Formula of regularized traces (Q416917)

Consider the operator \(\mathbb L\) generated by the \(2m\) order differential expression \(ly \equiv (-1)^mD^{2m}y+\sum^{2m-2}_{k=0}p_k(x)D^ky\) and by a boundary condition (a system of equations). Let \(\mathbb Q\) be the operator of multiplication by a real function \(q\in L_{\infty}({\mathbb{R}}_+)\). Assume that \(\mathbb L\) is self-adjoint in \(L_2({\mathbb{R}}_+)\), semibounded from below, and has a purely discrete spectrum \(\{\lambda_n\}^{+\infty}_{n=1}\). Then \(\mathbb L +\mathbb{Q}\) also has a purely discrete spectrum \(\{\mu_n\}^{+\infty}_{n=1}\). In this paper, under the assumptions that \(q\) has bounded support and \(\psi(x) = \frac{1}{x}\int^x_0q(t)\,dt\) has bounded variation at zero, it is proved that \(\sum^{\infty}_{n=1}[\mu_n-\lambda_n-\frac{c_n}{\pi}\int^{\infty}_0q(t)\,dt] = -\psi(0+) (\frac{m}{2}-\frac{1}{4}-\frac{\chi}{2m})\), where the \(c_n\) are determined by the \(\lambda_n\) and \(\chi\) by the degree of polynomials in the boundary condition.