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Formula for second regularized trace of a problem with spectral parameter dependent boundary condition. (English) Zbl 1259.34082
The Hilbert space valued boundary value problem $-y''(t)+Ay(t)+q(t)y(t)=\lambda y(t),\quad y(0)=0,\quad y'(\pi )-\lambda y(\pi )=0$
is represented by an operator $$L$$, where $$A$$ is a self-adjoint positive operator with $$A>E$$, $$E$$ the identity operator in the Hilbert space $$H$$, and $$q$$ is weakly measurable. For $$q=0$$, the operator $$L$$ is denoted by $$L_0$$. The eigenvalues of $$L$$ and $$L_0$$ form increasing sequences of real numbers, denoted by $$(\lambda _n)$$ and $$(\mu _n)$$, respectively. The second regularized trace is defined as $\begin{split} \lim_{m\to \infty }\left\{\sum _{n=1}^{n_m}\left(\lambda _n^2-\mu _n^2-\frac1\pi\int_0^\pi \operatorname{tr}q^2(t)\,dt\right)\right.+\\ \left.+\frac1{2\pi i}\int_{\Gamma _m}\sum _{k=2}^N\frac{(-1)^{k-1}}{k}\operatorname{tr}[(L_0Q+QL_0+Q^2)R_0(\lambda )]^k\,d\lambda \right\},\end{split}$ where $$Q=L-L_0$$, and $$R_0$$ is the resolvent of $$L_0^2$$. It is shown that, under suitable assumptions, the second regularized trace equals $-\frac{\operatorname{tr}q^2(0)}{4}- \frac{\operatorname{tr}Aq(0)+\operatorname{tr}Aq(\pi )}{2}+\frac{\operatorname{tr}q''(0)+\operatorname{tr}q''(\pi )}{8}.$

MSC:
 34L05 General spectral theory of ordinary differential operators 34B05 Linear boundary value problems for ordinary differential equations 34G10 Linear differential equations in abstract spaces 47E05 General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX) 34B09 Boundary eigenvalue problems for ordinary differential equations