High order corrections to the time-independent Born-Oppenheimer approximation II: Diatomic Coulomb systems

We study the bound states of diatomic molecular systems. We prove that if the nuclear masses are proportional to ε^?4 then certain eigenvalues and eigenvectors of the Hamiltonian have asymptotic expansions to arbitrarily high order in powers of ε, as ε→0. The zeroth through fourth order terms in the expansions for the eigenvalues are those of the well-known Born-Oppenheimer approximation. The fifth order term is zero.     

Discrete Hamiltonian symmetries and semiclassical quantization

Most quantum Hamiltonian systems exhibit discrete symmetries. Allowing for these is crucial when properly calculating the fluctuation properties of the quantal spectrum. These properties are then employed to distinguish between classically chaotic or non-chaotic quantum systems. In general, semiclassical quantization procedures do not take into account irreducible representations of the Hamiltonian. A procedure is presented to take these into account in semiclassical quantization schemes and calculate some of the energy eigenvalues belonging to a specific irreducible representation.     

Eigenvalue Statistics for Lattice Hamiltonian with Off-diagonal Disorder

This short note deals with a certain kind of lattice Hamiltonian with off-diagonal disorder. Based on the exponential decay of the fractional moment of the Green function, we are able to prove that the properly rescaled eigenvalues of the random Hamiltonian are distributed as a Poisson point process with intensity measure given by the density of states. One of the key step in this proof is the Minami-type estimate. As a crucial ingredient, we also use the Minami-type estimate to study some important properties of the random Hamiltonian, such as multiplicity of the eigenvalues and quantitative estimate of the localization centers.     

共轭线性对称性及其对PT-对称量子理论的应用

传统量子系统的哈密顿是自伴算子,哈密顿的自伴性不仅保证系统遵循酉演化和保持概率守恒,而且也保证了它自身具有实的能量本征值,这类系统称为自伴量子系统.然而,确实存在一些物理系统(如PT-对称量子系统),其哈密顿不是自伴的,这类系统称为非自伴量子系统.为了深入研究PT-对称量子系统,并考虑到算子PT的共轭线性性,首先讨论了共轭线性算子的一些性质,包括它们的矩阵表示和谱结构等;其次,分别研究了具有共轭线性对称性和完整共轭线性对称性的线性算子,通过它们的矩阵表示,给出了共轭线性对称性和完整共轭线性对称性的等价刻画;作为应用,得到了关于PT-对称及完整PT-对称算子的一些有趣性质,并通过一些具体例子,说明了完整PT-对称性对张量积运算不具有封闭性,同时说明了完整PT-对称性既不是哈密顿算子在某个正定内积下自伴的充分条件,也不是必要条件.     

Computing the pressure for Axiom-A attractors by time series and large deviations for the Lyapunov exponent

For the Axiom-A attractors a relation is given between the topological pressure and the spectrum of the generalized Lyapunov exponents. As a consequence, a simple formula is found to compute the topological entropy of the attractor by means of a time series. The results are used to compute the large deviations for positive Lyapunov exponents.     

Quantum States on Harmonic Lattices

We investigate bosonic Gaussian quantum states on an infinite cubic lattice in arbitrary spatial dimensions. We derive general properties of such states as ground states of quadratic Hamiltonians for both critical and non-critical cases. Tight analytic relations between the decay of the interaction and the correlation functions are proven and the dependence of the correlation length on band gap and effective mass is derived. We show that properties of critical ground states depend on the gap of the point-symmetrized rather than on that of the original Hamiltonian. For critical systems with polynomially decaying interactions logarithmic deviations from polynomially decaying correlation functions are found.     

The general relativistic hydrogen atom

The general relativistic Dirac equation is formulated in an arbitrary curved space-time using differential forms. These equations are applied to spherically symmetric systems with arbitrary charge and mass. For the case of a black hole (with event horizon) it is shown that the Dirac Hamiltonian is self-adjoint, has essential spectrum the whole real line and no bound states. Although rigorous results are obtained only for a spherically symmetric system, it is argued that, in the presence of any event horizon there will be no bound states. The case of a naked singularity is investigated with the results that the Dirac Hamiltonian is not self-adjoint. The self-adjoint extensions preserving angular momentum are studied and their spectrum is found to consist of an essential spectrum corresponding to that of a free electron plus eigenvalues in the gap (–mc ², +mc ²). It is shown that, for certain boundary conditions, neutrino bound states exist.Supported in part by the National Science Foundation     

