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TMF, 2002, Volume 132, Number 1, Pages 97–104 (Mi tmf349)  

This article is cited in 41 scientific papers (total in 41 papers)

Local Perturbations of the Schrödinger Operator on the Axis

R. R. Gadyl'shin

Bashkir State Pedagogical University

Abstract: We present necessary and sufficient conditions for the existence of eigenvalues of the Schrödinger operator on an axis under small local perturbations. In the case where the eigenvalues exist, we construct their asymptotic approximations.

Keywords: Schrödinger operator, perturbation, eigenvalue, asymptotic approximation

DOI: https://doi.org/10.4213/tmf349

Full text: PDF file (201 kB)
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English version:
Theoretical and Mathematical Physics, 2002, 132:1, 976–982

Bibliographic databases:

Received: 25.10.2001

Citation: R. R. Gadyl'shin, “Local Perturbations of the Schrödinger Operator on the Axis”, TMF, 132:1 (2002), 97–104; Theoret. and Math. Phys., 132:1 (2002), 976–982

Citation in format AMSBIB
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\pages 976--982
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Borisov, D, “Geometric coupling thresholds in a two-dimensional strip”, Journal of Mathematical Physics, 43:12 (2002), 6265  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    2. Rodriguez-Ceballos J.A., Zhevandrov P.N., “Shallow potential wells for the discrete Schrodinger equation”, Mathematical and Numerical Aspects of Wave Propagation, Waves 2003, 2003, 857–861  crossref  mathscinet  zmath  isi
    3. R. R. Gadyl'shin, “Local Perturbations of the Schrödinger Operator on the Plane”, Theoret. and Math. Phys., 138:1 (2004), 33–44  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    4. M. S. Smetanina, Yu. P. Chuburin, “Schrödinger Operator Levels for a Crystal Film with a Nonlocal Potential”, Theoret. and Math. Phys., 140:2 (2004), 1146–1150  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    5. Gadyl'shin RR, “On regular and singular perturbations of acoustic and quantum waveguides”, Comptes Rendus Mecanique, 332:8 (2004), 647–652  crossref  zmath  adsnasa  isi  scopus  scopus
    6. Borisov D, Exner P, “Exponential splitting of bound states in a waveguide with a pair of distant windows”, Journal of Physics A-Mathematical and General, 37:10 (2004), 3411–3428  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    7. R. R. Gadyl'shin, “Local Perturbations of Quantum Waveguides”, Theoret. and Math. Phys., 145:3 (2005), 1678–1690  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    8. A. R. Bikmetov, D. I. Borisov, “Discrete Spectrum of the Schrodinger Operator Perturbed by a Narrowly Supported Potential”, Theoret. and Math. Phys., 145:3 (2005), 1691–1702  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. A. R. Bikmetov, R. R. Gadyl'shin, “On the spectrum of the Schrödinger operator with large potential concentrated on a small set”, Math. Notes, 79:5 (2006), 729–733  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    10. D. I. Borisov, R. R. Gadyl'shin, “The spectrum of the Schrödinger operator with a rapidly oscillating compactly supported potential”, Theoret. and Math. Phys., 147:1 (2006), 496–500  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. D. I. Borisov, “Discrete spectrum of an asymmetric pair of waveguides coupled through a window”, Sb. Math., 197:4 (2006), 475–504  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    12. Borisov, D, “Distant perturbation asymptotics in window-coupled waveguides. I. The nonthreshold case”, Journal of Mathematical Physics, 47:11 (2006), 113502  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    13. D. I. Borisov, R. R. Gadyl'shin, “The spectrum of a self-adjoint differential operator with rapidly oscillating coefficients on the axis”, Sb. Math., 198:8 (2007), 1063–1093  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    14. Borisov, DI, “Distant perturbations of the Laplacian in a multi-dimensional space”, Annales Henri Poincare, 8:7 (2007), 1371  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    15. Borisov, D, “Asymptotic behaviour of the spectrum of a waveguide with distant perturbations”, Mathematical Physics Analysis and Geometry, 10:2 (2007), 155  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    16. Marin, AM, “High-frequency asymptotics of waves trapped by underwater ridges and submerged cylinders”, Journal of Computational and Applied Mathematics, 204:2 (2007), 356  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    17. D. I. Borisov, R. R. Gadyl'shin, “On the spectrum of a periodic operator with a small localized perturbation”, Izv. Math., 72:4 (2008), 659–688  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    18. R. R. Gadyl'shin, I. Kh. Khusnullin, “Schrödinger operator on the axis with potentials depending on two parameters”, St. Petersburg Math. J., 22:6 (2011), 883–894  mathnet  crossref  mathscinet  zmath  isi
