Irregular boundary value problems revisited
The author gives a survey on results that have been obtained during the last century for strongly irregular boundary eigenvalue problems of the form \[ \begin{aligned} l(y)=y^{(n)}+\sum_{j=0}^{n-2}p_j(x)y^{(j)}=\lambda y, \;\;0\leq x\leq 1,\\ U_{\nu}(y)=U_{\nu 0}(y)+U_{\nu 1}(y)=0,\;\;1\leq \nu \leq n, \end{aligned} \] with complex-valued coefficients \(p_j\in L[0,1]\) and \[ U_{\nu 0}(y)=a_{\nu}y^{(k_{\nu})}(0)+\sum_{h=0}^{k_{\nu}-1}\alpha_{\nu h}y^{(h)}(0),\;\;U_{\nu 1}(y)=b_{\nu}y^{(k_{\nu})}(1)+\sum_{h=0}^{k_{\nu}-1}\beta_{\nu h}y^{(h)}(1), \] and \(a_{\nu}, b_{\nu}, \alpha_{\nu h}, \beta_{\nu h}\in \mathbb{C}\). He especially focuses on the difference concerning the convergence properties of the expansion of a given function into a series of eigenfunctions and associated functions of the above problems between the irregular case and the regular case. Also, several related results on some further classes of non-selfadjoint boundary value problems are introduced.
scientific article; zbMATH DE number 6108483
scientific article