Diagonalizable Hamiltonians Research Articles

Dynamical symmetry algebras of two superintegrable two-dimensional systems

A complete classification of 2D quadratically superintegrable systems with scalar potential on two-dimensional conformally flat spaces has been performed over the years and 58 models, divided into 12 equivalence classes, have been obtained. We will re-examine two pseudo-Hermitian quantum systems E 8 and E 10 from such a classification by a new approach based on extra sets of ladder operators. They correspond in fact to two of those equivalence classes. Those extra ladder operators are exploited to obtain the generating spectrum algebra and the dynamical symmetry one. We will relate the generators of the dynamical symmetry algebra to the Hamiltonian, thus demonstrating that the latter can be written in an algebraic form. We will also link them to the integrals of motion providing the superintegrability property. This demonstrates how the underlying dynamical symmetry algebra allows to write the integrals in terms of its generators and therefore explains the symmetries. Furthermore, we will exploit those algebraic constructions to generate extended sets of states and give the action of the ladder operators on them. We will present polynomials of the Hamiltonian and the integrals of motion that vanish on some of those states, then demonstrating that the sets of states not only contain eigenstates, but also generalized states which are beyond the well-known eigenstates of diagonalizable Hamiltonians and satisfy more complicated polynomial identities. Our approach provides a natural framework for such states.

Statistical–mechanical theory of topological indices

Topological indices (TI) are algebraic invariants of molecular graphs representing the topology of a molecule, which are very valuable in quantitative structure–property relations (QSPR). Here we prove that TI are the partition functions of such molecules when the temperature of the thermal bath at which they are submerged is very high. These partition functions are obtained by describing molecular electronic properties through tight-binding Hamiltonians (TBH), where the hopping parameters are topological properties describing atom–atom interactions. We prove that the TBH proposed here are non-Hermitian diagonalizable Hamiltonians which can be replaced by symmetric ones. In this way we propose a statistical–mechanical theory for TI, which is exemplified by deriving the Randić, Zagreb, Balaban, Wiener and ABC indices. The work also illuminates how to improve QSPR models using the current theoretical framework as well as how to derive statistical–mechanical parameters of molecular graphs.

Completely solvable models with nearest-neighbor interactions on one-dimensional lattices

The most general exclusion single species reaction–diffusion models are investigated, which are defined on a one-dimensional lattice, have nearest-neighbor interactions, and their corresponding two-site Hamiltonians are diagonalizable or converted to a Jordan form by a direct product of matrices. The diagonalizable Hamiltonians represent a family of 3-parameter stochastic processes, while those which can be transformed into a Jordan form have 4 free parameters. For the diagonalizable Hamiltonians, all of the eigenvalues and eigenvectors and some correlation functions are explicitly calculated. Also, both the entropy and the entropy production rate in such systems are studied. For those Hamiltonians which can be converted to a Jordan form, some parts of the spectrum are obtained.

Tight binding approach for cuprate superconductors–Energy of the mean field and the correlation

We construct a mean field theory different from the wave vector representation, based on the observation that the ordinary mean field theory for superconductivity involves an inconsistency in the description of high Tcd-wave superconductivity. By using a cell- and ion-position (site position) representation, we obtain a basis for the tight-binding approximation (TBA) of the energy gap parameter and the energy of the pair condensate system. The mean field Hamiltonian is not diagonalized exactly by the Bogoliubov transformation, but a diagonalizable Hamiltonian may be defined by a truncation of that. A modified BCS model is introduced accompanying this Hamiltonian. We calculate the energy of the mean field and, moreover, the correlation energy of the condensate system in the TBA. We use the Gaussian orbital method to show that the gap parameter has a d-wave-like node structure in the form defined in the wave vector representation, but the gap in the site position representation and the quasiparticle energy do not have a node structure. The correction of the gap parameter by the off-site pair interaction is shown to be the first-order perturbation and the correction of the mean field energy is the second-order perturbation. The ratio of the correlation energy to the mean field energy is on the order of the carrier concentration in this TBA.

