5. International Conference on Life and Engineering Sciences, Antalya, Turkey, 19 - 22 May 2022, pp.123
We consider a matrix operator ,ܸ + (Δ(݈, ܸ) = (−ܪ in ܴௗ , ݀ ≥ 2, ଵ ଶ < ݈ < 1 , where ܸ is the multiplication operator by a symmetric ݏݔݏ matrix ܸ(ݔ (which is periodic with respect to an arbitrary lattice. It is well-known that the eigenvalues of ܪ are divided into two groups: stable (non-resonance) and unstable (resonance). In this study, we obtain the asymptotic formula for the unstable eigenvalue, more precisely, when the eigenvalue of ܪ belongs to a part of resonance domain called single resonance domain which has asymptoticaly full measure on the resonance domain. Roughly, the number of unstable eigenvalues which do not belong to the single resonance domain is small compared to the number of the ones in the single resonance domain. In [1], we obtained asymptotic formulas for the stable eigenvalues of ܪ .The unstable eigenvalues requires a detailed and carefull analysis in higher dimensions. In [2], we found the asymptotic formula for unstable eigenvalue in tems of the eigenvalues of a matrix ܥ and in this study, we give a detailed analysis of this matrix ܥ in a single resonance domain and obtain higher order asymptotics for this group of eigenvalues.
Keywords: eigenvalues, matrix potential, periodic, polyharmonic, resonance