A. A. Shkalikov

Strongly definitizable linear pencils in Hilbert space

Selfadjoint linear pencils ΛF−G are considered which have discrete spectrum and neither F nor G is definite. Several characterizations are given of a “strongly definitizable” property when F and G are bounded, and also when both operators are unbounded. The theory is applied to analysis of the stability of a linear second order initial-boundary value problem with boundary conditions dependent on the eigenvalue parameter.

Trace Formula for Sturm--Liouville Operators with Singular Potentials

AbstractSuppose that u(x) is a function of bounded variation on the closed interval [0,π], continuous at the endpoints of this interval. Then the Sturm—Liouville operator Sy=−y″+q(x) with Dirichlet boundary conditions and potential q(x)=u′(x) is well defined. (The above relation is understood in the sense of distributions.) In the paper, we prove the trace formula

A sturm-liouville problem with physical and spectral parameters in boundary conditions

\sum\limits_{k = 1}^\infty {\left( {\lambda _k^2 - k^2 + b_{2k} } \right)} = - \frac{1}{8}\sum {h_j^2 } , b_k = \frac{1}{{\pi }}\int_0^\pi cos kx du (x),

Exponential Stability of Semigroups Related to Operator Models in Mechanics

where the λk are the eigenvalues of S and hj are the jumps of the function u(x). Moreover, in the case of local continuity of q(x) at the points 0 and π the series

Strongly elliptic operators with singular coefficients

\sum\nolimits_{k = 1}^\infty {\left( {\lambda _{\,k} - k^2 } \right)}

On a Model Problem for the Orr–Sommerfeld Equation with Linear Profile

is summed by the mean-value method, and its sum is equal to