Exact gaussian states under quadratic non-hermitian deformations

Source abstract

Abstract We develop an exact deformation calculus for single-mode
Hermite--Gaussian states under the quadratic analytic map
\[
T_\phi=\exp\{2\phi(\partial_x)\},
\qquad
\phi(z)=c_1+c_2z+c_3z^2 .
\]
The construction is not proposed as a general replacement for
non-Hermitian quantum mechanics; rather, it provides an exactly
solvable sector within the standard metric/quasi-Hermitian framework.
For the literal similarity realization
\[
H_\phi=T_\phi^{-1}HT_\phi,
\]
the physical inner product is generated by
$M_\phi=T_\phi^\dagger T_\phi$, time evolution is governed by the
Schr"odinger equation with $H_\phi$, and physical observables are
self-adjoint with respect to the metric inner product.

Separately, the deformation-adapted Hermite--Gaussian realization
leads to closed coordinate- and Fourier-space expressions for
coherent, squeezed, and squeezed-coherent states. Using the Hermite
generating function and Mehler kernel, we obtain explicit Gaussian
formulae controlled by
$\kappa=e^{i\theta}\tanh r$,
$\Delta_\phi=1-\kappa+8c_3\kappa$, and
$\Gamma_\phi=1+\kappa+8c_3\kappa$, together with their Wigner
profiles. In these dimensionless coefficients, $c_3$ is understood
in oscillator-length units, as specified in the main text. The pure
quadratic sector gives an exactly solvable benchmark for non-unitary
Gaussian filtering and effective resolution broadening in
continuous-variable optical models.