Irregular eigenvalue-problems on the half-axis
Singular differential operators \(L:D\to L^ 2({\mathbb{R}}^+_ 0)\) generated by the differential expression \(\ell y:=y^{(n)}+\sum^{n}_{k=2}p_ ky^{(n-k)}\) and boundary conditions \(U_ i\), \(1\leq i\leq m\) for \(D=\{y| y\in C^{n-1}({\mathbb{R}}^+_ 0)\cap L^ 2({\mathbb{R}}^+_ 0),y^{(n-1)}\) a.c., \(U_ iy=0\), \(1\leq i\leq m\), \(\ell y\in L^ 2({\mathbb{R}}^+_ 0)\}\) are investigated where \(m,n/2\leq m\leq n-1\), is an integer. For compact intervals the resp. operators are called irregular and have been considered by Hromov, Freiling and the author. For the singular case is proved that the resolvent of L is an integral operator, but not being bounded resp. the eigenvalue-parameter \(\lambda\). An analogon of the expansion-formula of Weyl-Titchmarsh is derived for a certain well-defined class of analytic functions.
scientific article; zbMATH DE number 4067363
scientific article