Eigenvalues of Non-selfadjoint Operators: A Comparison of Two Approaches

Abstract

The central problem we 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.

Improved Eigenvalue Bounds for Schrödinger Operators with Slowly Decaying Potentials

“…Finally we should also mention that there are versions of (1.1) concerning sums of eigenvalues (e.g. [13,6,7,15,14]), but these will not be discussed here. Several works also deal with a class of potentials outside the L q -scale (e.g.…”

Some Remarks on Upper Bounds for Weierstrass Primary Factors and Their Application in Spectral Theory

“…where R p is an explicitly known constant and a n (K) denotes the nth approximation number of K. For other appearances of Γ p in eigenvalue estimates (sometimes with a different notation), see, e.g. [4,8,12,25,22,9,11,16,32,13,24,14,15,17,23]. The results from below will allow us to compute the Γ p 's numerically (apparently, this has not been done before).…”

On the critical value of the coupling constant in exterior elliptic problems

“…where the domain of H 0 consists of functions u ∈ H s+2 (Ω) satisfying the last two boundary conditions in (5). Obviously, [0, ∞) belongs to the continuous spectrum of H β = H 0 − βV (x) since one can use the same Weyl sequence for operator H β − λ, λ > 0, as the one in the case of the Dirichlet or Neumann boundary conditions.…”