Spectrum Of Hamiltonian Research Articles

THE POLARON REVISITED

In recent years, the spectral properties of the translation invariant Nelson model has been studied. Some of the results obtained did not extend to the related polaron model for technical reasons related to the typical assumption of boundedness of the phonon dispersion relation in the polaron model. In this paper, we work with a large class of linearly coupled translation invariant models which includes both the Nelson model and H. Fröhlich's polaron model. The problems considered are chosen based on relevance for the polaron model. A key input is an analysis of the behavior of the bottom of the spectrum of the fiber Hamiltonians at large total momentum.

A Bethe ansatz solvable model for superpositions of Cooper pairs and condensed molecular bosons

We introduce a general Hamiltonian describing coherent superpositions of Cooper pairs and condensed molecular bosons. For particular choices of the coupling parameters, the model is integrable. One integrable manifold, as well as the Bethe ansatz solution, was found by Dukelsky et al. [J. Dukelsky, G.G. Dussel, C. Esebbag, S. Pittel, Phys. Rev. Lett. 93 (2004) 050403]. Here we show that there is a second integrable manifold, established using the boundary quantum inverse scattering method. In this manner we obtain the exact solution by means of the algebraic Bethe ansatz. In the case where the Cooper pair energies are degenerate we examine the relationship between the spectrum of these integrable Hamiltonians and the quasi-exactly solvable spectrum of particular Schrödinger operators. For the solution we derive here the potential of the Schrödinger operator is given in terms of hyperbolic functions. For the solution derived by Dukelsky et al., loc. cit. the potential is sextic and the wavefunctions obey PT -symmetric boundary conditions. This latter case provides a novel example of an integrable Hermitian Hamiltonian acting on a Fock space whose states map into a Hilbert space of PT -symmetric wavefunctions defined on a contour in the complex plane.

Annealed asymptotics for the parabolic Anderson model with a moving catalyst

This paper deals with the solution u to the parabolic Anderson equation ∂ u / ∂ t = κ Δ u + ξ u on the lattice Z d . We consider the case where the potential ξ is time-dependent and has the form ξ ( t , x ) = δ 0 ( x − Y t ) with Y t being a simple random walk with jump rate 2 d ϱ . The solution u may be interpreted as the concentration of a reactant under the influence of a single catalyst particle Y t . In the first part of the paper we show that the moment Lyapunov exponents coincide with the upper boundary of the spectrum of certain Hamiltonians. In the second part we study intermittency in terms of the moment Lyapunov exponents as a function of the model parameters κ and ϱ .

Spectra of Hamiltonians of molecule pseudorelativistic electrons in spaces of functions with permutational and point symmetry

We study the spectral properties of the Hamiltonian Hn of n pseudorelativistic electrons in the Coulomb field of k fixed nuclei in spaces of functions having arbitrary given types of permutational and point symmetry. For this operator, we establish the location of the essential spectrum, obtain two-sided estimates of the discrete spectrum counting function in terms of the counting functions of the discrete spectrum of some two-particle nonrelativistic operators, and find the leading term of the spectral asymptotics.

QUASI-EXACTLY-SOLVABLE QUANTUM LATTICE SOLITONS

We extend the exactly solvable Hamiltonian describing f quantum oscillators considered recently by J. Dorignac et al. We introduce a new interaction which we choose to be quasi-exactly solvable. The properties of the spectrum of this new Hamiltonian are studied as functions of the new coupling constant. We point out that both the original and the by us modified Hamiltonians are related to adequate Lie structures.

On the Essential Spectrum of Many-Particle Pseudorelativistic Hamiltonians with Permutational Symmetry Account

In this paper we formulate our results on the essential spectrum of many-particle pseudorelativistic Hamiltonians without magnetic and external potential fields in the spaces of functions, having arbitrary type of the permutational symmetry. We discover location of the essential spectrum for all and for some cases we establish new properties of the lower bound of this spectrum, which are useful for study of the discrete spectrum. In this paper we formulate our results on the essential spectrum of many-particle pseudore- lativistic Hamiltonians without magnetic and external potential fields in spaces of functions, having arbitrary type of the permutational symmetry. We discover the location of the essential spectrum for all (Theorem 1) and for some cases we establish new properties of the lower bound

LargeNspectrum of two matrices in a harmonic potential and BMN energies

The large N spectrum of the quantum mechanical hamiltonian of two hermitean matrices in a harmonic potential is studied in a framework where one of the matrices is treated exactly and the other is treated as a creation operator impurity in the background of the first matrix. For the free case, the complete set of invariant eigenstates and corresponding energies are obtained. When g_{YM}^2 interactions are added, it is shown that the full string tension corrected spectrum of BMN loops is obtained.

