In a recent (and still growing) series of papers the author
generalized the concept of the bosonic number operator in such a
manner that the underlying generalized creation and annihilation
operators are not mutually adjoint while the underlying eigenvectors
may still form a biorthogonal basis in a suitable Hilbert space. He
calls the resulting quantum dynamics ``pseudo-bosonic" and, in the
present continuation of the series, he studies a two-parametric toy
model, pointing out some of its interesting and unusual properties
including the observation that the biorthogonal basis in question is
an ``irregular", unbounded-operator map of an orthonormal basis.


MR2780946 Bagarello, F. Two-parameters pseudo-bosons. Internat. J.
Theoret. Phys. 50 (2011), no. 4, 1060--1065. 81Sxx