Abstract A family of exactly solvable quantum square wells with discrete coordinates and with certain non-stationary Hermiticity-violating Robin boundary conditions is proposed and studied. Manifest non-Hermiticity of the model in conventional Hilbert space H friendly is required to coexist with the unitarity of system in another, ad hoc Hilbert space H physical . Thus, quantum mechanics in its non-Hermitian interaction picture (NIP) representation is to be used. We must construct the time-dependent states (say, ψ(t)) as well as the time-dependent observables (say, Λ(t)). Their evolution in time is generated by the operators denoted, here, by the respective symbols G(t) (a Schrödinger-equation generator) and Σ(t) (a Heisenberg-equation generator, a.k.a. quantum Coriolis force). The unitarity of evolution in H physical is then guaranteed by the reality of spectrum of the energy observable alias Hamiltonian H(t) = G(t) + Σ(t). The applicability of these ideas is illustrated via an N by N matrix model. At N = 2, closed formulae are presented not only for the measurable instantaneous energy spectrum but also for all of the eligible time-dependent physical inner-product metrics Θ(N=2)(t), for the related Dyson maps Ω(N=2)(t), for the Coriolis force Σ(N=2)(t) as well as, in the very ultimate step of the construction, for the truly nontrivial Schrödinger-equation generator G (N=2)(t).
Abstract A family of exactly solvable quantum square wells with discrete coordinates and with certain non-stationary Hermiticity-violating Robin boundary conditions is proposed and studied. Manifest non-Hermiticity of the model in conventional Hilbert space H friendly is required to coexist with the unitarity of system in another, ad hoc Hilbert space H physical . Thus, quantum mechanics in its non-Hermitian interaction picture (NIP) representation is to be used. We must construct the time-dependent states (say, ψ(t)) as well as the time-dependent observables (say, Λ(t)). Their evolution in time is generated by the operators denoted, here, by the respective symbols G(t) (a Schrödinger-equation generator) and Σ(t) (a Heisenberg-equation generator, a.k.a. quantum Coriolis force). The unitarity of evolution in H physical is then guaranteed by the reality of spectrum of the energy observable alias Hamiltonian H(t) = G(t) + Σ(t). The applicability of these ideas is illustrated via an N by N matrix model. At N = 2, closed formulae are presented not only for the measurable instantaneous energy spectrum but also for all of the eligible time-dependent physical inner-product metrics Θ(N=2)(t), for the related Dyson maps Ω(N=2)(t), for the Coriolis force Σ(N=2)(t) as well as, in the very ultimate step of the construction, for the truly nontrivial Schrödinger-equation generator G (N=2)(t).
A new Hermitizable quantum model is proposed in which the bound-state energies are real and given as roots of an elementary trigonometric expression while the wave function components are expressed as superpositions of two Chebyshev polynomials. As an -site lattice version of square well with complex Robin-type two-parametric boundary conditions the model is unitary with respect to the Hilbert space metric Θ which becomes equal to the most common Diracʼs metric in the conventional textbook Hermitian–Hamiltonian limit. This metric is constructed in closed form at all .
In the conventional (so-called Schrödinger-picture) formulation of quantum theory the operators of observables are chosen self-adjoint and time-independent. In the recent innovation of the theory, the operators can be not only non-Hermitian but also time-dependent. The formalism (called non-Hermitian interaction-picture, NIP) requires a separate description of the evolution of the time-dependent states ψ(t) (using Schrödinger-type equations) as well as of the time-dependent observables Λj(t), j=1,2,…,K (using Heisenberg-type equations). In the unitary-evolution dynamical regime of our interest, both of the respective generators of the evolution (viz., in our notation, the Schrödingerian generator G(t) and the Heisenbergian generator Σ(t)) have, in general, complex spectra. Only the spectrum of their superposition remains real. Thus, only the observable superposition H(t)=G(t)+Σ(t) (representing the instantaneous energies) should be called Hamiltonian. In applications, nevertheless, the mathematically consistent models can be based not only on the initial knowledge of the energy operator H(t) (forming a “dynamical” model-building strategy) but also, alternatively, on the knowledge of the Coriolis force Σ(t) (forming a “kinematical” model-building strategy), or on the initial knowledge of the Schrödingerian generator G(t) (forming, for some reason, one of the most popular strategies in the literature). In our present paper, every such choice (marked as “one”, “two” or “three”, respectively) is shown to lead to a construction recipe with a specific range of applicability.
A discrete N-point Runge–Kutta version H(N)(λ) of one of the simplest non-Hermitian square-well Hamiltonians with real spectrum is studied. Its possible Hermitizations mediated by nontrivial (often called “non-Dirac”) metrics Θ≠I are considered as a source of nonequivalent standard probabilistic interpretations of this quantum model. A complete set of these alternative, multiparametric metrics Θ=Θ(a,b,…)(N)(λ) defining all the eligible Hamiltonian-dependent representations of the physical Hilbert space of states is constructed, in closed form, for any coupling λ∊(−1,1) and for any matrix dimension N.
