Research Article
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Year 2021, , 44 - 50, 30.06.2021
https://doi.org/10.47000/tjmcs.851839

Abstract

References

  • [1] Andrews, B., Clutterbuck, J., Hauer, D., The fundamental gap for a one-dimensional Schr¨odinger operator with Robin boundary conditions, arXiv:2002.06900 [math.CA], (2020).
  • [2] Ashbaugh M. S., Benguria, R., Optimal lower bound for the gap between the first two eigenvalues of one-dimensional Schr¨odinger operators with symmetric single-well potentials and related results, Proceedings of the American Mathematical Society, 105(1989), 419–424.
  • [3] Ashbaugh, M. S., Kielty, D., Spectral gaps of 1-D Robin Sch¨odinger operators with single well potentials, Journal of Mathematical Physics, 61 (9)(2020), 091507.
  • [4] Bas¸kaya, E., Asymptotics of eigenvalues for Sturm-Liouville problem with eigenparameter-dependent boundary conditions, New Trends in Mathematical Sciences, 6(2)(2018), 247–257.
  • [5] Capała, K., Dybiec, B., Multimodal stationary states in symmetric single-well potentials driven by Cauchy noise, Journal of Statistical Mechanics: Theory and Experiment, 2019(2019), 033206.
  • [6] Chen, W.-C., Cheng, Y.-H., Remarks on the one-dimensional sloshing problem involving the p-Laplacian operator, Turkish Journal of Mathematics, 44(2020), 1376–1387.
  • [7] Chen, D.- Y., Huang, M.- J., Comparison theorems for the eigenvalue gap of Schr¨odinger operators on the real line, Annales Henri Poincar´e, 13(2011), 85–101.
  • [8] Ciesla, M., Capała, K., Dybiec, B., Multimodal stationary states under Cauchy noise, Physical Review E, 99(2019), 052118.
  • [9] Coşkun, H., Bas¸kaya, E., Asymptotics of eigenvalues of regular Sturm-Liouville problems with eigenvalue parameter in the boundary condition for integrable potential, Mathematica Scandinavica, 107(2017), 209–223.
  • [10] Coşkun, H., Bayram, N., Asymptotics of eigenvalues for regular Sturm-Liouville problems with eigenvalue parameter in the boundary condition, Journal of Mathematical Analysis and Applications, 306(2)(2005), 548–566.
  • [11] Coşkun, H., Kabatas¸, A., Asymptotic approximations of eigenfunctions for regular Sturm-Liouville problems with eigenvalue parameter in the boundary condition for integrable potential, Mathematica Scandinavica, 113(1)(2013), 143–160.
  • [12] Coşkun, H., Kabatas¸, A., Green’ s function of regular Sturm-Liouville problem having eigenparameter in one boundary condition, Turkish Journal of Mathematics and Computer Science, 4(2016), 1–9.
  • [13] Coşkun, H., Kabatas¸, A., Bas¸kaya, E., On Green’ s function for boundary value problem with eigenvalue dependent quadratic boundary condition, Boundary Value Problems, 71(2017).
  • [14] Coşkun, H., Bas¸kaya, E., Kabatas¸, A., Instability intervals for Hill’ s equation with symmetric single well potential, Ukrainian Mathematical Journal, 71(6)(2019), 977–983.
  • [15] Fulton, C., Two point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proceedings of the Royal Society of Edinburgh, 77A(1977), 293–308.
  • [16] Fulton, C., Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions, Proceedings of the Royal Society of Edinburgh, 87A(1980), 1–34.
  • [17] Guliyev, N. J., Schr¨odinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter, Journal of Mathematical Physics, 60(6)(2019), 063501.
  • [18] Guliyev, N. J., Inverse square singularities and eigenparameter dependent boundary conditions are two sides of the same coin, arXiv:2001.00061 [math-ph], 2019.
  • [19] Guliyev, N. J., Essentially isospectral transformations and their applications, Annali di Matematica Pura ed Applicata, 199(2020), 1621–1648.
  • [20] Guliyev, N. J., On two-spectra inverse problems, Proceedings of the American Mathematical Society, 148(2020), 4491–4502.
  • [21] Haaser, N. B., Sullivian, J. A., Real Analysis, Van Nostrand Reinhold Company, New York, 1991.
  • [22] Hinton, D. B., Eigenfunction expansions for a singular eigenvalue problem with eigenparameter in the boundary condition, SIAM Journal on Mathematical Analysis, 12(1981), 572-584.
  • [23] Horvath, M., On the first two eigenvalues of Sturm-Liouville Operators, Proceedings of the American Mathematical Society, 131(4)(2002), 1215–1224.
  • [24] Huang, M. J., The first instability interval for Hill equations with symmetric single well potentials, Proceedings of the American Mathematical Society, 125(1997), 775–778.
  • [25] Huang, M. J., Tsai, T. M., The eigenvalue gap for one-dimensional Schrodinger operators with symmetric potentials, Proceedings of the Royal Society of Edinburgh Section A, 139(2009), 359–366.
  • [26] Kerner, J., T¨aufer, M., On the spectral gap of one-dimensional Schrödinger operators on large intervals, arXiv:2012.09060 [math.SP], 2020.
  • [27] Mandrysz, M., Dybiec, B., Energetics of single-well undamped stochastic oscillators, Physical Review E, 99(1)(2019), 012125.
  • [28] Messori, C., Deep into the Water: Exploring the Hydro-Electromagnetic and QuantumElectrodynamic Properties of Interfacial Water in Living Systems, Open Access Library Journal, 6 (2019), e5435.

