Pseudo-Hermitian β-Hermite Ensemble with an Unbound Metric

Jean-Pierre Antoine, Camillo Trapanı

Motivated by the recent developments of pseudo-Hermitian quantum mechanics, we analyze the structure of unbounded metric operators in a Hilbert space. It turns out that such operators generate a canonical lattice of Hilbert spaces, that is, the simplest case of a partial inner product space ( pip -space). Next, we introduce several generalizations of the notion of similarity between operators and explore to what extent they preserve spectral properties. Then we apply some of the previous results to operators on a particular pip -space, namely a scale of Hilbert spaces generated by a metric operator. Finally, we reformulate the notion of pseudo-Hermitian operators in the preceding formalism.

Ebru Ergün, M. Sağlam

Marcelo Losada, Ignacio S. Gomez, Federico Holik

Ali Mostafazadeh

I extend the formulation of pseudo-Hermitian quantum mechanics to η(+)-pseudo-Hermitian Hamiltonian operators H with an unbounded metric operator η(+). In particular, I give the details of the construction of the physical Hilbert space, observables and equivalent Hermitian Hamiltonian for the case that H has a real and discrete spectrum and its eigenvectors belong to the domain of η(+) and consequently √η(+).

Ali Mostafazadeh

Fabio Bagarello, Miloslav Znojil

A close parallelism between the notions of nonlinear pseudobosons and those of an apparent non-Hermiticity of observables as shown in part I (2011 J. Phys. A: Math. Theor. 44 415305) is demonstrated to survive the transition to the quantum models using an unbounded metric in the so-called standard Hilbert space of states.