| Popis: |
We consider a matrix operator,ܸ + (Δ(݈, ܸ) = (−ܪin ܴௗ , ݀ ≥ 2,ଵଶ< ݈ < 1 , where ܸ is the multiplication operator by a symmetric ݏݔݏ matrixܸ(ݔ (which is periodic with respect to an arbitrary lattice. It is well-known that theeigenvalues 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, moreprecisely, when the eigenvalue of ܪ belongs to a part of resonance domain called singleresonance 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 issmall compared to the number of the ones in the single resonance domain. In [1], we obtainedasymptotic formulas for the stable eigenvalues of ܪ .The unstable eigenvalues requires adetailed and carefull analysis in higher dimensions. In [2], we found the asymptotic formulafor unstable eigenvalue in tems of the eigenvalues of a matrix ܥ and in this study, we give adetailed analysis of this matrix ܥ in a single resonance domain and obtain higher orderasymptotics for this group of eigenvalues.Keywords: eigenvalues, matrix potential, periodic, polyharmonic, resonance |
