## The cryptohermitian smeared-coordinate representation of wave functions

Miloslav Znojil

The one-dimensional real line of coordinates is replaced, for simplification or approximation purposes, by an N-plet of the so called Gauss-Hermite grid points. These grid points are interpreted as the eigenvalues of a tridiagonal matrix $$\mathfrak{q}_0$$ which proves rather complicated. Via the “zeroth” Dyson-map $$\Omega_0$$ the “operator of position” $$\mathfrak{q}_0$$ is then further simplified into an isospectral matrix $$Q_0$$ which is found optimal for the purpose. As long as the latter matrix appears non-Hermitian it is not an observable in the manifestly “false” Hilbert space $${\cal H}^{(F)}:=\mathbb{R}^N$$. For this reason the optimal operator $$Q_0$$ is assigned the family of its isospectral avatars $$\mathfrak{h}_\alpha$$, $$\alpha=(0,)\,1,2,…$$. They are, by construction, selfadjoint in the respective $$\alpha-$$dependent image Hilbert spaces $${\cal H}^{(P)}_\alpha$$ obtained from $${\cal H}^{(F)}$$ by the respective “new” Dyson maps $$\Omega_\alpha$$. In the ultimate step of simplification, the inner product in the F-superscripted space is redefined in an {\it ad hoc}, $\alpha-$dependent manner. The resulting “simplest”, S-superscripted representations $${\cal H}^{(S)}_\alpha$$ of the eligible physical Hilbert spaces of states (offering different dynamics) then emerge as, by construction, unitary equivalent to the (i.e., indistinguishable from the) respective awkward, P-superscripted and $$\alpha-$$subscripted physical Hilbert spaces.

http://arxiv.org/abs/1107.1770
Quantum Physics (quant-ph); Mathematical Physics (math-ph)