Double completeness of the system of eigenfunctions and associated functions of the regge problem

On spectra of a certain class of quadratic operator pencils with one-dimensional linear part

“…However, this has already been proved [see (2.21)]. [36][37][38][39][40][41]. This problem occurs in scattering theory when the potential is supposed to have a finite support: Here, λ is a spectral parameter and the potential q is real-valued and belongs to L 2 (0, a).…”

“…The so-called Regge problem (see [36][37][38][39][40][41] etc.) can also be reduced to spectral problems for such pencils.…”

Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and their Applications

“…The results in Section 2.1 are taken from [186]. For further aspects of the theory related to the Regge problem we refer the reader to [122], [115] [255], [256], [150], [142], [243], [148] and [251]. In particular, statement 4 of Theorem 2.1.2 was obtained in [251,Theorem 6].…”

“…Under the above assumption on the behaviour of the potential at a, [150] also obtained two-fold series expansions into eigenfunctions and associated functions of the Regge problem for certain classes of sufficiently smooth functions. It was shown in [142] that the condition that the potential q satisfies a ∈ supp q is necessary for two-fold completeness in L 2 (0, a). Also sufficient conditions for two-fold completeness were given in [142].…”

“…It was shown in [142] that the condition that the potential q satisfies a ∈ supp q is necessary for two-fold completeness in L 2 (0, a). Also sufficient conditions for two-fold completeness were given in [142]. More recently, it was shown in [248,Theorem 3] that under the assumption that a ∈ supp q and that all of its eigenvalues λ n , n ∈ Z are simple with eigenfunctions y n , then the system {(y n , λ n y n ) : n ∈ Z} is complete and minimal in W exp((a − x) −α )| is bounded and bounded away from 0 as x → a − 0 for some α > 0, an asymptotic formula for the eigenvalues can be found in [243,III.4].…”