Trace of order \((-1)\) for a string with singular weight
The spectral theory of the Sturm-Liouville equation \[ (Ly :=) \, -y'' + q(x) y = \lambda \rho(x) y \, (=: \lambda Vy) \] on \([0, \pi]\) with self-adjoint (separated) boundary conditions is well studied for a long time if \(q\) is a real and \(\rho\) is a positive function. In the present paper this equation is generalized to a setting where \(q\) is a real and \(\rho\) a complex (not necessarily positive) distribution in the space \(W_2^{-1}[0, \pi]\). After a precise definition of \(L\) and \(V\) it is observed that \(L^{-1}V\) is a compact operator in \({\mathcal H}_1\), i.e. in the Sobolev space \(W_2^{1}[0, \pi]\) restricted by the essential boundary conditions. As the first main result it is shown that the s-numbers of \(L^{-1}V\) satisfy the estimates \(\Sigma_{j=1}^{n}s_j \leq C \ln n\) and \(s_n \leq C \frac{\ln n}{n}\) and the same estimates hold true for \(|\lambda_n|^{-1}\) where \(\lambda_n\) are the eigenvalues of the operator pencil \(L - \lambda V\). Such estimates are modified for a setting where the so-called generalized antiderivative of \(\rho\) belongs to \(W_2^\theta[0,\pi]\) with \(\theta \in [0,1]\). In the next step it is assumed that \(L^{-1}V\) is a trace class operator. Then, it is shown that the series \(\Sigma_{n=1}^{\infty} \lambda_n^{-1}\) converges absolutely and this trace of order \(-1\) can be expressed in terms of \(\rho\) and the integral kernel of \(L^{-1}\). Finally, such traces are calculated explicitly for some examples.
scientific article; zbMATH DE number 6822727
scientific article