Real-Space Renormalization Group Approach for the Corner Hamiltonian

We present a real-space renormalization group approach for the corner Hamiltonian, which is relevant to the reduced density matrix in the density matrix renormalization group. A set of self-consistent equations that the renormalized Hamiltonian should satisfy in the thermodynamic limit is also derived from the fixed point of the recursion relation for the corner Hamiltonian. We demonstrate the renormalization group algorithm for the $S=1/2$ XXZ spin chain and show that the results are consistent with the exact solution. We further examine the renormalization group for the S=1 Heisenberg spin chain and then discuss the nature of the eigenvalue spectrum of the corner Hamiltonian for the non-integrable model.

Darboux transformation of the Green’s function of a regular Sturm-Liouville problem

Darboux transformations of a regular Sturm-Liouville problem are considered. A relationship between the Green’s functions of the initial and transformed problems is found. Second-order reducible and irreducible Darboux transformations are analyzed. It is demonstrated that a new second-order irreducible supersymmetry can exist in the regular case for which the spectrum of the intermediate Hamiltonian differs radically from the spectrum of the initial Hamiltonian. The results obtained are illustrated by the example of a particle in a box.

Determination of Non-adiabatic Scattering Wave Functions in a Born-Oppenheimer Model

We study non--adiabatic transitions in scattering theory for the time dependent molecular Schroedinger equation in the Born--Oppenheimer limit. We assume the electron Hamiltonian has finitely many levels and consider the propagation of coherent states with high enough total energy. When two of the electronic levels are isolated from the rest of the electron Hamiltonian's spectrum and display an avoided crossing, we compute the component of the nuclear wave function associated with the non--adiabatic transition that is generated by propagation through the avoided crossing. This component is shown to be exponentially small in the square of the Born--Oppenheimer parameter, due to the Landau-Zener mechanism. It propagates asymptotically as a free Gaussian in the nuclear variables, and its momentum is shifted. The total transition probability for this transition and the momentum shift are both larger than what one would expect from a naive approximation and energy conservation.

The scaled Hermite–Weber basis in the spectral and pseudospectral pictures

Computational efficiencies of the discrete (pseudospectral, collocation) and continuous (spectral, Rayleigh–Ritz, Galerkin) variable representations of the scaled Hermite–Weber basis in finding the energy eigenvalues of Schrödinger operators with several potential functions have been compared. It is well known that the so-called differentiation matrices are neither skew-symmetric nor symmetric in a pseudospectral formulation of a differential equation, unlike their Rayleigh–Ritz counterparts. In spite of this fact, it is shown here that the spectra of matrix Hamiltonians generated by Hermite collocation method may be determined by way of diagonalizing symmetric matrices. Furthermore, the symmetric matrix elements do not require the evaluation of Hermite polynomials at the grid points. Surprisingly, the present numerical results suggest that the convergence rates of collocation and Rayleigh–Ritz methods are entirely the same.

Linear stability of Einstein-Gauss-Bonnet static spacetimes: Tensor perturbations

We study the stability under linear perturbations of a class of static solutions of Einstein-Gauss-Bonnet gravity in $D=n+2$ dimensions with spatial slices of the form ${\ensuremath{\Sigma}}_{\ensuremath{\kappa}}^{n}\ifmmode\times\else\texttimes\fi{}{\mathbb{R}}^{+}$, ${\ensuremath{\Sigma}}_{\ensuremath{\kappa}}^{n}$ an $n$ manifold of constant curvature $\ensuremath{\kappa}$. Linear perturbations for this class of spacetimes can be generally classified into tensor, vector and scalar types. The analysis in this paper is restricted to tensor perturbations. We show that the evolution equations for tensor perturbations can be cast in Schr\"odinger form, and obtain the exact potential. We use $S$ deformations to analyze the Hamiltonian spectrum, and find an S-deformed potential that factors in a convenient way, allowing us to draw definite conclusions about stability in every case. It is found that there is a minimal mass for a $D=6$ black hole with a positive curvature horizon to be stable. For any $D$, there is also a critical mass above which black holes with negative curvature horizons are unstable.

Decoherence time in self-induced decoherence

A general method for obtaining the decoherence time in self-induced decoherence is presented. In particular, it is shown that such a time can be computed from the poles of the resolvent or of the initial conditions in the complex extension of the Hamiltonian's spectrum. Several decoherence times are estimated: $10^{-13}-$ $10^{-15}s$ for microscopic systems, and $10^{-37}-10^{-39}s$ for macroscopic bodies. For the particular case of a thermal bath, our results agree with those obtained by the einselection (environment-induced decoherence) approach.