Asymptotic Eigenvalues of Regular Sturm-Liouville Problems with Spectral Parameter-Dependent Boundary Conditions and Symmetric Single Well Potential

Year 2021, , 44 - 50, 30.06.2021
https://doi.org/10.47000/tjmcs.851839

Abstract

In this study, we find asymptotic estimates of eigenvalues for regular
Sturm-Liouville problems having the eigenvalue parameter in all boundary
conditions with the symmetric single well potential that is symmetric to the midpoint of the related interval and nonincreasing on the first semi-region of the related interval.

References

  • [1] Andrews, B., Clutterbuck, J., Hauer, D., The fundamental gap for a one-dimensional Schr¨odinger operator with Robin boundary conditions, arXiv:2002.06900 [math.CA], (2020).
  • [2] Ashbaugh M. S., Benguria, R., Optimal lower bound for the gap between the first two eigenvalues of one-dimensional Schr¨odinger operators with symmetric single-well potentials and related results, Proceedings of the American Mathematical Society, 105(1989), 419–424.
  • [3] Ashbaugh, M. S., Kielty, D., Spectral gaps of 1-D Robin Sch¨odinger operators with single well potentials, Journal of Mathematical Physics, 61 (9)(2020), 091507.
  • [4] Bas¸kaya, E., Asymptotics of eigenvalues for Sturm-Liouville problem with eigenparameter-dependent boundary conditions, New Trends in Mathematical Sciences, 6(2)(2018), 247–257.
  • [5] Capała, K., Dybiec, B., Multimodal stationary states in symmetric single-well potentials driven by Cauchy noise, Journal of Statistical Mechanics: Theory and Experiment, 2019(2019), 033206.
  • [6] Chen, W.-C., Cheng, Y.-H., Remarks on the one-dimensional sloshing problem involving the p-Laplacian operator, Turkish Journal of Mathematics, 44(2020), 1376–1387.
  • [7] Chen, D.- Y., Huang, M.- J., Comparison theorems for the eigenvalue gap of Schr¨odinger operators on the real line, Annales Henri Poincar´e, 13(2011), 85–101.
  • [8] Ciesla, M., Capała, K., Dybiec, B., Multimodal stationary states under Cauchy noise, Physical Review E, 99(2019), 052118.
  • [9] Coşkun, H., Bas¸kaya, E., Asymptotics of eigenvalues of regular Sturm-Liouville problems with eigenvalue parameter in the boundary condition for integrable potential, Mathematica Scandinavica, 107(2017), 209–223.
  • [10] Coşkun, H., Bayram, N., Asymptotics of eigenvalues for regular Sturm-Liouville problems with eigenvalue parameter in the boundary condition, Journal of Mathematical Analysis and Applications, 306(2)(2005), 548–566.
  • [11] Coşkun, H., Kabatas¸, A., Asymptotic approximations of eigenfunctions for regular Sturm-Liouville problems with eigenvalue parameter in the boundary condition for integrable potential, Mathematica Scandinavica, 113(1)(2013), 143–160.
  • [12] Coşkun, H., Kabatas¸, A., Green’ s function of regular Sturm-Liouville problem having eigenparameter in one boundary condition, Turkish Journal of Mathematics and Computer Science, 4(2016), 1–9.
  • [13] Coşkun, H., Kabatas¸, A., Bas¸kaya, E., On Green’ s function for boundary value problem with eigenvalue dependent quadratic boundary condition, Boundary Value Problems, 71(2017).
  • [14] Coşkun, H., Bas¸kaya, E., Kabatas¸, A., Instability intervals for Hill’ s equation with symmetric single well potential, Ukrainian Mathematical Journal, 71(6)(2019), 977–983.
  • [15] Fulton, C., Two point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proceedings of the Royal Society of Edinburgh, 77A(1977), 293–308.
  • [16] Fulton, C., Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions, Proceedings of the Royal Society of Edinburgh, 87A(1980), 1–34.
  • [17] Guliyev, N. J., Schr¨odinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter, Journal of Mathematical Physics, 60(6)(2019), 063501.
  • [18] Guliyev, N. J., Inverse square singularities and eigenparameter dependent boundary conditions are two sides of the same coin, arXiv:2001.00061 [math-ph], 2019.
  • [19] Guliyev, N. J., Essentially isospectral transformations and their applications, Annali di Matematica Pura ed Applicata, 199(2020), 1621–1648.
  • [20] Guliyev, N. J., On two-spectra inverse problems, Proceedings of the American Mathematical Society, 148(2020), 4491–4502.
  • [21] Haaser, N. B., Sullivian, J. A., Real Analysis, Van Nostrand Reinhold Company, New York, 1991.
  • [22] Hinton, D. B., Eigenfunction expansions for a singular eigenvalue problem with eigenparameter in the boundary condition, SIAM Journal on Mathematical Analysis, 12(1981), 572-584.
  • [23] Horvath, M., On the first two eigenvalues of Sturm-Liouville Operators, Proceedings of the American Mathematical Society, 131(4)(2002), 1215–1224.
  • [24] Huang, M. J., The first instability interval for Hill equations with symmetric single well potentials, Proceedings of the American Mathematical Society, 125(1997), 775–778.
  • [25] Huang, M. J., Tsai, T. M., The eigenvalue gap for one-dimensional Schrodinger operators with symmetric potentials, Proceedings of the Royal Society of Edinburgh Section A, 139(2009), 359–366.
  • [26] Kerner, J., T¨aufer, M., On the spectral gap of one-dimensional Schrödinger operators on large intervals, arXiv:2012.09060 [math.SP], 2020.
  • [27] Mandrysz, M., Dybiec, B., Energetics of single-well undamped stochastic oscillators, Physical Review E, 99(1)(2019), 012125.
  • [28] Messori, C., Deep into the Water: Exploring the Hydro-Electromagnetic and QuantumElectrodynamic Properties of Interfacial Water in Living Systems, Open Access Library Journal, 6 (2019), e5435.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Elif Başkaya 0000-0001-6118-9183

