Fundamental length in quantum theories withPT-symmetric Hamiltonians. II. The case of quantum graphs

Abstract

PT −symmetrization of quantum graphs is proposed as an innovation where an adjustable, tunable nonlocality is admitted. The proposal generalizes the PT −symmetric square-well models of Ref.[1] (with real spectrum and with a variable fundamental length θ) which are reclassified as the most elementary quantum q−pointed-star graphs with minimal q = 2. Their equilateral q = 3, 4, . . . 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 λ ∈ (−1, 1). Inside this interval the set of closed-form metrics Θ (N ) j (λ) is constructed, defining independent eligible local (at j = 0) or increasingly nonlocal (at j = 1, 2, . . .) inner products in the respective physical Hilbert spaces of states H (N ) j (λ). In this way each graph is assigned a menu of non-equivalent, optional probabilistic quantum interpretations.

Quantum network approach to spin interferometry driven by Abelian and non-Abelian fields

“…The fulfillment of these boundary conditions guarantees that the resulting Schrödinger operator is Hermitian [35,36,43]. We note in passing that this condition can be relaxed to account for non-Hermiatian Hamiltonians with PT -symmetry [49]. The continuity condition implies that the wave function assumes a certain value at a vertex, regardless of the bond from which it is approached.…”

An exactly solvable quantum-lattice model with a tunable degree of nonlocality

“…One of the reasons lies in the virtually prohibitive technical obstacles [9]. In contrast, the recent transition to PT -symmetric difference Schrödinger equations has been accompanied by the comparatively quick success in finding the comparatively extensive sets of metrics compatible with a given H [10,32,34].…”

“…Lemma 4. Hamiltonians H (N) (a) of equation ( 4) may be assigned the heptadiagonal family of metrics (34). The cutoff-insensitive matrix elements of pseudometric (35) are given by the formula…”

The Coulomb potential and the paradoxes of PT symmetrization

“…As another emergent concept one should list fundamental length, i.e., a quantity θ defined, in the simplified discrete models, as the number of diagonals in the metric which is required to possess a band-matrix form, Θ (S) mn = 0 for |m − n| > θ. In this context one might mention the first papers devoted to the study of PT −symmetric quantum graphs [29] in which one might search for a connection between the fragile parts of the spectrum and the topological characteristics of the underlying graph structure.…”