Asymptotic behavior of the spectra of second-order non-self-adjoint systems of differential operators

Abstract

i. Let J = (0, i), H = L2(J), ~ s [0, 2); n > 0 is an integer.In the space H n = L=(J) n, we consider the differential operator with matrix coefficients2. The distribution of the eigenvalues (e.va.) of the operator P under Dirichlet-type boundary conditions have been investigated in [i].In [i] one has assumed that the e.va. of the matrix A(t) (t e 7) vary on the fixed rays Let p1(t), ..., pv(t) denote a complete collection of positive e.va. of the matrix A(t), indexed in nondecreasing order; let N(~) =card{i: I~:I~<X}.

Spectral properties of degenerate elliptic operators with matrix coefficients

“…Analogous result for the second order differential operators was established in [4,13]. However we note that the scheme of works [4,13] is not applicable for the case m > 1 even if condition (1.4) is satisfied.…”

“…However we note that the scheme of works [4,13] is not applicable for the case m > 1 even if condition (1.4) is satisfied. An significant moment of methods of current work is that we "single out" in an explicit form a principle part of "generalized resolvent" as operator acting from…”

“…Obtained asymptotic formulas here even for the case m = 1 related to more general class of operators then those in works [4,13].…”

“…and p.d.o. that are far from selfadjoint were studied in [3,4,[13][14][15][16][17][18]; only [3,4,13] are devoted to degenerate elliptic problems.…”