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On the Delocalization of a Quantum Particle under the Adiabatic Evolution Generated by a One-Dimensional Schrödinger Operator

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Abstract

The one-dimensional nonstationary Schrödinger equation is discussed in the adiabatic approximation. The corresponding stationary operator \(H\) depending on time as a parameter has a continuous spectrum \(\sigma_c=[0,+\infty)\) and finitely many negative eigenvalues. In time, the eigenvalues approach the edge of \(\sigma_c\) and disappear one by one. The solution under consideration is close at some moment to an eigenfunction of \(H\). As long as the corresponding eigenvalue \(\lambda\) exists, the solution is localized inside the potential well. Its delocalization with the disappearance of \(\lambda\) is described.

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Funding

This work was supported by the Russian Foundation for Basic Research under grant 20-01-00451 A.

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Authors and Affiliations

  1. Saint Petersburg State University, St. Petersburg, 199034, Russia

    V. A. Sergeev & A. A. Fedotov

  2. Euler International Mathematical Institute, St. Petersburg, 191011, Russia

    V. A. Sergeev

Authors
  1. V. A. Sergeev
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  2. A. A. Fedotov
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Corresponding author

Correspondence to V. A. Sergeev.

Additional information

Translated from Matematicheskie Zametki, 2022, Vol. 112, pp. 752–769 https://doi.org/10.4213/mzm13776.

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Sergeev, V.A., Fedotov, A.A. On the Delocalization of a Quantum Particle under the Adiabatic Evolution Generated by a One-Dimensional Schrödinger Operator. Math Notes 112, 726–740 (2022). https://doi.org/10.1134/S0001434622110098

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