Eigenvalues of non-selfadjoint operators: a comparison of two approaches. (English) Zbl 1280.47005

Demuth, Michael; Hansmann, Marcel; Katriel, Guy

Demuth, Michael (ed.) et al., Mathematical physics, spectral theory and stochastic analysis. Basel: Birkhäuser/Springer (ISBN 978-3-0348-0590-2/hbk; 978-3-0348-0591-9/ebook). Operator Theory: Advances and Applications 232. Advances in Partial Differential Equations, 107-163 (2013).

The central problem which the authors consider is the distribution of eigenvalues of closed linear operators which are not selfadjoint, with a focus on those operators which are obtained as perturbations of selfadjoint linear operators. Two methods are explained and elaborated. One approach uses complex analysis to study a holomorphic function whose zeros can be identified with the eigenvalues of the linear operator. The second method is an operator theoretic approach involving the numerical range. General results obtained by the two methods are derived and compared. Applications to non-selfadjoint Jacobi and Schrödinger operators are considered. Some possible directions for future research are discussed.
For the entire collection see [Zbl 1264.00036].

Reviewer: Bilender P. Allahverdiev (Isparta)

MSC:

47A10	Spectrum, resolvent
47A75	Eigenvalue problems for linear operators
47B36	Jacobi (tridiagonal) operators (matrices) and generalizations
81Q12	Nonselfadjoint operator theory in quantum theory including creation and destruction operators

Keywords:

non-selfadjoint operators; eigenvalues; Schrödinger operators; Jacobi operators; complex potentials