A numerical basis for the accurate representation of the continuum spectrum of atomic Hamiltonians

We present a method for the accurate calculation of the complete spectrum of the Schrodinger equation in terms of B-splines polynomial basis. The method is capable to represent numerically the bound and continuum spectrum of complex atomic systems. The theoretical method is discussed, and an application to hydrogenic Hamiltonian is given.

Suppression of Modes in the Random Phase Approximation

A general but simple method is proposed to eliminate the quantum fluctuations generated by selected one-body operators in the excitation spectrum of a discrete random phase approximation (RPA) Hamiltonian. This method provides an outstanding tool for the removal of the contaminating spurious effects originated from symmetry violations. It can be also applied as a mode filter for analyzing RPA response functions.

Optical control of nonradiative decay in polyatomic molecules

We propose an approach for elimination of radiationless transitions in the excited states of polyatomic molecules. Our method is based on the concept of electronically localized eigenstates of coupled vibronic Hamiltonians---the occurrence of stationary eigenfunctions of essentially pure electronic character in the (otherwise strongly mixed) spectra of excited manifold Hamiltonians of systems undergoing nonradiative decay. The origin of this phenomenon and its generality are explained within the framework of the theory of scars of unstable periodic orbits. Optimal control theory is combined with the notion of electronically localized states to prepare the system in an excited superposition that is immune to nonradiative decay. The approach is applied to suppress ultrafast radiationless transitions in two model systems that differ widely in coupling strength and electronic structure.

GROUND-STATE PROPERTIES OF BILAYER ν=2 QUANTUM HALL STATES

We construct a Hartree-Fock-Bogoliubov (HFB) Hamiltonian Haldane-sphere and study excitation spectra and ground-state properties in bilayer ν=2 QH systems. It is found that the HFB approximation can systematically give SU (2) symmetrical excitation spectra spectra well agree with those from exact diagonalization calculations in large tunneling-energy region. In this region, has been considered to be a pseudospin-squeezed ground state is favored theory.

Operator averages in finite-dimensional N-electron model spaces: a diagrammatic approach

An N-electron Hamiltonian projected onto a K-orbital, antisymmetric and spin-adapted model space (in quantum chemistry known as the full configuration interaction space) may be expressed as a linear combination of products of generators of the unitary group . Consequently, spectral density distribution moments and other quantities which characterize global properties of the Hamiltonian spectra in are defined by the averages over the model space of products of the generators. This paper presents a set of new recurrent formulae for the averages and describes a simple graphical method of their evaluation.

Quasifree States in Some One-Dimensional Quantum Spin Models

We use numerical methods to investigate the SUq(N) Perk-Schultz spin chain at the special quantum parameter value q = -e iπ/N . We discover simple laws applicable to a considerable part of the Hamiltonian spectrum, which in particular contains the energy of the ground state and the nearest excitations. The phenomenological formulas obtained resemble formulas for the spectrum of the free-fermion model. We formulate several hypotheses, some of which can be justified by constructing exact solutions of the system of Bethe-ansatz equations for finite-length chains. We obtain two sets of solutions of these equations. The first corresponds to the special value of the quantum parameter q and, in particular, describes the model ground state, which is antiferromagnetic. The second set of solutions describes a part of the spectrum belonging to the sectors where the numbers n i of particles of different types (i = 0, 1...,N - 1) do not exceed unity for all the types except one. For this set, we obtain a simple spectrum at arbitrary values of q. It is hypothesized that this spectrum and the solutions of the Bethe-ansatz equations found in a closed form are intimately related to the existence of a special eigenstate for the transfer matrix of the auxiliary inhomogeneous SU q (N-1) vertex model that is involved in constructing the system of Bethe-ansatz equations of a matrioshka structure. Indirect arguments based on combinatorial properties of the wave function of the relevant state are given to support this hypothesis.

The Hamiltonians of Pseudorelativistic Atoms with Finite-Mass Nuclei: The Structure of the Discrete Spectrum

We study the structure of the discrete spectrum of pseudorelativistic Hamiltonians H for atoms and positive ions with finite-mass nuclei and with n electrons, where n ≥ 1 is arbitrary. The center-of-mass motion cannot be separated, and hence we study the spectrum of the restriction H P of H to the subspace of states with given value P of the total momentum of the system. For the operators H P we discover a) two-sided estimates for the counting function of the discrete spectrum σ d (H P ) of H P in terms of the counting functions of some effective two-particle operators; b) the leading term of the spectral asymptotics of σ d (H P ) near the lower bound inf σess(H P ) of the essential spectrum of H P . The structure of the discrete spectrum of such systems was known earlier only for n=1.