Pseudomodes for Schrödinger operators with complex potentials

Abstract

For one-dimensional Schrödinger operators with complex-valued potentials, we construct pseudomodes corresponding to large pseudoeigenvalues. We develop a first systematic non-semi-classical approach, which results in a substantial progress in achieving optimal conditions and conclusions as well as in covering a wide class of previously inaccessible potentials, including discontinuous ones. Applications of the present results to higher-dimensional Schrödinger operators are also discussed.

Pseudospectra of the Damped Wave Equation with Unbounded Damping

“…for instance [2], due to different powers of λ, and the non-semiclassical one for Schrödinger operators, cf. [10], due to the "λ-dependent potential" q + 2λa + λ 2 . Since more than a local behavior of the damping a is required in our approach, the spirit of the performed estimates is closer to the non-semiclassical case, nonetheless, the quadratic dependence on λ in the "potential" brings new obstacles and effects.…”

“…For the forthcoming estimates, it is crucial to understand the structure of the functions ψ k and remainders r n , which is the content of the following two lemmas; detailed proofs (with minor modifications in notations) are in [10,Appendix]. (3.13) where (with some c ω ∈ C)…”

“…This paper deals with a more subtle pseudospectral analysis, which reveals highly non-normal character of the semigroup generator G, see (1.2), similarly as for Schrödinger operators with complex potentials, see e.g. [2,4,8,10]. Thus this type of the "overdamping" is responsible also for strong spectral instabilities.…”

Generalized Riesz systems and orthonormal sequences in Krein spaces

“…The operator C is called a C-symmetry operator and this notion is widely used in PT -symmetric approach in Quantum Mechanics [10]. 17) is unbounded, then we understood (17) as the identity JQf = −QJf , where f ∈ D(Q) and J leaves D(Q) invariant. From now on, we will adopt this simplifying notation.…”

“…The eigenstates of H lose the property of being Riesz basis in the original Hilbert space L 2 (R) but they still form a complete set in L 2 (R) [17,20]. Moreover, they form a sequence which is orthonormal with respect to the indefinite inner product [•, •].…”