Quantum star-graph analogues of PT-symmetric square wells

We recall the solvable PT-symmetric quantum square well on an interval of x ∈ (-L, L):= [G.sup.(2)] (with an α-dependent non-Hermiticity given by Robin boundary conditions) and generalize it. In essence, we just replace the support interval [G.sup.(2)] (reinterpreted as an equilateral two-pointed star graph with Kirchhoff matching at the vertex x = 0) with a q-pointed equilateral star graph [G.sup.(q)] endowed with the simplest complex-rotation-symmetric external α-dependent Robin boundary conditions. The remarkably compact form of the secular determinant is then deduced. Its analysis reveals that (i) at any integer q = 2, 3,..., there exists the same q-independent and infinite subfamily of the real energies, and (ii)at any special q = 2, 6, 10,..., there exists another, additional, q-dependent infinite subfamily of the real energies. In the spirit of the recently proposed dynamical construction of the Hilbert space of a quantum system, the physical bound-state interpretation of these eigenvalues is finally proposed. PACS Nos: 03.65.Ca, 03.65.Db, 03.65.Ta, 03.70.+k Nous prenons un potentiel quantique carre a symetrie PT sur un intervalle x [elemento de] (-L, L):= [G.sup.(2)] (non hermitique et dependant de [alfa] a cause des conditions aux limites de Robin) et nous le generalisons. Globalement, nous remplacons simplement l'intervalle de support support [G.sup.(2)] (reinterprete comme un graphe etoile a deux branches avec couplage de Kirchoff au vertex x = 0), par un graphe etoile equilateral a q branches [G.sup.(q)], dote des conditions aux limites externes de Robin dependantes de [alfa] a symetrie de rotation complexe la plus simple. Nous en obtenons la forme remarquablement compacte du determinant seculaire. Son analyse revele que : (i) pour tout entier q = 2,3,..., il existe la meme sous-famille infinie d'energies propres reelles independantes de q et (ii)qu'a toute valeur entiere particuliere, q = 2, 6, 10,..., il existe une sousfamille infinie additionnelle d' energies propres reelles qui dependent de q. Dans l'optique de propositions recentes pour construire dynamiquement l' espace de Hilbert d' un systeme quantique, nous avancons une interpretation en termes d' etats lies physiques pour ces energies propres. [Traduit par la Redaction]
1. Introduction The heuristic use of the concept of PT symmetry, that is, of the parity-times-time-reversal symmetry of quantum Hamiltonians H and (or) of the toy-model wave functions ψ(x) (with, [...]