On the Asymptotic Behaviour of the Unstable Bloch Eigenvalues of a Polyharmonic Matrix Operator

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Karakılıç S., Özcan S., Akduman S.

ICOMAA 2024, İstanbul, Türkiye, 8 - 11 Mayıs 2024, ss.1

  • Yayın Türü: Bildiri / Özet Bildiri
  • Basıldığı Şehir: İstanbul
  • Basıldığı Ülke: Türkiye
  • Sayfa Sayıları: ss.1
  • Dokuz Eylül Üniversitesi Adresli: Evet

Özet

We explore the asymptotic behaviour of the so-called unstable Bloch eigenvalues of the Polyharmonic matrix operator  with , in the single resonance domain which is a subset of resonance domain– the set of eigenvalues situated close to the diffraction hyperplanes. The single resonance domain approaches full measure asymptotically across the entire resonance domain. In our analysis, we discover a significant trend: as energy levels increase, the eigenvalues are related to those of a Sturm-Liouville operator. Our methodology builds upon perturbation theoretic techniques developed by Veliev, which is presented in [1].

Keywords: Perturbation theory, system of poyharmonic operators, eigenvalue, asymptotic, resonance domain.

References:

1.      Veliev, O. A., Multidimensional periodic Schrödinger operator: Perturbation

theory and applications, Springer, Vol. 263 (2015).

2.      Karakılıç, S. and Akduman, S., On the eigenvalues of a polyharmonic

matrix operator near diffraction planes, AIP conference Proceedings 2321,

030017 (2021); https://doi.org/10.1063/5.0040407.

3.      Karakılıç, S., Perturbation of the Non-Resonance Eigenvalue of a

Polyharmonic Matrix Operator, DEU FMD, 22(66) (2020) 725-733;

https://doi.org/10.21205/deufmd.2020226607.

4.      Veliev, O. A., On the spectrum of the Schrödinger operator with periodic

potential, Dokl. Akad. Nauk SSSR., Vol. 268, No. 6 (1983).