Differential operators admitting various rates of spectral projection growth

Abstract

Abstract. We consider families of non-self-adjoint perturbations of the selfadjoint Schrödinger operators with single-well potentials. The norms of spectral projections of these operators are found to grow at intermediate rates from arbitrarily slowly to exponentially rapidly.

On Dual Definite Subspaces in Krein Space

“…Consider the sequence f n (x) = e p(x) g n (x), where a real-valued odd function p ∈ C 2 (R) satisfies Assumption II in [14]. The sequence { f n } is complete in L 2 (R) [14,Lemma 3.6] and f n are simple eigenfunctions of the P-symmetric operator…”

“…The sequence { f n } is complete in L 2 (R) [14,Lemma 3.6] and f n are simple eigenfunctions of the P-symmetric operator…”

A Bound on the Pseudospectrum for a Class of Non-normal Schrödinger Operators

“…Notice that the strategy of the proof of Theorem 4.1 also applies to more general operators for which the norms of the spectral projections are known such as those considered in [Hen14a,MSV13] …”

Towards Generalized Riesz Systems Theory

“…This interest grew considerably due to the recent progress in theoretical physics of PT -symmetric (pseudo-Hermitian) Hamiltonians [10,11,24]. Studies of pseudo-Hermitian operators carried out in [22,23,27] show that, even if the eigenvalues of a Hamiltonian are real, the Riesz basis property of its eigenstates is, in many cases, lost.…”

“…The tameness of eigenstates of non self-adjoint operators H with a purely discrete real spectrum allows one to discover additional properties of H. In particular, a polynomially bounded behavior of the corresponding resolvent was established in [13,Theorem 3]. However, in major part, eigenstates of pseudo-Hermitian Hamiltonians form wild systems that are much more complicated for the investigation [13,22,23].…”