The cryptohermitian smeared-coordinate representation of wave functions

Abstract

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 ${\mathcal{H}}^{(F)}:={\mathbb{R}}^N$. 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 ${\mathcal{H}}_{\alpha }^{(P)}$ obtained from ${\mathcal{H}}^{(F)}$ 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 ${\mathcal{H}}_{\alpha }^{(S)}$ 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.

13. pp, 3 figs

References in corpus (4)

• Making Sense of Non-Hermitian Hamiltonians
• Cryptogauge symmetry and cryptoghosts for crypto-Hermitian Hamiltonians
• Energy-levels crossing and radial Dirac equation: Supersymmetry and quasi-parity spectral signatures
• Cryptohermitian Hamiltonians on graphs

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