Publication Date June 30, 2021
Published in Issue Year 2021

Cite

APA Başkaya, E. (2021). Asymptotic Eigenvalues of Regular Sturm-Liouville Problems with Spectral Parameter-Dependent Boundary Conditions and Symmetric Single Well Potential. Turkish Journal of Mathematics and Computer Science, 13(1), 44-50. https://doi.org/10.47000/tjmcs.851839
AMA Başkaya E. Asymptotic Eigenvalues of Regular Sturm-Liouville Problems with Spectral Parameter-Dependent Boundary Conditions and Symmetric Single Well Potential. TJMCS. June 2021;13(1):44-50. doi:10.47000/tjmcs.851839
Chicago Başkaya, Elif. “Asymptotic Eigenvalues of Regular Sturm-Liouville Problems With Spectral Parameter-Dependent Boundary Conditions and Symmetric Single Well Potential”. Turkish Journal of Mathematics and Computer Science 13, no. 1 (June 2021): 44-50. https://doi.org/10.47000/tjmcs.851839.
EndNote Başkaya E (June 1, 2021) Asymptotic Eigenvalues of Regular Sturm-Liouville Problems with Spectral Parameter-Dependent Boundary Conditions and Symmetric Single Well Potential. Turkish Journal of Mathematics and Computer Science 13 1 44–50.
IEEE E. Başkaya, “Asymptotic Eigenvalues of Regular Sturm-Liouville Problems with Spectral Parameter-Dependent Boundary Conditions and Symmetric Single Well Potential”, TJMCS, vol. 13, no. 1, pp. 44–50, 2021, doi: 10.47000/tjmcs.851839.
ISNAD Başkaya, Elif. “Asymptotic Eigenvalues of Regular Sturm-Liouville Problems With Spectral Parameter-Dependent Boundary Conditions and Symmetric Single Well Potential”. Turkish Journal of Mathematics and Computer Science 13/1 (June 2021), 44-50. https://doi.org/10.47000/tjmcs.851839.
JAMA Başkaya E. Asymptotic Eigenvalues of Regular Sturm-Liouville Problems with Spectral Parameter-Dependent Boundary Conditions and Symmetric Single Well Potential. TJMCS. 2021;13:44–50.
MLA Başkaya, Elif. “Asymptotic Eigenvalues of Regular Sturm-Liouville Problems With Spectral Parameter-Dependent Boundary Conditions and Symmetric Single Well Potential”. Turkish Journal of Mathematics and Computer Science, vol. 13, no. 1, 2021, pp. 44-50, doi:10.47000/tjmcs.851839.
Vancouver Başkaya E. Asymptotic Eigenvalues of Regular Sturm-Liouville Problems with Spectral Parameter-Dependent Boundary Conditions and Symmetric Single Well Potential. TJMCS. 2021;13(1):44-50.