Linear Differential Operators Naimark Pdf Download

We consider the Dirichlet problem for a fourth order linear differential equation with discontinuous weight function in the half space of the complex plane. We derive the Weierstrass eigenvalue asymptotic formula for this problem and investigate the conditions on the data under which the eigenvalues accumulate at the origin. The obtained results are applied for study of the Weierstrass eigenvalue asymptotic formula for a fourth order linear differential equation with transmission conditions for a regular slit on a manifold, and a reflection on a curve. Also, we derive a Weierstrass asymptotic expression for a Sturm-Liouville problem with a non-compact symmetric operator.Keywords: Weierstrass asymptotic expression; Weierstrass eigenvalue; Weierstrass conditions; Weierstrass eigenvalue asymptotic formula; Weierstrass svp; Sturm-Liouville problem; fourth order linear differential equation with discontinuous weight function; Sturm-Liouville problem with non-compact symmetric operator

Thus, the general problem of finding the spectrum of a fourth order linear differential operator with discontinuous weight function and transmission conditions can be reduced to the problem of finding the spectrum of a certain first order linear differential operator with the transmission conditions.

In the following sections, we briefly summarize some important definitions and properties of the Krein spaces [3, 4] and linear operators and linear differential expressions [5-7]. We introduce the particular case of the differential operator with discontinuous weight function and transmission conditions. We show the existence of a fundamental solution of the problem and describe its properties. These results are based on the results from the paper of N.K.

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