A. M. Savchuk

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

On the properties of maps connected with inverse Sturm-Liouville problems

\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),

Asymptotic formulas for fundamental solutions of the Dirac system with complex-valued integrable potential

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

On the interpolation of analytic mappings

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

Trace of order (−1) for a string with singular weight

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