Asymptotic Behavior of the Eigenvalues of the Laplacian with Two Distant Perturbations on the Plane (the Case of Arbitrary Multiplicity)

Abstract

The Laplacian with a pair of distant perturbations is studied in two-dimensional space. Perturbations are understood as real finite continuous potentials. The discrete spectrum of the perturbed Laplacian is studied when the distance between the potentials increases. The presence of its eigenvalues and eigen-functions that correspond to them is considered for various cases of multiplicities of the limiting eigenvalue. The first case of the considered multiplicity is the double limiting eigenvalue. By this we mean the simple and isolated Laplacian eigenvalue with the first potential, as well as the simple and isolated Laplacian eigenvalue with the second potential. The second case under consideration is the case of arbitrary multiplicity of the limiting eigenvalue. By this we mean the Laplacian eigenvalue with the first potential of arbitrary multiplicity and the Laplacian eigenvalue with the second potential also of arbitrary multiplicity. In both cases under consideration (of multiplicity two and arbitrary), the first terms of formal asymptotic expansions of the eigenvalues and eigenfunctions of the perturbed Laplacian are constructed. A complex exponential-power structure of the constructed asymptotic is demonstrated. Also, in both cases under consideration, symmetry with respect to zero of the first corrections of the asymptotics of the eigenvalues of the perturbed Laplacian is shown

Please cite this article in English as:

Golovina A.M. Asymptotic behavior of the eigenvalues of the Laplacian with two distant perturbations on the plane (the case of arbitrary multiplicity). Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2023, no. 3 (108), pp. 4--19 (in Russ.). DOI: https://doi.org/10.18698/1812-3368-2023-3-4-19

References

[1] Davies E.B. The twisting trick for double well Hamiltonians. Commun. Math. Phys., 1982, vol. 85, no. 3, pp. 471--479. DOI: https://doi.org/10.1007/BF01208725

[2] Hoegh-Krohn R., Mebknout M. The 1/r expansion for the critical multiple well problem. Commun. Math. Phys., 1983, vol. 91, no. 1, pp. 66--73. DOI: https://doi.org/10.1007/BF01206050

[3] Klaus M., Simon B. Coupling constant thresholds in nonrelativistic quantum mechanics. I. Short-range two-body case. Ann. Phys., 1980, vol. 130, iss. 2, pp. 251--281. DOI: https://doi.org/10.1016/0003-4916(80)90338-3

[4] Morgan J.D. III, Simon B. Behavior of molecular potential energy curves for large nuclear separations. Int. J. Quantum. Chem., 1980, vol. 17, iss. 6, pp. 1143--1166. DOI: https://doi.org/10.1002/qua.560170609

[5] Graffi S., Grecchi V., Harrell E.M. II, et al. The 1/R expansion for H₂⁺ analyticity, summability, and asymptotics. Ann. Phys., 1985, vol. 165, iss. 2, pp. 441--483. DOI: https://doi.org/10.1016/0003-4916(85)90305-7

[6] Harrell E.M. Double wells. Commun. Math. Phys., 1980, vol. 75, no. 3, pp. 239--261. DOI: https://doi.org/10.1007/BF01212711

[7] Klaus M. Some remarks on double-wells in one and three dimensions. Ann. Inst. Henri Poincare, 1981, vol. 34, no. 4, pp. 405--417.

[8] Reity O.K. Asymptotic expansions of the potential curves of the relativistic quantum-mechanical two-Coulomb-center problem. Proceeding of Institute of Mathematics of NAS of Ukraine, 2002, vol. 43-2, pp. 672--675.

[9] Klaus M., Simon B. Binding of Schrodinger particles through conspiracy of potential wells. Ann. Inst. Henri Poincare, 1979, vol. 30, no. 2, pp. 83--87.

[10] Borisov D. Asymptotic behaviour of the spectrum of a waveguide with distant perturbations. Math. Phys. Anal. Geom., 2007, vol. 10, no. 2, pp. 155--196.DOI: https://doi.org/10.1007/s11040-007-9028-1

[11] Borisov D. Distant perturbation of the Laplacian in a multi-dimensional space. Ann. Henri Poincare, 2007, vol. 8, no. 7, pp. 1371--1399. DOI: https://doi.org/10.1007/s00023-007-0338-4

[12] Golovina A.M. Discrete eigenvalues of periodic operators with distant perturbations. J. Math. Sci., 2013, vol. 189, no. 3, pp. 342--364. DOI: https://doi.org/10.1007/s10958-013-1192-1

[13] Borisov D.I., Golovina A.M. On the resolvents of periodic operators with distant perturbations. Ufimskiy matematicheskiy zhurnal [Ufa Mathematical Journal], 2012, vol. 4, no. 2, pp. 65--73 (in Russ.).

[14] Golovina A.M. Investigations in the spectral properties of operators with distant perturbations (survey). Matematika i matematicheskoe modelirovanie [Mathematics and Mathematical Modeling], 2015, vol. 2, pp. 1--22 (in Russ.). DOI: 10.7463/mathm.0215.0776859

[15] Borisov D.I., Golovina A.M. On occurrence of resonances from multiple eigenvalues of the Schrödinger operator in a cylinder with distant perturbations. J. Math Sci., 2021, vol. 258, no. 1, pp. 1--12. DOI: https://doi.org/10.1007/s10958-021-05532-x

[16] Borisov D.I., Konyrkulzhaeva M.N. On infinite system of resonances and eigenvalues with exponential asymptotics generated by distant perturbations. Ufimskiy matematicheskiy zhurnal [Ufa Mathematical Journal], 2020, vol. 12, no. 4, pp. 3--18 (in Russ.). DOI: http://dx.doi.org/10.13108/2020-12-4-3

[17] Borisov D.I., Golovina A.M. On finitely many resonances emerging under distant perturbations in multi-dimensional cylinders. J. Math. Anal. Appl., 2021, vol. 496, no. 2, art. 124809. DOI: https://doi.org/10.1016/j.jmaa.2020.124809

[18] Golovina A.M. On Laplacian discrete spectrum behavior with two distant perturbations on the plane in the case of a double limiting eigenvalue. Matematika i matematicheskoe modelirovanie [Mathematics and Mathematical Modeling], 2022, vol. 2, pp. 1--13 (in Russ.). DOI: https://doi.org/10.24108/mathm.0222.0000301

[19] Golovina A.M. Asymptotic behavior of the eigenvalues of a periodic operator with two distant perturbations on the axis. Matematika i matematicheskoe modelirovanie [Mathematics and Mathematical Modeling], 2022, vol. 1, pp. 21--30 (in Russ.). DOI: https://doi.org/10.24108/mathm.0122.0000300

[20] Tikhonov A.N., Samarskiy A.A. Uravneniya matematicheskoy fiziki [Equations of mathematical physics]. Moscow, Nauka Publ., 2004.