An effective spin Hamiltonian approach to heavy electron metamagnetism

ABSTRACT

We describe a minimal model for metamagnetism, based on a quantum spin Hamiltonian with a single energy scale. Within this model, the metamagnetic critical field is proportional to the temperature where a peak in the linear susceptibility occurs which in turn is proportional to the temperatures where the nonlinear susceptibilities also peak. The thermodynamic properties are derived in a straightforward manner and bear a striking resemblance to observations in such strongly correlated systems as heavy fermion materials. We also consider extensions of the model by including effects such as a mean field and a tilt of the quantisation axis to encompass observed deviations from a minimal metamagnetic behaviour.     

Quantum Harmonic Oscillator Systems with Disorder

We study many-body properties of quantum harmonic oscillator lattices with disorder. A sufficient condition for dynamical localization, expressed as a zero-velocity Lieb-Robinson bound, is formulated in terms of the decay of the eigenfunction correlators for an effective one-particle Hamiltonian. We show how state-of-the-art techniques for proving Anderson localization can be used to prove that these properties hold in a number of standard models. We also derive bounds on the static and dynamic correlation functions at both zero and positive temperature in terms of one-particle eigenfunction correlators. In particular, we show that static correlations decay exponentially fast if the corresponding effective one-particle Hamiltonian exhibits localization at low energies, regardless of whether there is a gap in the spectrum above the ground state or not. Our results apply to finite as well as to infinite oscillator systems. The eigenfunction correlators that appear are more general than those previously studied in the literature. In particular, we must allow for functions of the Hamiltonian that have a singularity at the bottom of the spectrum. We prove exponential bounds for such correlators for some of the standard models.     

Reduction of the XXZ model with twisted boundary conditions

The XXZ model with twisted boundary conditions is considered. The method of energy spectrum calculation based on the functional equation for the transfer matrix is analyzed. The Hamiltonian eigenvalues are obtained in an explicit form.     

Polynomial integrability of the Hamiltonian systems with homogeneous potential of degree − 3

In this paper, we study the polynomial integrability of natural Hamiltonian systems with two degrees of freedom having a homogeneous potential of degree k given either by a polynomial, or by an inverse of a polynomial. For k=−2,−1,…,3,4, their polynomial integrability has been characterized. Here, we have two main results. First, we characterize the polynomial integrability of those Hamiltonian systems with homogeneous potential of degree −3. Second, we extend a relation between the nontrivial eigenvalues of the Hessian of the potential calculated at a Darboux point to a family of Hamiltonian systems with potentials given by an inverse of a homogeneous polynomial. This relation was known for such Hamiltonian systems with homogeneous polynomial potentials. Finally, we present three open problems related with the polynomial integrability of Hamiltonian systems with a rational potential.     

Non-Relativistic Treatment of a Generalized Inverse Quadratic Yukawa Potential

A bound state solution is a quantum state solution of a particle subjected to a potential such that the particle's energy is less than the potential at both negative and positive infinity. The particle's energy may also be negative as the potential approaches zero at infinity. It is characterized by the discretized eigenvalues and eigenfunctions,which contain all the necessary information regarding the quantum systems under consideration. The bound state problems need to be extended using a more precise method and approximation scheme. This study focuses on the non-relativistic bound state solutions to the generalized inverse quadratic Yukawa potential. The expression for the non-relativistic energy eigenvalues and radial eigenfunctions are derived using proper quantization rule and formula method, respectively. The results reveal that both the ground and first excited energy eigenvalues depend largely on the angular momentum numbers, screening parameters, reduced mass, and the potential depth. The energy eigenvalues,angular momentum numbers,screening parameters,reduced mass,and the potential depth or potential coupling strength determine the nature of bound state of quantum particles. The explored model is also suitable for explaining both the bound and continuum states of quantum systems.     

Computation of bound states for 2D potentials using discrete basis

Using a suitable Laguerre basis set that ensures a tridiagonal matrix representation of the reference Hamiltonian, we were able to evaluate in closed form the matrix representation of the associated Hamiltonian for two exactly solvable 2D potentials. This enabled us to treat analytically the full Hamiltonian and compute the associated bound states spectrum as the eigenvalues of the associated analytical matrix representing their Hamiltonians. Finally we compared our results satisfactorily with those obtained using the Gauss quadrature numerical integration approach.

