Fundamental length in quantum theories with $\mathcal{P}\mathcal{T}$-symmetric Hamiltonians. II. The case of quantum graphs

$\mathcal{P}\mathcal{T}$-symmetrization of quantum graphs is proposed as an innovation where an adjustable, tunable nonlocality is admitted. The proposal generalizes the $\mathcal{P}\mathcal{T}$-symmetric square-well models of Ref. [M. Znojil, Phys. Rev. D 80, 045022 (2009).] (with real spectrum and with a variable fundamental length $\theta$) which are reclassified as the most elementary quantum $q$-pointed-star graphs with minimal $q=2$. Their equilateral $q=3,4,\dots$ generalizations are considered, with interactions attached to the vertices. Runge-Kutta discretization of coordinates simplifies the quantitative analysis by reducing our graphs to star-shaped lattices of $N=qK+1$ points. The resulting bound-state spectra are found real in an $N$-independent interval of couplings $\lambda \in (-1,1)$. Inside this interval the set of closed-form metrics ${\Theta }_j^{(N)}(\lambda )$ is constructed, defining independent eligible local (at $j=0$) or increasingly nonlocal (at $j=1,2,\dots$) inner products in the respective physical Hilbert spaces of states ${\mathcal{H}}_j^{(N)}(\lambda )$. In this way each graph is assigned a menu of nonequivalent, optional probabilistic quantum interpretations.

Figure 1

The spectrum of $H^{(7)}(\lambda )$.

Figure 2

The spectrum of $H^{(10)}(\lambda )$.

Figure 3

The three lowest eigenvalues of the matrix $2{\Theta }_0^{(7)}+{\mathcal{P}}_1^{(7)}$.

Figure 4

The spectrum of the metric ${\Theta }_{[3]}^{(7)}=3{\Theta }_0^{(7)}+{\mathcal{P}}_1^{(7)}$.

Figure 5

The lowest two numerical eigenvalues of metric ${\Theta }_{[5/2]}^{(7)}(\lambda )$.

Figure 6

The lowest two numerical eigenvalues of matrix ${\Theta }_{[38/16]}^{(7)}(\lambda )$.