Feasibility and method of multi-step Hermitization of crypto-Hermitian quantum Hamiltonians

Abstract: In the popular

c a l P T -

symmetry-based formulation of quantum mechanics of closed systems one can build unitary models using non-Hermitian Hamiltonians (i.e.,

H e q H^{d} a g g e r

) which are Hermitizable (so that one can write, simultaneously,

H = H^{d} d a g g e r

). The essence of the trick is that the reference Hilbert space

c a l R

(in which we use the conventional inner product

l a n g l e p s i_{a} | p s i_{b} a n g l e

and write

H e q H^{d} a g g e r

) is declared unphysical. The necessary Hermiticity of the Hamiltonian

H = H^{d} d a g g e r

can be then achieved by the mere metric-mediated amendment

l a n g l e p s i_{a} | T h e t a | p s i_{b} a n g l e

to the inner product. This converts

c a l R

into a correct physical Hilbert space

c a l H

. The feasibility of the construction is based on a factorization postulate

T h e t a = c a l P C

where, usually,

c a l P

is parity and

c a l C

is charge. In our paper we propose a more general factorization recipe in which one constructs

T h e t a = Z_{N} Z_{N - 1} l d o t s Z_{1}

, at any

N

, in terms of suitable auxiliary pre-metric operators

Z_{k}

.

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