Abstract:
We consider the one-dimensional stationary Schrödinger equation with a smooth double-well potential. We obtain a criterion for the double localization of wave functions, exponential splitting of energy levels, and the tunneling transport of a particle in an asymmetric potential and also obtain asymptotic formulas for the energy splitting that generalize the formulas known in the case of a mirror-symmetric potential. We consider the case of higher energy levels and the case of energies close to the potential minimums. We present an example of tunneling transport in an asymmetric double well and also consider the problem of tunnel perturbation of the discrete spectrum of the Schrödinger operator with a single-well potential. Exponentially small perturbations of the energies occur in the case of local potential deformations concentrated only in the classically forbidden region. We also calculate the leading term of the asymptotic expansion of the tunnel perturbation of the spectrum.
Keywords:
tunneling, quasi-intersection of energy levels, one-dimensional Schrödinger equation, semiclassical approximation.
Citation:
E. V. Vybornyi, “Tunnel splitting of the spectrum and bilocalization of eigenfunctions in an asymmetric double well”, TMF, 178:1 (2014), 107–130; Theoret. and Math. Phys., 178:1 (2014), 93–114
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\by E.~V.~Vybornyi
\paper Tunnel splitting of the~spectrum and bilocalization of eigenfunctions in an~asymmetric double well
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\jour Theoret. and Math. Phys.
\yr 2014
\vol 178
\issue 1
\pages 93--114
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Linking options:
https://www.mathnet.ru/eng/tmf8531
https://doi.org/10.4213/tmf8531
https://www.mathnet.ru/eng/tmf/v178/i1/p107
This publication is cited in the following 10 articles:
D. J. Nader, E. Serrano‐Ensástiga, “Critical Energies and Wigner Functions of the Stationary States of the Bose Einstein Condensates in a Double‐Well Trap”, Adv Quantum Tech, 2025
S. Yu. Dobrokhotov, A. V. Tsvetkova, “Global Asymptotics for Functions of Parabolic Cylinder and Solutions of the Schrödinger Equation with a Potential in the Form of a Nonsmooth Double Well”, Russ. J. Math. Phys., 30:1 (2023), 46
P. A. Golovinski, V. A. Dubinkin, “Quantum States of the Kapitza Pendulum”, Russ Phys J, 65:1 (2022), 21
G. A. Durkin, “Quantum speedup at zero temperature via coherent catalysis”, Phys. Rev. A, 99:3 (2019), 032315
M. Karasev, E. Vybornyi, “Bi-orbital states in hyperbolic traps”, Russ. J. Math. Phys., 25:4 (2018), 500–508
M. Karasev, E. Novikova, E. Vybornyi, “Bi-states and 2-level systems in rectangular Penning traps”, Russ. J. Math. Phys., 24:4 (2017), 454–464
N. Mukherjee, A. K. Roy, “Quantum confinement in an asymmetric double-well potential through energy analysis and information entropic measure”, Ann. Phys.-Berlin, 528:5 (2016), 412–433
M. V. Karasev, E. M. Novikova, E. V. Vybornyi, “Non-Lie Top Tunneling and Quantum Bilocalization in Planar Penning Trap”, Math. Notes, 100:6 (2016), 807–819
K. A. Sveshnikov, A. V. Tolokonnikov, “H 2 + in a lattice of cavities: Ammonia-like splitting of the lowest level”, Moscow Univ. Phys., 70:3 (2015), 181
E. V. Vybornyi, “Energy splitting in dynamical tunneling”, Theoret. and Math. Phys., 181:2 (2014), 1418–1427
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