Spectral asymptotics and trace formulas for differential operators with unbounded coefficients (Q1040805)

In \(L^2(0,\pi)\), consider the differential operators \[ L=L_0+V,\qquad L_0y=-y''+(\nu^2-1/4)r^{-2}y\qquad (\nu\geq 1/2) \] with Dirichlet boundary conditions, where the generally complex-valued potential \(V\in L^2(0,\pi)\) satisfies \[ \int_0^\pi r^\varepsilon (\pi-r)^\varepsilon |V(r)|\,dr<\infty ,\qquad \varepsilon \in[0,1]. \] Let \((\lambda _n)_{n=1}^\infty \) and \((\mu_n)_{n=1}^\infty \) be the suitably enumerated eigenvalues of \(L_0\) and \(L\), respectively, where the \(\lambda _n\) are related to the zeros of the Bessel function \(J_\nu\) via \(J_\nu(\sqrt \lambda \pi)=0\). It is shown that \[ \mu_n=\lambda _n+\sum_{k=1}^\infty \alpha _k^{(n)}, \] where \[ \alpha _k^{(n)}=\frac{(-1)^{k+1}}{2\pi i}\oint_{|z-\lambda |=n/(2\pi a_0)} (z-\lambda _n)\text{tr}[R_0(z)V]^kR_0(z)\,dz \] with \(R_0(z)=(L_0-z)^{-1}\). If \(0\leq \varepsilon _1\) and \(m\) is the smallest positive integer such that \(m>(1+\varepsilon )(1-\varepsilon )^{-1}\), it is shown that \[ \sum_{n=1}^\infty \left[\mu_n-\lambda _n-\sum_{k=1}^m\alpha _k^{(n)}\right]=0. \]