Exact gaussian states under quadratic non-hermitian deformations

We develop an exact deformation calculus for single-mode Hermite--Gaussian states under the quadratic analytic map \[ T_φ=exp\{2φ(∂_x)\}, φ(z)=c_1+c_2z+c_3z^2 . \] 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 \[ H_φ=T_φ⁻¹HT_φ, \] the physical inner product is generated by \(M_φ=T_φ^ T_φ\), time evolution is governed by the Schr"odinger equation with \(H_φ\), 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 \(κ=eiθ r\), \(Δ_φ=1-κ+8c_3κ\), and \(Γ_φ=1+κ+8c_3κ\), 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.