An energy spectrum $\{E_n\}$ is assumed prescribed by the
self-adjoint Schr\"{o}dinger-type Hamiltonian $\tilde{h}$ defined,
on the real line of $x$, by Eq.~(8). As long as it contains an
optional mass-function $A(x)$, the change of variables (13) is
performed replacing $x \to z(x)$. In a way known from Ref.~[11] this
transforms the initial eigenvalue problem into the exactly solvable
confluent hypergeometric differential Schr\"{o}dinger Eq.~(17).
Naturally, the original Dirichlet boundary conditions imposed at $x
= \pm  \infty$ are translated into the ``new" boundary conditions
imposed at $z_\pm = z(\pm  \infty)$. In Tables 1 and 2 a few
examples are listed. In the former cases the original model (with
$E_n \sim n$ since $(z_-,z_+)= (-\infty,\infty)$) is, by
construction, isospectral to harmonic oscillator. In the latter
cases (with the choice of $A(x)\equiv 1/\sqrt{m(x)}$ such that
$(z_-,z_+)< (-\infty,\infty)$ and, hence, $E_n \sim n^2$, this
isospectrality is broken.

In section 1 (NOT recommended to be read first) the authors
emphasize that the warning ``be careful" as mediated by Table~2 is
particularly important in the context of the recent growth of
popularity of such representations of observables in quantum
mechanics (as admirably reviewed, e.g., by Ref.~[6]) which the
present reviewer would recommend calling cryptohermitian (cf. also
MZ, Three-Hilbert-space formulation of Quantum Mechanics, SIGMA 5
(2009), 001, for the reasons).


MR2763430 Midya, Bikashkali; Dube, P. P.; Roychoudhury, Rajkumar
Non-isospectrality of the generalized Swanson Hamiltonian and
harmonic oscillator. J. Phys. A 44 (2011), no. 6, 062001, 8 pp.
81Qxx