Quantum Mechanics on S 1 Based on Dirac Formalism

We deal with quantum mechanics on S 1 (embedded to R 2 ) based on Dirac formalism. To conclude that the dynamical system is a quantum mechanical system we have to show selfadjointness of the dynamical variables. First we show selfadjointness of the momentum operators and then study their spectra. We find that the continuous spectrum of each momentum operator coincides with the set of all real numbers. Second we discuss selfadjointness of the Hamiltonian for a free quantum mechanical particle moving on S 1 together with its spectrum. We see that the spectrum of the Hamiltonian consists of eigenvalues only. Finally we apply these results to the Cauchy problem for the Schrödinger equation for a free quantum mechanical particle moving on S 1 .     

Peculiarities of the electron energy spectrum in the Coulomb field of a superheavy nucleus

We consider the peculiarities of the electron energy spectrum in the Coulomb field of a superheavy nucleus and discuss the long history of an incorrect interpretation of this problem in the case of a pointlike nucleus and its current correct solution. We consider the spectral problem in the case of a regularized Coulomb potential. For some special regularizations, we derive an exact equation for the point spectrum in the energy interval (-m,m) and find some of its solutions numerically. We also derive an exact equation for charges yielding bound states with the energy E = -m; some call them supercritical charges. We show the existence of an infinite number of such charges. Their existence does not mean that the oneparticle relativistic quantum mechanics based on the Dirac Hamiltonian with the Coulomb field of such charges is mathematically inconsistent, although it is physically unacceptable because the spectrum of the Hamiltonian is unbounded from below. The question of constructing a consistent nonperturbative second-quantized theory remains open, and the consequences of the existence of supercritical charges from the standpoint of the possibility of constructing such a theory also remain unclear.     

Countably Periodic Gibbs Measures of the Ising Model on the Cayley Tree

We diagonalize the metric Hamiltonian and evaluate the energy spectrum of the corresponding quasiparticles for a scalar field coupled to a curvature in the case of an N-dimensional homogeneous isotropic space. The energy spectrum for the quasiparticles corresponding to the diagonal form of the canonical Hamiltonian is also evaluated. We construct a modified energy–momentum tensor with the following properties: for the conformal scalar field, it coincides with the metric energy–momentum tensor; the energies of the particles corresponding to its diagonal form are equal to the oscillator frequency; and the number of such particles created in a nonstationary metric is finite. We show that the Hamiltonian defined by the modified energy–momentum tensor can be obtained as the canonical Hamiltonian under a certain choice of variables.     

二次算子族及非负无穷维Hamilton算子的谱分布

本文讨论了一类在弦和梁的微小振动中出现的二次算子族L(λ)=λ~2MλK-A的谱分布问题,进而将所得结论与无穷维Hamilton算子联系起来,利用无穷维Hamilton算子的特殊结构,得到了一类非负无穷维Hamilton算子的谱分布,这为无穷维Hamilton算子的半群方法提供了理论保证.     

Conley conjecture and local Floer homology

In this paper we connect algebraic properties of the pair-of-pants product in local Floer homology and Hamiltonian dynamics. We show that for an isolated periodic orbit, the product is non-uniformly nilpotent and use this fact to give a simple proof of the Conley conjecture for closed manifolds with aspherical symplectic form. More precisely, we prove that on a closed symplectic manifold, the mean action spectrum of a Hamiltonian diffeomorphism with isolated periodic orbits is infinite.     

Jordanian deformation of the open XXX spin chain

We find the general solution of the reflection equation associated with the Jordanian deformation of the SL(2)-invariant Yang R-matrix. A special scaling limit of the XXZ model with general boundary conditions leads to the same K-matrix. Following the Sklyanin formalism, we derive the Hamiltonian with the boundary terms in explicit form. We also discuss the structure of the spectrum of the deformed XXX model and its dependence on the boundary conditions.     

The dressed nonrelativistic electron in a magnetic field

We consider a nonrelativistic electron interacting with a classical magnetic field pointing along the x₃‐axis and with a quantized electromagnetic field. When the interaction between the electron and photons is turned off, the electronic system is assumed to have a ground state of finite multiplicity. Because of the translation invariance along the x₃‐axis, we consider the reduced Hamiltonian associated with the total momentum along the x₃‐axis and, after introducing an ultraviolet cutoff and an infrared regularization, we prove that the reduced Hamiltonian has a ground state if the coupling constant and the total momentum along the x₃‐axis are sufficiently small. We determine the absolutely continuous spectrum of the reduced Hamiltonian and, when the ground state is simple, we prove that the renormalized mass of the dressed electron is greater than or equal to its bare one. We then deduce that the anomalous magnetic moment of the dressed electron is nonnegative. Copyright © 2006 John Wiley & Sons, Ltd.     

Spectral Properties of an Energy-Dependent Hamiltonian

We study spectral properties of a quantum Hamiltonian with a complex-valued energy-dependent potential related to a model introduced in physics of nuclear reactions[30]and we prove that the principle of limiting absorption holds at any point of a large subset of the essential spectrum.When an additional dissipative or smallness hypothesis is assumed on the potential,we show that the principle of limiting absorption holds at any point of the essential spectrum.     

上三角无穷维Hamilton算子的剩余谱

将点谱划分为四个部分,得到上三角无穷维Hamilton算子的点谱σ_p(H)关于虚轴对称的充要条件.在此基础上,结合无穷维Hamilton算子的谱结构,得到无穷维Hamilton算子剩余谱的完全描述,从而实现了利用其内部算子刻画剩余谱.     

