On the singular Rayleigh functional

Abstract

The present note is devoted to the study of spectral properties of the operator function L(A) = A(A)-I,where A(A) is a continuous self-adjoint compact operator function in a Hilbert space 7-/with inner product (*, ,) and I is the identity operator on ~. Moreover, we assume that the following condition holds:for any nonzero vector u E "H, the functionincreases in a neighborhood of any of its zeros (if they exist). Clearly, under this condition the function f~ can have at most one zero (possibly multiple).The spectral properties of self-adjoint operator functions with property (2) were studied by means of the Rayleigh functional in [1-3]. However, we note that in these papers the Rayleigh functional was defined under the following assumptions: first, for any nonzero u E 7"i, the function f~ has a zero p(u) e (a, b); second, either the operator function L(A) is continuously differentiable and the zero p(u) of the functionThe motivation for studying operator functions of the form (1)-(2) is explained in [4,5], in which the asymptotic behavior as e --. 0 of the spectrum of the Neumann problem in thick singularly degenerate unions depending on a small parameter ~ was investigated. In these papers it was shown that the limit (~ = 0) spectral problem is equivalent to the spectral problem for the operator function S(A)=(A2-F1)A § AE(Trn-r/2,~rn+r/2), hEN, where A and B are self-adjoint compact operators on ~, A > 0, and B > 0. We can also reduce the quadratic operator pencil T(A) = A21 + A(aA + B) + A + C, considered in [6], to an operator function of the form (1); here A = A* >> 0, the inverse operator A-' is compact, and the operators B and C are symmetric and A is compact. The spectra of the pencils T(,~) and TI(A) = -(I + coincide, and the pencil TI(A) on the interval (a3,-l/a) satisfies conditions (1)-(2).We can readily see that the methods of [1-3] cannot be applied to the operator functions S(A) and T(A), and the method suggested in [4,5,7] is not general for the operator functions of the form (1)-(2).In the present note we construct a singular Rayleigh functional and apply it to obtain the following results:1) the spectrum of the operator function L(A) on (a, b) is discrete; 2) the only possible accumulation point of the eigenvalues is the point A --b; we obtain a criterion for accumulation; 3) variational principles are proved for the eigenvalues.

Asymptotic analysis of a spectral problem in a periodic thick junction of type 3:2:1

“…From results on existence and concentration of the spectrum of self-adjoint discontinuous operator-functions that were proved in [8,15,21] it follows the following theorem.…”

“…m!k"1), then we can reduce the resulting spectral problem to a spectral problem for some discontinuous self-adjoint operator-function. The spectrum of such operator-functions was studied in the papers [8,15,21].The type of the junction de"nes also the construction scheme of an extension operator P C : H( C )PH( 6D \ ), which plays an important role in the proofs of the convergence theorems. For the domains of the type under consideration, there exists 322 T. A. Mel'nyk…”

“…m!k"1), then we can reduce the resulting spectral problem to a spectral problem for some discontinuous self-adjoint operator-function. The spectrum of such operator-functions was studied in the papers [8,15,21].…”

On the spectrum of a weak class of operator pencils of waveguide type

“…In this case the situation is considerably different from the cases considered by the other authors. A subclass of one parameter quadratic operator pencils of the form A(k) and similar spectral problems were considered in [6] and [7], where some shifted (on the dimension of subspaces) variational principles were established (see [7,Theorem 3]). The pencils considered here are the limit case (ε = 0) of the spectral problems depending on a parameter ε, investigated in [11] and [12].…”

Eigenvalues of Two Parameter Polynomial Operator Pencils of Waveguide Type

“…More general operator functions defined on the interval [α, β[⊂ R with values in the class of bounded self-adjoint operators were studied in [5], where the classical conditions given below for overdamped spectral problems in general are not satisfied (see also [9]). These conditions are:…”