Lowest Eigenvalues in Quantum Double Well Problem,” Regul. Chaotic Dyn. 11 (2), 167–180 (2006)] at the physical level of rigor. The crucial role in our considerations is played by an asymptotic formula for the Maupertuis action (as a function of energy) on a periodic trajectory of the classical system (a libration) lying near a doubly asymptotic trajectory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. M. Polyakov, “Quark Confinement and Topology of Gauge Theories,” Nuclear. Phys. B. 120, 429–458 (1977).
E. Gildener and A. Patrascioiu, “Pseudoparticle Contributions to the Energy Spectrum of a One-Dimensional System,” Phys. Rev. D. 16, 425–443 (1977).
S. Coleman, The Uses of Instantons, The Whys of Subnuclear Physics (Proc. 1977 International School of Subnuclear Physics, Erice, A. Zichichi, ed., 1979).
E. Harrell, “On the Rate of Asymptotic Eigenvalue Degeneracy,” Comm. Math. Phys. 60(1), 73–95 (1978).
E. Harrell, “Double Wells,” Comm. Math. Phys. 75(3), 239–261 (1980).
G. Jona-Lasinio, F. Martinelli, and E. Scoppola, “New Approach to the Semiclassical Limit of Quantum Mechanics. I. Multiple Tunnelings in One Dimension,” Comm. Math. Phys. 80(2), 223–254 (1981).
J. M. Combes, P. Duclos, and R. Seiler, “Krein’s Formula and One-Dimensional Multiple-Well,” J. Funct. Anal. 52(2), 257–301 (1983).
B. Simon, “Semiclassical Analysis of Low Lying Eigenvalues. II. Tunneling,” Ann. of Math. 120(1), 89–118 (1984).
B. Helffer and J. Sjöstrand, “Multiple Wells in the Semiclassical Limit. I,” Comm. Partial Differential Equations 9(4), 337–408 (1984).
B. Helffer and J. Sjöstrand, “Puits Multiples en limite semi-classique. II. Interaction moléculaire. Symétries. Perturbation,” Ann. Inst. H. Poincaré. Phys. Théor. 42(2), 127–212 (1985).
V. P. Maslov, “Global Exponential Asymptotic Behavior of Solutions of the Tunnel Equations and the Problem of Large Deviations,” Trudy Mat. Inst. Steklova 163, 150–180 (1984) [in Russian].
B. Helffer, Semi-Classical Analysis for the Schrödinger Operator and Applications, Lect. Notes. Math. 1336 (Springer, Berlin, 1988).
S. Yu. Slavyanov, “The Asymptotic Behavior of Singular Sturm-Liouville Problems with Respect to a Large Parameter in the Case of Nearby Transition Points,” Differ. Uravn. 5(2), 313–325 (1969) [in Russian].
T. F. Pankratova, “Quasimodes and Splitting of Eigenvalues,” Dokl. Akad. Nauk SSSR 276(4), 795–799 (1984) [Soviet Math. Dokl. 29 (3), 597–601 (1984)].
L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Non-Relativistic Theory) (Izdat. “Nauka”, Moscow, 1974; Pergamon, Oxford Press, 1977).
M. V. Fedoryuk [Fedorjuk], “Asymptotics of the Discrete Spectrum of the Operator w″(x)λ 2 p(x)w(x),” (Russian) Mat. Sb. (N.S.) 68(110) (1), 81–110 (1965) [in Russian].
S. Yu. Dobrokhotov, V. N. Kolokol’tsov, and V. P. Maslov, “Splitting of the Low Energy Levels of the Schrödinger Equation, and the Asymptotic Behavior of the Fundamental Solution of the Equation \(hu_t = \tfrac{1} {2}h^2 \Delta u - V(x)u\),” Teoret. Mat. Fiz. 87(3), 323–375 (1991) [Theoret. and Math. Phys. 87 (3), 561–599 (1991)].
S. Yu. Dobrokhotov and V. N. Kolokol’tsov, “On the Amplitude of the Splitting of Lower Energy Levels of the Schrödinger Operator with Two Symmetric Wells,” Teoret. Mat. Fiz. 94(3), 426–434 (1993) [Theoret. and Math. Phys. 94 (3), 300–305 (1993)].
S. Albeverio, S. Yu. Dobrokhotov, and E. S. Semenov, “On Formulas for Splitting the Upper and Lower Energy Levels of the One-Dimensional Schrödinger Operator,” Teoret. Mat. Fiz. 138(1), 116–126 (2004) [Theoret. and Math. Phys. 138 (1), 98–106 (2004)]; a correction: Teoret. Mat. Fiz. 141 (3), 485 (2004) [Theoret. and Math. Phys. 141 (3), 1750 (2004)].
J. Brüning, S. Yu. Dobrokhotov, and E. S. Semenov, “Unstable Closed Trajectories, Librations and Splitting of the Lowest Eigenvalues in Quantum Double Well Problem,” Regul. Chaotic Dyn. 11(2), 167–180 (2006).
S. V. Bolotin, “Libration Motions of Natural Dynamical Systems,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. 6, 72–77 (1978) [Moscow Univ. Math. Bull. 33, 49–53 (1978)].
V. I. Arnol’d, V. V. Kozlov, and A. I. Nejshtadt, “Mathematical Aspects of Classical and Celestial Mechanics,” Dynamical Systems III. Series: Encyclopaedia of Mathematical Sciences 3, (2006), Springer.
S. V. Bolotin and P. H. Rabinowitz, “A Variational Construction of Chaotic Trajectories for a Reversible Hamiltonian System,” J. Differential Equations 148, 364–387 (1998).
L. P. Shil’nikov, “On a Poincaré-Birkhoff Problem,” Mat. Sb. 74, 378–397 (1967) [Math. USSR Sbornik. 3, 353–371 (1967)].
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Anikin, A.Y. Asymptotic behavior of the Maupertuis action on a libration and tunneling in a double well. Russ. J. Math. Phys. 20, 1–10 (2013). https://doi.org/10.1134/S1061920813010019
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1061920813010019