PACS numbers: 03.65.Ge, 34.20.Cf, 03.65.Nk, 34.20.Gj     

Determination of bound states and resonances of the Debye-potential by complex scaling

For a statically screened Coulomb-potential, two particle bound and resonant states are determined which correspond to complex poles of the S-matrix. The complex scaling method is used in order to calculate both the real eigenvalues of the Hamiltonian and the complex resonances in the same way. The eigenfunctions are described by complex power series; for the coefficients a recursion formula is given.     

Exact Solutions of Non-Autonomous Quantum Systems with Quadratic Hamiltonian

Based on algebraic dynamics, we present an algorithm to obtain exact solutions of the Schrodinger equation of non-autonomous quantum systems with Hamiltonian expressed in quadratic function of creation and annihilation operators of bosons. The Hamiltonian is treated as a linear function of generators of a symplectic group. Similar to the canonical transformation of classical dynamics, we employ a set of gauge transformations to gradually transform the Hamiltonian to a linear function of Cartan operators. The exact solutions are obtained by inverse gauge transformations. When the system is autonomous, this algorithm can obtain the normal mode of the Hamiltonian, as well as the eigenstates and eigenvalues.     

Mathematical theory of nonrelativistic matter and radiation

We consider a system of finitely many nonrelativistic electrons bound in an atom or molecule which are coupled to the electromagnetic field via minimal coupling or the dipole approximation. Among a variety or results, we give sufficient conditions for the existence of a ground state (an eigenvalue at the bottom of the spectrum) and resonances (eigenvalues of a complex dilated Hamiltonian) of such a system. We give a brief outline of the proofs of these statements which will appear at full length in a later work.Dedicated to the memory of J. Schwinger, whose understanding of Quantum Electrodynamics was profound     

Solution of a one-dimensional torsion Schrödinger equation with a general periodic potential

Relations are found for the calculation of eigenvalues and eigenfunctions of the Hamiltonian of internal rotational motion in molecules in the basis of plane waves. The dependences of kinematic coefficient F(φ) and potential V(φ) on dihedral angle φ are represented by Fourier series for both symmetric and asymmetric functions, as well as for general periodic functions. If a molecule has symmetry elements, the found solution transforms to that previously known.     

Constructive proof of localization in the Anderson tight binding model

We prove that, for large disorder or near the band tails, the spectrum of the Anderson tight binding Hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues. The corresponding eigenfunctions are exponentially well localized. These results hold in arbitrary dimension and with probability one. In one dimension, we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder. Our techniques extend to other physical systems which exhibit localization phenomena, such as infinite systems of coupled harmonic oscillators, or random Schrödinger operators in the continuum.Work supported in part by National Science Foundations grant MCS-8108814 (A03).Work supported in part by National Science Foundation grant DMR 81-00417.     

Nonautonomous Hamiltonians

We present a theory of resonances for a class of nonautonomous Hamiltonians to treat the structural instability of spatially localized and time-periodic solutions associated with an unperturbed autonomous Hamiltonian. The mechanism of instability is radiative decay, due to resonant coupling of the discrete modes to the continuum modes by the time-dependent perturbation. This results in a slow transfer of energy from the discrete modes to the continuum. The rate of decay of solutions is slow and hence the decaying bound states can be viewed as metastable. The ideas are closely related to the authors' work on (i) a time-dependent approach to the instability of eigenvalues embedded in the continuous spectra, and (ii) resonances, radiation damping, and instability in Hamiltonian nonlinear wave equations. The theory is applied to a general class of Schrödinger equations. The phenomenon of ionization may be viewed as a resonance problem of the type we consider and we apply our theory to find the rate of ionization, spectral line shift, and local decay estimates for such Hamiltonians.     

Über den Grundzustand überschwerer Atome

We consider a bound relativistic electron in a Coulomb-Field with charge numbers greater than 137. By careful examination of the self-adjointness property of the Hamiltonian we determine the energy eigenvalues; they depend on a parameter which can be interpreted as a cut-off radius of the potential leading to quantitative predictions of the energy of the spectrum, especially of the lowest state.