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Spectrum of periodic elliptic operators with distant perturbations in space


Author: A. M. Golovina
Translated by: S. Kislyakov
Original publication: Algebra i Analiz, tom 25 (2013), nomer 5.
Journal: St. Petersburg Math. J. 25 (2014), 735-754
MSC (2010): Primary 35B20
DOI: https://doi.org/10.1090/S1061-0022-2014-01314-3
Published electronically: July 18, 2014
MathSciNet review: 3184606
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Abstract: A periodic selfadjoint differential operator of even order and with distant perturbations in a multidimensional space is treated. The role of perturbations is played by arbitrary localized operators. The localization is described by specially chosen weight functions. The behavior of the spectrum of the perturbed operator is studied under the condition that the distance between the domains where the perturbation are localized tends to infinity. It is shown that there exists a simple isolated eigenvalue of the perturbed operator that tends to a simple isolated eigenvalue of the limit operator. Series expansions are obtained for this eigenvalue of the perturbed operator and for the corresponding eigenfunction. Uniform convergence for these series is shown and formulas for their terms are deduced.


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  • 1. D. I. Borisov, Asymptotic behavior of the spectrum of a waveguide with distant perturbation, Math. Phys. Anal. Geom. 10 (2007), no. 2, 155-196. MR 2342630 (2009b:35049)
  • 2. D. I. Borisov and P. Exner, Exponential splitting of bound in a waveguide with a pair of distant windows, J. Phys. A 37 (2004), no. 10, 3411-3428. MR 2039856 (2005b:81030)
  • 3. D. I. Borisov, Distant perturbation of the Laplacian in a multi-dimensional space, Ann. Inst. H. Poincaré 8 (2007), no. 7, 1371-1399. MR 2360440 (2009b:47034)
  • 4. E. B. Davies, The twisting trick for double well Hamiltonians, Comm. Math. Phys. 85 (1982), no. 3, 471-479. MR 678157 (84a:81008)
  • 5. E. M. Harrell, Double wells, Comm. Math. Phys. 75 (1980), no. 3, 239-261. MR 581948 (81j:81010)
  • 6. R. Hoegh-Krohn and M. Mebkhout, The multiple well problem asymptotic behavior of the eigenvalues and resonances, Trends and Developments in the Eightes (Bielefeld, 1982/83), World Sci. Publ., Singapore, 1985, pp. 244-272. MR 853751 (88b:81034)
  • 7. H. Tamura, Existence of bound states for double well potentials and the Efimov effect, Functional-Analytic Methods for Partial Differential Equations (Tokyo, 1989), Lecture Notes in Math., vol. 1450, Springer, Berlin, 1990, pp. 173-186. MR 1084608 (91j:35088)
  • 8. T. Aktosun, M. Klaus, and C. van der Mee, On the number of bound states for the one-dimensional Schrödinger equation, J. Math. Phys. 39 (1998), no. 9, 4249-4256. MR 1643229 (99j:81026)
  • 9. J. D. Morgan III and B. Simon, Behavior of molecular potential energy curves for large nuclear separations, Intern. J. Quantum Chemistry 17 (1980), no. 2, 1143-1166.
  • 10. S. Graffi, V. Grecel, E. V. Harrell and H. J. Silverstone, The $ \frac {1}{R}$-expansion for $ H_2^+$: analyticity, summability and asymptotics, Ann. Phys. 165 (1985), no. 2, 441-483. MR 816800 (87i:81049)
  • 11. R. Ahlrichs, Convergence properties of the intermolecular Force series $ (\frac {1}{R}$-expansion), Theor. Chimica Acta 41 (1976), no. 1, 7-15.
  • 12. M. Klaus, Some remarks on double-wells in one and three dimensions, Ann. Inst. H. Poincaré Sect. A 34 (1981), no. 4, 405-417. MR 625171 (82i:81026)
  • 13. M. Klaus and B. Simon, Binding of Schrödinger particles through conspiracy of potential wells, Ann. Inst. H. Poincaré Sect. A 30 (1979), no. 2, 83-87. MR 535366 (80h:81012)
  • 14. Y. Pinchover, On the localization of binding for Schrödinger operators and its extension to elliptic operators, J. Anal. Math. 66 (1995), 57-83. MR 1370346 (96m:35061)
  • 15. E. M. Harrell and M. Klaus, On the double-well problem for Dirac operators, Ann. Inst. H. Poincaré 38 (1983), no. 2, 153-166. MR 705337 (84k:81031)
  • 16. O. K. Reity, Asymptotic expansions of the potential curves of the relativistic quantum-mechanical two-Coulomb-center problem, Symmetry in Nonlinear Mathematical Physics. Pt. 1, 2 (Kyiv, 2001), Pr. Inst. Math. Nats. Akad. Nauk Ukr. Mat. Zastos., vol. 43, Natsional. Akad. Nauk Ukraini, Inst. Mat., Kiev, 2002, pp. 672-675. MR 1918384
  • 17. S. Kondej and I. Veselić, Lower bound on the lowest spectral gap of singular potential Hamiltonians, Ann. Inst. H. Poincaré 8 (2007), no. 1, 109-134. MR 2299195 (2008a:81066)
  • 18. D. I. Borisov, On the spectrum of a two-dimensional periodic operator with a small localized perturbation, Izv. Ross. Akad. Nauk Ser. Mat. 75 (2011), no. 2, 29-64; English transl., Izv. Math. 75 (2011), no. 3, 471-505. MR 2847781 (2012g:35049)
  • 19. A. M. Golovina, On the resolvent of elliptic operators with distant perturbations in the space, Rus. J. Math. Phys. 19 (2012), no. 2, 182-192. MR 2926322
  • 20. -, Resolvent of operators with distant perturbations, Mat. Zametki 91 (2012), no. 3, 464-466; English transl., Math. Notes 91 (2012), no. 3, 435-438.
  • 21. R. R. Gadyl'shin, On local perturbations of the Schrödinger operator on the axis, Teoret. Mat. Fiz. 132 (2002), no. 1, 97-104; English transl., Teoret. and Math. Phys. 132 (2002), no. 1, 976-982. MR 1956680 (2003m:34205)
  • 22. D. I. Borisov, The discrete spectrum of a pair of asymmetric window-coupled waveguides, Mat. Sb. 197 (2006), no. 4, 3-32; English transl., Sb. Math. 197 (2006), no. 3-4, 475-504. MR 2263787 (2007g:35162)
  • 23. S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35-92. MR 0162050 (28:5252)
  • 24. T. Kato, Perturbation theory for linear operators, Grundlehren Math. Wiss., Bd. 132, Springer-Verlag, New York, 1966 MR 0203473 (34:3324)
  • 25. A. I. Markushevich, Theory of analytic functions, Nauka, Moscow, 1968. (Russian)

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Additional Information

A. M. Golovina
Affiliation: Department of Physics and Mathematics, M. Akmulla Bashkir State Pedagogocal University, ul. Oktyabr$’$skoi revolyutsii 3a, Ufa 450055, Russia
Email: nastya{\textunderscore}gm@mail.ru

DOI: https://doi.org/10.1090/S1061-0022-2014-01314-3
Keywords: Selfadjoint operator, distant perturbations, spectrum, eigenvalue, eigenfunction, asymptotics
Received by editor(s): May 19, 2012
Published electronically: July 18, 2014
Additional Notes: Supported by RFBR and by the Federal Thematic Program “Scientific and educational personnel for innovative Russia”, 2009-2013 (contract no. 14.B37.21.0358).
Article copyright: © Copyright 2014 American Mathematical Society

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