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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


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Theoretical and Mathematical Physics, 2002, 132:1, 976–982

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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
\by R.~R.~Gadyl'shin
\paper Local Perturbations of the Schr\"odinger Operator on the Axis
\jour TMF
\yr 2002
\vol 132
\issue 1
\pages 97--104
\jour Theoret. and Math. Phys.
\yr 2002
\vol 132
\issue 1
\pages 976--982

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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
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    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
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    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
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    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
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    33. D.I. Borisov, “Discrete spectrum of thin $\mathcal{PT}$-symmetric waveguide”, Ufa Math. J., 6:1 (2014), 29–55  mathnet  crossref  isi  elib
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    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
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