Diagonalization of Hamiltonian in the photogravitation-al restricted three body problem with P-R drag

In this paper, restricted, three-body problem (RTBP) is generalised to study the non-linear stability of equilibrium points in the photogravitational RTBP with P-R drag. In the present study, both primaries are considered as a source of radiation and effect of P-R drag. Hence the problem will contain four parameters q1, q2, W1 and W2. At first, the Lagrangian and the Hamiltonian of the problem were computed, then the Lagrangian function is expanded in power series of the coordinates of the triangular equilibrium points x and y. Lastly, diagonalized the quadratic term of the Hamiltonian of the problem, which is obtained by expanding original Lagrangian or Hamiltonian by Taylor's series about triangular equilibrium point. Finally, the study concluded that the diagonalizable Hamiltonian is H2=ω1I1-ω2I2.

Non-self-adjoint Hamiltonians with complex eigenvalues

Motivated by what one observes dealing with PT-symmetric quantum mechanics, we discuss what happens if a physical system is driven by a diagonalizable Hamiltonian with not all real eigenvalues. In particular, we consider the functional structure related to systems living in finite-dimensional Hilbert spaces, and we show that certain intertwining relations can be deduced also in this case if we introduce suitable antilinear operators. We also analyze a simple model, computing the transition probabilities in the broken and in the unbroken regime.

Automatic Hermiticity

We study a diagonalizable Hamiltonian that is not at first hermitian. Requirement that a measurement shall not change one Hamiltonian eigenstate into another one with a different eigenvalue imposes that an inner product must be defined so as to make the Hamiltonian normal with regard to it. After a long time development with the non-hermitian Hamiltonian, only a subspace of possible states will effectively survive. On this subspace the effect of the anti-hermitian part of the Hamiltonian is suppressed, and the Hamiltonian becomes hermitian. Thus hermiticity emerges automatically, and we have no reason to maintain that at the fundamental level the Hamiltonian should be hermitian. If the Hamiltonian is given in a local form, a conserved probability current density can be constructed with two kinds of wave functions. We also point out a possible misestimation of a past state by extrapolating back in time with the hermitian Hamiltonian. It is a seeming past state, not a true one.

Non-linear supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: I. General properties

We study complex potentials and related non-diagonalizable Hamiltonians with special emphasis on formal definitions of associated functions and Jordan cells. The non-linear SUSY for complex potentials is considered and the theorems characterizing its structure are presented. We define the class of complex potentials invariant under SUSY transformations for (non-)diagonalizable Hamiltonians and formulate several results concerning the properties of associated functions. We comment on the applicability of these results for softly non-Hermitian PT-symmetric Hamiltonians. The role of SUSY (Darboux) transformations in increasing/decreasing of Jordan cells in SUSY partner Hamiltonians is thoroughly analyzed and summarized in the Index Theorem. The properties of non-diagonalizable Hamiltonians as well as the Index Theorem are illustrated in the solvable examples of non-Hermitian reflectionless Hamiltonians. The rigorous proofs are relegated to part II of this paper. At last, some peculiarities in resolution of identity for discrete and continuous spectra with a zero-energy bound state at threshold are discussed.

Pseudo-supersymmetric quantum mechanics and isospectral pseudo-Hermitian Hamiltonians

We examine the properties and consequences of pseudo-supersymmetry for quantum systems admitting a pseudo-Hermitian Hamiltonian. We explore the Witten index of pseudo-supersymmetry and show that every pair of diagonalizable (not necessarily Hermitian) Hamiltonians with discrete spectra and real or complex-conjugate pairs of eigenvalues are isospectral and have identical degeneracy structure except perhaps for the zero eigenvalue if and only if they are pseudo-supersymmetric partners. This implies that pseudo-supersymmetry is the basic framework for generating non-Hermitian PT-symmetric and non- PT-symmetric Hamiltonians with a real spectrum via a Darboux transformation, and shows that every diagonalizable Hamiltonian H with a discrete spectrum and real or complex-conjugate pairs of eigenvalues may be factored as H= L ♯ L where L is a linear operator with pseudo-adjoint L ♯. In particular, this factorization applies to PT-symmetric and Hermitian Hamiltonians. The non-degenerate two-level systems provide a class of Hamiltonians that are pseudo-Hermitian. We demonstrate the implications of our general results for this class in some detail.