    19. I. Kh. Khusnullin, “A perturbed boundary eigenvalue problem for the Schrödinger operator on an interval”, Comput. Math. Math. Phys., 50:4 (2010), 646–664  mathnet  crossref  mathscinet  adsnasa  isi
    20. Gadyl'shin R.R., “On Regular and Singular Perturbations of the Eigenelements of the Laplacian”, Integral Methods in Science and Engineering, Analytic Methods, 1, 2010, 135–148  crossref  mathscinet  zmath  isi
    21. Romero Rodriguez M.I., Zhevandrov P., “Trapped Modes and Resonances for Water Waves over a Slightly Perturbed Bottom”, Russian Journal of Mathematical Physics, 17:3 (2010), 307–327  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    22. D. I. Borisov, “On the spectrum of a two-dimensional periodic operator with a small localized perturbation”, Izv. Math., 75:3 (2011), 471–505  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    23. Borisov D., Cardone G., “Planar waveguide with “twisted” boundary conditions: Discrete spectrum”, J Math Phys, 52:12 (2011), 123513  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    24. Borisov D., Veselic I., “Low Lying Spectrum of Weak-Disorder Quantum Waveguides”, J Stat Phys, 142:1 (2011), 58–77  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    25. Borisov D., Cardone G., “Planar waveguide with “twisted” boundary conditions: Small width”, J Math Phys, 53:2 (2012), 023503  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    26. Borisov D., Krejcirik D., “The effective Hamiltonian for thin layers with non-Hermitian Robin-type boundary conditions”, Asymptot Anal, 76:1 (2012), 49–59  mathscinet  zmath  isi  elib
    27. D. I. Borisov, “On a $\mathcal{PT}$-symmetric waveguide with a pair of small holes”, Proc. Steklov Inst. Math. (Suppl.), 281, suppl. 1 (2013), 5–21  mathnet  crossref  isi  elib
    28. Aya H., Cano R., Zhevandrov P., “Scattering and Embedded Trapped Modes for an Infinite Nonhomogeneous Timoshenko Beam”, J. Eng. Math., 77:1 (2012), 87–104  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    29. Magnolia Marin A., Dario Ortiz R., Arturo Rodriguez-Ceballos J., “Trapping Waves by a Submerged Cylinder”, Indian J. Pure Appl. Math., 43:3 (2012), 197–210  crossref  zmath  isi  scopus  scopus
    30. A. M. Golovina, “On the spectrum of elliptic operators with distant perturbation in the space”, St. Petersburg Math. J., 25:5 (2014), 735–754  mathnet  crossref  mathscinet  zmath  isi  elib
    31. Borisov D., Veselic I., “Low Lying Eigenvalues of Randomly Curved Quantum Waveguides”, J. Funct. Anal., 265:11 (2013), 2877–2909  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    32. Borisov D., Exner P., Golovina A., “Tunneling Resonances in Systems Without a Classical Trapping”, J. Math. Phys., 54:1 (2013), 012102  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    33. D.I. Borisov, “Discrete spectrum of thin $\mathcal{PT}$-symmetric waveguide”, Ufa Math. J., 6:1 (2014), 29–55  mathnet  crossref  isi  elib
    34. D.I. Borisov, R. Kh. Karimov, T. F. Sharapov, “Initial length scale estimate for waveguides with some random singular potentials”, Ufa Math. J., 7:2 (2015), 33–54  mathnet  crossref  isi  elib
    35. D.I. Borisov, “The Emergence of Eigenvalues of a $\mathcal{PT}$-Symmetric Operator in a Thin Strip”, Math. Notes, 98:6 (2015), 872–883  mathnet  crossref  crossref  mathscinet  isi  elib
    36. A. R. Bikmetov, V. F. Vil'danova, I. Kh. Khusnullin, “On perturbation of a Schrödinger operator on axis by narrow potentials”, Ufa Math. J., 7:4 (2015), 24–31  mathnet  crossref  isi  elib
    37. Borisov D., Golovina A., Veselic I., “Quantum Hamiltonians with Weak Random Abstract Perturbation. I. Initial Length Scale Estimate”, Ann. Henri Poincare, 17:9 (2016), 2341–2377  crossref  mathscinet  zmath  isi  elib  scopus
    38. Bikmetov A.R., Gadyl'shin R.R., “On local perturbations of waveguides”, Russ. J. Math. Phys., 23:1 (2016), 1–18  crossref  mathscinet  zmath  isi  scopus
    39. Borisov D.I., Dmitriev S.V., “On the Spectral Stability of Kinks in 2D Klein-Gordon Model with Parity-Time-Symmetric Perturbation”, Stud. Appl. Math., 138:3 (2017), 317–342  crossref  mathscinet  zmath  isi  scopus
    40. I. Kh. Khusnullin, “Vozmuschenie volnovoda uzkim potentsialom”, Tr. IMM UrO RAN, 23, no. 2, 2017, 274–284  mathnet  crossref  elib
    41. M. S. Smetanina, “Asimptotika urovnei operatora Shrëdingera dlya kristallicheskoi plenki s nelokalnym potentsialom”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 28:4 (2018), 462–473  mathnet  crossref  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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