On the metric operator for the imaginary cubic oscillator

Abstract

We show that the eigenvectors of the PT-symmetric imaginary cubic oscillator are complete, but do not form a Riesz basis. This results in the existence of a bounded metric operator having intrinsic singularity reflected in the inevitable unboundedness of the inverse. Moreover, the existence of non-trivial pseudospectrum is observed. In other words, there is no quantum-mechanical Hamiltonian associated with it via bounded and boundedly invertible similarity transformations. These results open new directions in physical interpretation of PT-symmetric models with intrinsically singular metric, since their properties are essentially different with respect to self-adjoint Hamiltonians, for instance, due to spectral instabilities.Comment: 7 pages; completely rewritten, new result

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

“…More precisely, we will prove a bound on the pseudospectrum of the operator H = −∆+V , where Re V (x) ≥ c|x| 2 −d for some c, d > 0 on L 2 (R n ), which complements the results of [SK12,Nov14].…”

“…Let us compare Theorem 4.1 to the results of [SK12] . As noted in the introduction, it was shown there that for every δ > 0 there exist constants C 1 , C 2 > 0 such that for all ε > 0…”

“…For c = 0 it was shown by Krejčiřík and Siegl [SK12] that the pseudospectrum of H c always contains an unbounded component. More precisely, they showed that for every δ > 0 there exist constants C 1 , C 2 > 0 such that for all ε > 0…”

“…As in the previous section we want to estimate e −tH B | Ran(I−Pm) for m ∈ N. We know that the eigenfunctions of H B form a complete set in L 2 (R) and the algebraic eigenspaces are one-dimensional [SK12,Tai06]. Thus, we can use Lemma 3.1 of [Dav05]:…”

Hermitian–Non-Hermitian Interfaces in Quantum Theory

“…Certainly, the latter family is not small. Pars pro toto it contains Hamiltonians of relativistic quantum mechanics [41,42], the well-known P -symmetric imaginary cubic oscillator [43][44][45][46] (which appears, after a more detailed scrutiny, strongly nonlocal [31,47]), its power-law generalizations [10][11][12]48] as well as exactly solvable models [49][50][51][52], models with methodical relevance in the context of supersymmetry [53,54], realistic and computation-friendly interacting-boson models of heavy nuclei [2], benchmark candidates for classification of quantum catastrophes [55][56][57], and so forth.…”