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Trace of order \((-1)\) for a string with singular weight. (English. Russian original) Zbl 1384.34027

Math. Notes 102, No. 2, 164-180 (2017); translation from Mat. Zametki 102, No. 2, 197-215 (2017).
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.

MSC:

34B09 Boundary eigenvalue problems for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
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