A discrete N-point Runge–Kutta version H(N)(λ) of one of the simplest non-Hermitian square-well Hamiltonians with real spectrum is studied. Its possible Hermitizations mediated by nontrivial (often called “non-Dirac”) metrics Θ≠I are considered as a source of nonequivalent standard probabilistic interpretations of this quantum model. A complete set of these alternative, multiparametric metrics Θ=Θ(a,b,…)(N)(λ) defining all the eligible Hamiltonian-dependent representations of the physical Hilbert space of states is constructed, in closed form, for any coupling λ∊(−1,1) and for any matrix dimension N.
A non-Hermitian operator H defined in a Hilbert space with inner product ⟨ · | · ⟩ may serve as the Hamiltonian for a unitary quantum system if it is η -pseudo-Hermitian for a metric operator (positive-definite automorphism) η . The latter defines the inner product ⟨ · | η · ⟩ of the physical Hilbert space H η of the system. For situations where some of the eigenstates of H depend on time, η becomes time-dependent. Therefore, the system has a non-stationary Hilbert space. Such quantum systems, which are also encountered in the study of quantum mechanics in cosmological backgrounds, suffer from a conflict between the unitarity of time evolution and the unobservability of the Hamiltonian. Their proper treatment requires a geometric framework which clarifies the notion of the energy observable and leads to a geometric extension of quantum mechanics (GEQM). We provide a general introduction to the subject, review some of the recent developments, offer a straightforward description of the Heisenberg-picture formulation of the dynamics for quantum systems having a time-dependent Hilbert space, and outline the Heisenberg-picture formulation of dynamics in GEQM.
Overview In the centennial year of the Schrödinger Equation (1926–2026), this work introduces a novel conceptual framework where quantum dynamics emerges from the intrinsic topological properties of a discrete, non-orientable space-time lattice. The core of the model is the "Möbius Pixel": a fundamental 0D/0+1D quantum unit defined on a non-orientable manifold. By employing the formalism of Pseudo-Hermitian Quantum Mechanics, we demonstrate how the antilinear symmetries inherent in surfaces like the Möbius strip and the Klein bottle naturally generate real energy spectra and complex dynamical structures. Key Scientific Contributions Emergent Dynamics: Derivation of a modified Schrödinger equation from a discrete pixel lattice (pi, pi*), featuring emergent Dirac and Majorana-type coupling terms. Topological Foundation: A rigorous link between non-orientable geometry and the Nambu-like doubled representation of quantum states. Pseudo-Hermiticity: Use of the metric operator η and the involutory operator C to ensure spectral stability and positive-definite inner products in the presence of torsional couplings. Unified Vision: A "symphonic" perspective where fundamental interactions and mass are interpreted as rotovibrational resonances and topological inertia of the lattice. Technical Abstract We construct a toy-model framework in which quantum dynamics emerges from the topological structure of non-orientable surfaces. Using a discrete pixel lattice, we derive a pseudo-Hermitian Hamiltonian whose continuum limit reproduces a Schrödinger equation with additional Dirac and Majorana-type terms. The reality of the spectrum is guaranteed by the pseudo-Hermiticity condition. In this framework, the observable wavefunction emerges as an energy-resolved sector of a higher-dimensional pseudo-Hermitian system. Technical Focus: Pseudo-Hermitian Stability and Lagrangian Spectrum This framework is centered on the formalization of a stable, real-valued spectrum for non-orientable pixel dynamics. The core mathematical challenge addressed is the preservation of a positive-definite inner product in the presence of torsional (antilinear) terms. 1. The Metric Operator η and Involutory C To ensure spectral stability, we define the Hamiltonian H as pseudo-Hermitian with respect to the metric operator η: H† = η H η⁻¹ We introduce a linear, involutory operator C (C² = I) that commutes with the Hamiltonian [H, C] = 0. This operator C is essential for constructing a strictly positive-definite inner product: ⟨ψ | φ⟩_C = ⟨ψ | η C | φ⟩ This construction guarantees that the "Möbius Pixel" states reside in a physical Hilbert space with a consistent probabilistic interpretation, neutralizing the instabilities typically associated with non-Hermitian torsional couplings. 2. Lagrangian Extension and Hamiltonian H The internal dynamics is derived from a 0+1D Lagrangian density accounting for vibrational (n) and topological (k) degrees of freedom. The extension to the full Hamiltonian H includes: Mass-topological Term: m_top c² Internal Excitation Spectrum: Defined by the functional f(n,k), representing the energy levels of the discrete pixel. Torsional Coupling: An antilinear term λ₄ Φ T inducing mixing between the pi and pi* sectors. The resulting spectrum E_n,k remains real under the stability condition: (m_top c² + λ₃ f(n,k))² ≥ (λ₄ Φ_k)² This inequality provides a natural threshold for the lattice, where Planck-scale incompressibility acts as a physical cut-off. Call for Formalization and Collaboration I have established the foundational structure for η and C within the context of the C-formalization. I am looking for collaborators (Theoretical Physicists, Mathematicians) to: Generalize the C operator for an N-pixel lattice interaction. Refine the Lagrangian to include full Lorentz invariance in the continuum limit. Analyze Torsional Modes as symmetry-protected sectors in a 4D pseudo-Hermitian field theory.