On a new method for solving irregular problems
The author deals with the Regge problem \[ L(\lambda, D)u:= \lambda^2u(x)- u''(x)+ Q(x) u(x)= f(x),\quad x\in (0,1),\tag{1} \] \[ L_1(\lambda) u:=\lambda u(1)+ u'(1)= f_1,\quad L_2u:= u(0)= f_2,\tag{2} \] where \(f(x)\) and \(Q(x)\) are given complex-valued functions, \(f_1\), \(f_2\) are given complex numbers. This problem arises in scattering theory and it is irregular in the sense of Birkhoff. It is shown that (1), (2) is coercive inside the right half-plane and it is coercive with a defect inside the left half-plane. In the papers by \textit{T. Regge} [Nuovo Cimento, X. Ser. 8, 671-679 (1958; Zbl 0080.41903) and 9, 491-503 (1958; Zbl 0081.43002)] completeness of eigenfunctions and asymptotics of eigenvalues are established in the case of the real-valued potential \(Q(x)\). Here, the author suggests a new approach to the investigation of the problem (1), (2). He presents algebraic conditions for coercive solvability with a defect with respect to the spectral parameter. The twofold completeness of root functions of the Regge problem is proven.
scientific article; zbMATH DE number 1215720
scientific article