Spectral inclusion and spectral pollution results are proved for sequences of linear operators of the form T 0 + iγsn on a Hilbert space, where sn is strongly convergent to the identity operator and γ > 0. We work in both an abstract setting and a more concrete Sturm-Liouville framework. The results provide rigorous justification for a method of computing eigenvalues in spectral gaps.

We consider Schrödinger operators of the form $$H_R = - \,\text {{d}}^2/\,\text {{d}}x^2 + q + i \gamma \chi _{[0,R]}$$ H R = - d 2 / d x 2 + q + i γ χ [ 0 , R ] for large $$R>0$$ R > 0 , where $$q \in L^1(0,\infty )$$ q ∈ L 1 ( 0 , ∞ ) and $$\gamma > 0$$ γ > 0 . Bounds for the maximum magnitude of an eigenvalue and for the number of eigenvalues are proved. These bounds complement existing general bounds applied to this system, for sufficiently large R.

No abstract

We prove a general Mosco convergence theorem for bounded Euclidean domains satisfying a set of mild geometric hypotheses. For bounded domains, this notion implies norm-resolvent convergence for the Dirichlet Laplacian which in turn ensures spectral convergence. A key element of the proof is the development of a novel, explicit Poincaré-type inequality. These results allow us to construct a universal algorithm capable of computing the eigenvalues of the Dirichlet Laplacian on a wide class of rough domains. Many domains with fractal boundaries, such as the Koch snowflake and certain filled Julia sets, are included among this class. Conversely, we construct a counterexample showing that there does not exist a universal algorithm of the same type capable of computing the eigenvalues of the Dirichlet Laplacian on an arbitrary bounded domain.

We consider Schrödinger operators of the form H R = − d 2 / dx 2 + q + iγχ [0,R] for large R > 0, where q ∈ L 1 (0, ∞) and γ > 0. Bounds for the maximum magnitude of an eigenvalue and for the number of eigenvalues are proved. These bounds complement existing general bounds applied to this system, for sufficiently large R.