Corrigendum to A Krein space approach to PT symmetry

Abstract

We comment on some inaccuracies in the above paper ([I]); they do not affect the applications to P T -symmetric Hamiltonians given therein.It is necessary to modify two assumptions. The first is required in Corollary 3.2 if the spectrum of the family A ε , 0 ≤ ε ≤ 1, does not only consist of eigenvalues. The second concerns Corollary 3.4 and Theorem 3.5, where it has to be assumed that A 0 is also self-adjoint (or, more generally, normal) with respect to some Hilbert space inner product (·,The application in Section 4 of [I] to the operator −d 2 /dx 2 + V with P Tsymmetric potential V is not affected by these alterations; in this case all the above assumptions are satisfied.For the convenience of the reader, we do not only point out the modifications, but we repeat the full correct statements of the respective theorems. We begin with Corollary 3.2 in [I] where the last sentence should be modified as follows:

Local Definitizability of $${{T^{[\ast]}T}}$$ and $${{TT^{{[\ast]}}}}$$

“…[13] as well). It was shown therein that the spectrum of −T T [ * ] consists of isolated eigenvalues and that-with possible exception of a finite number of points-these eigenvalues form a real sequence which accumulates to +∞ and alternates between eigenvalues of positive and negative type.…”

Mathematical and physical aspects of complex symmetric operators

“…Among the rigorous explanations of the reality of the spectrum of a non-selfadjoint operator, perturbation arguments play a leading role. In particular, perturbation theory in Kreȋn space was succesfully used by Langer and Tretter [102,103]. The thesis of Nesemann [114] contains A Kreȋn space is a vector space K endowed with an inner product {•, •}, such that there exists a direct sum orthogonal decomposition…”

$\mathcal{CPT}$ -Symmetric Discrete Square Well

“…In such a purely theoretical setting (as reviewed, e.g., in [3,6,12]) as well as in its very recent experimental verifications [27] it makes sense to choose V (x) in an exactly solvable and simplest possible form, say, of a square well [2,33,34,36,37,39,51] or of a point interaction [1,29] or of their perturbations [19,20]. In all of the similar non-selfadjoint models it is generically difficult to complement the "easy" results concerning energies by the "difficult" predictions of the role and physical interpretation of any other observable quantity (like, e.g., of the coordinate [26]).…”

“…In such a purely theoretical setting (as reviewed, e.g., in Refs. [3]) as well as in its very recent experimental verifications [14] it makes sense to choose V (x) in an exactly solvable and simplest possible form, say, of a square well [15] or of a point interaction [16] or of their perturbations [17].…”