Zur Vollständigkeit des Systems der Eigenfunktionen irregulärer Eigenwertprobleme mit \(\lambda\)-abhängigen Randbedingungen
We consider irregular eigenvalue problems \[ (1)\quad y^{(n)}+\sum^{n}_{\nu =2}p_{\nu}(x)y^{(n-\nu)}=\lambda y, \] \[ (2)\quad \sum^{n-1}_{j=0}\alpha_{\nu j}(\lambda)y^{(j)}(0)=0,\quad 1\leq \nu \leq m>n-m, \] \[ \sum^{n- 1}_{j=1}[\alpha_{\nu j}(\lambda)y^{(j)}(0)+\beta_{\nu j}(\lambda)y^{(j)}(1)]=0,\quad m+1\leq \nu \leq n, \] with polynomials \(\alpha_{\nu j}\) and \(\beta_{\nu j}\) and summable functions \(p_{\nu}\). Eigenvalue problems of this type are irregular - that means that the Green's function of these problems is growing exponentially for \(| \lambda | \to \infty.\) Assuming that the boundary conditions (2) have different weights, we give asymptotic estimates for the eigenvalues of problem (1),(2), and we prove that the system of eigenfunctions and associated functions of (1), (2) is complete in \(L_ 2[0,1]\).
scientific article; zbMATH DE number 3851525