Asymptotic behaviour of resonance eigenvalues of the Schrödinger operator with a matrix potential

Yıl 2020, Cilt: 69 Sayı: 1, 486 - 510, 30.06.2020

Setenay Akduman Sedef Karakılıç Didem Coşkan

https://doi.org/10.31801/cfsuasmas.577438

Öz

We will discuss the asymptotic behaviour of the eigenvalues of a Schrödinger operator with a matrix potential defined by the Neumann boundary condition in L₂^{m}(F), where F is a d-dimensional rectangle and the potential is an m×m matrix with m≥2, d≥2 , when the eigenvalues belong to the resonance domain, roughly speaking they lie near the planes of diffraction.

Anahtar Kelimeler

Schrödinger operator, Neumann condition, Resonance eigenvalue, Perturbation theory

Kaynakça

• Atılgan, Ş., Karakılıç, S. and Veliev, O. A., Asymptotic Formulas for the Eigenvalues of the Schrödinger Operator, Turk J Math., 26 (2002) 215--227.
• Berezin, F. A. and Shubin, M. A., The Schrödinger Equation, Kluwer Academic Publishers, Dordrecht, 1991.
• Coskan, D. and Karakılıç, S., High energy asymptotics for eigenvalues of the Schrödinger operator with a matrix potential, Mathematical Communications, 16(2) (2011).
• Feldman, J., Knoerrer, H. and Trubowitz, E., The Perturbatively Stable Spectrum of the Periodic Schrödinger Operator, Invent. Math., 100 (1990) 259--300.
• Feldman, J., Knoerrer, H. and E. Trubowitz, The Perturbatively Unstable Spectrum of the Periodic Schrödinger Operator, Comment. Math. Helvetica, 66(1991) 557--579.
• Friedlanger, L., On the Spectrum for the Periodic Problem for the Schrödinger Operator, Communications in Partial Differential Equations, 15(1990) 1631--1647.
• Hald, O. H. and McLaughlin, J.R., Inverse Nodal Problems: Finding the Potential from Nodal Lines, Memoirs of AMS, 572, 119 (1996) 0075--9266.
• Karakılıç, S. and Akduman, S., Eigenvalue Asymptotics for the Schrödinger Operator with a Matrix Potential in a Single Resonance Domain, Filomat, 29(1) (2015) 21--38.
• Karakılıç, S., Atılgan, Ş. and Veliev, O. A., Asymptotic Formulas for the Eigenvalues of the Schrödinger Operator with Dirichlet and Neumann Boundary Conditions, Reports on Mathematical Physics (ROMP), 55(2) (2005) 221--239.
• Karakılıç, S., Veliev, O. A. and Atılgan, Ş., Asymptotic Formulas for the Resonance Eigenvalues of the Schrödinger Operator, Turkish Journal of Mathematics, 29(4) (2005) 323--347.
• Karpeshina, Y., Perturbation Theory for the Schrödinger Operator with a non-smooth Periodic Potential, Math. USSR-Sb, 71 (1992) 701--123.
• Karpeshina, Y., Perturbation series for the Schrödinger Operator with a Periodic Potential near Planes of Diffraction, Communication in Analysis and Geometry, 4(3) (1996) 339--413.
• Karpeshina, Y., On the Spectral Properties of Periodic Polyharmonic Matrix Operators, Indian Acad. Sci. (Math. Sci.), 112(1) (2002) 117--130.
• Kato, T., Perturbation Theory for Linear Operators, Springer Berlin, 1980.
• Reed, M. and Simon, B., Methods of Modern Mathematical Physics, 3rd ed., New York, San Francisco, London: Academic Press, vol. IV 1987.
• Naimark, M. A., Dawson, E. R. and Everitt, W. N., Linear Differential Operators, Part I, Elementary Theory of Linear Differential Operators, with additional material by the author, Frederick Ungar Publ., Co., New York, 196--6 1967.
• Veliev, O. A., On the spectrum of the Schrödinger operator with periodic potential, Dokl. Akad. Nauk SSSR., Vol. 268, No. 6 (1983).
• Veliev, O. A., Asymptotic Formulas for the Eigenvalues of the Periodic Schrödinger Operator and the Bethe-Sommerfeld Conjecture, Functsional Anal. i Prilozhen, 21(2) (1987) 1--15.
• Veliev, O. A., The Spectrum of Multidimensional Periodic Operators, Teor. Functional Anal. i Prilozhen, 49 (1988) 17--34.
• Veliev, O. A., Asymptotic Formulas for the Bloch Eigenvalues Near Planes of Diffraction, Reports on Mathematical Physics(ROMP), 58(3) (2006) 445--464.
• Veliev, O. A., Perturbation Theory for the Periodic Multidimensional Schrödinger Operator and the Bethe-Sommerfeld Conjecture, International Journal of Contemporary Mathematical Sciences, 2(2) (2007) 19--87.
• Veliev, O. A., Multidimensional periodic Schrödinger operator: Perturbation theory and applications.,Springer, Vol. 263 2015.