Structure of the spectrum of infinite dimensional Hamiltonian operators

This paper deals with the structure of the spectrum of infinite dimensional Hamiltonian operators.It is shown that the spectrum,the union of the point spectrum and residual spectrum,and the continuous spectrum are all symmetric with respect to the imaginary axis of the complex plane. Moreover,it is proved that the residual spectrum does not contain any pair of points symmetric with respect to the imaginary axis;and a complete characterization of the residual spectrum in terms of the point spectrum is then given.As applications of these structure results,we obtain several necessary and sufficient conditions for the residual spectrum of a class of infinite dimensional Hamiltonian operators to be empty.     

The Asymptotic Form of the Lower Landau Bands in a Strong Magnetic Field

The asymptotic form of the bottom part of the spectrum of the two-dimensional magnetic Schrödinger operator with a periodic potential in a strong magnetic field is studied in the semiclassical approximation. Averaging methods permit reducing the corresponding classical problem to a one-dimensional problem on the torus; we thus show the almost integrability of the original problem. Using elementary corollaries from the topological theory of Hamiltonian systems, we classify the almost invariant manifolds of the classical Hamiltonian. The manifolds corresponding to the bottom part of the spectrum are closed or nonclosed curves and points. Their geometric and topological characteristics determine the asymptotic form of parts of the spectrum (spectral series). We construct this asymptotic form using the methods of the semiclassical approximation with complex phases. We discuss the relation of the asymptotic form obtained to the magneto-Bloch conditions and asymptotics of the band spectrum.     

Spectrum and States of the BCS Hamiltonian in a Finite Domain. III. BCS Hamiltonian with Mean-Field Interaction

We investigate the spectrum of a model Hamiltonian with BCS and mean-field interaction in a finite domain under periodic boundary conditions. The model Hamiltonian is considered on the states of pairs and waves of density charges and their excitations. It is represented as the sum of three operators that describe noninteracting pairs, the interaction between pairs, and the interaction between pairs and waves of density charges. The last two operators tend to zero in the thermodynamic limit, and the spectrum of the model Hamiltonian coincides with the spectrum of noninteracting pairs with chemical potential shifted by mean-field interaction. It is shown that the model and approximating Hamiltonians coincide in the thermodynamic limit on their ground and excited states and both have two branches of eigenvalues and eigenvectors.     

Investigation of the energy operator spectrum for a two-magnon system in a one-dimensional anisotropic heisenberg ferromagnet with second neighbor interactions

We consider a physical system of two magnons in an anisotropic one-dimensional Heisenberg ferromagnet with interactions up to the second neighbor. We study the spectrum and bound states of the two-magnon Hamiltonian. We find that under certain conditions, the spectrum differs from that of the isotropic Hamiltonian by one extra or one missing bound state.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 1, pp. 155–161, April, 1996.Translated by A. M. Semikhatov.     

Limiting absorption principle at low energies for a mathematical model of weak interaction: The decay of a boson

We study the spectral properties of a Hamiltonian describing the weak decay of spin 1 massive bosons into the full family of leptons. We prove that the considered Hamiltonian is self-adjoint, with a unique ground state and we derive a Mourre estimate and a limiting absorption principle above the ground state energy and below the first threshold, for a sufficiently small coupling constant. As a corollary, we prove absence of eigenvalues and absolute continuity of the energy spectrum in the same spectral interval. To cite this article: J.-M. Barbaroux, J.-C. Guillot, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

The Possibility of Forming Coupled Pairs in the Periodic Anderson Model

We investigate the generalized periodic Anderson model describing two groups of strongly correlated (d- and f-) electrons with local hybridization of states and d-electron hopping between lattice sites from the standpoint of the possible appearance of coupled electron pairs in it. The atomic limit of this model admits an exact solution based on the canonical transformation method. The renormalized energy spectrum of the local model is divided into low- and high-energy parts separated by an interval of the order of the Coulomb electron-repulsion energy. The projection of the Hamiltonian on the states in the low-energy part of the spectrum leads to pair-interaction terms appearing for electrons belonging to d- and f-orbitals and to their possible tunneling between these orbitals. In this case, the terms in the Hamiltonian that are due to ion energies and electron hopping are strongly correlated and can be realized only between states that are not twice occupied. The resulting Hamiltonian no longer involves strong couplings, which are suppressed by quantum fluctuations of state hybridization. After linearizing this Hamiltonian in the mean-field approximation, we find the quasiparticle energy spectrum and outline a method for attaining self-consistency of the order parameters of the superconducting phase. For simplicity, we perform all calculations for a symmetric Anderson model in which the energies of twice occupied d- and f-orbitals are assumed to be the same.     

Scaling estimates for solutions and dynamical lower bounds on wavepacket spreading

We establish quantum dynamical lower bounds for discrete one-dimensional Schrödinger operators in situations where, in addition to power-law upper bounds on solutions corresponding to energies in the spectrum, one also has lower bounds following a scaling law. As a consequence, we obtain improved dynamical results for the Fibonacci Hamiltonian and related models.     

Lowest energy states in nonrelativistic QED: Atoms and ions in motion

Within the framework of nonrelativisitic quantum electrodynamics we consider a single nucleus and N electrons coupled to the radiation field. Since the total momentum P is conserved, the Hamiltonian H admits a fiber decomposition with respect to P with fiber Hamiltonian H(P). A stable atom, respectively ion, means that the fiber Hamiltonian H(P) has an eigenvalue at the bottom of its spectrum. We establish the existence of a ground state for H(P) under (i) an explicit bound on P, (ii) a binding condition, and (iii) an energy inequality. The binding condition is proven to hold for a heavy nucleus and the energy inequality for spinless electrons.     

Properties of the heavy-fermion spectrum in the canted phase of antiferromagnetic intermetallides

We study the effect of the magnetic-field-induced canting of magnetic sublattices on the energy structure of heavy fermion quasiparticles in intermetallides with antiferromagnetic-type ordering. We work in the framework of an effective Hamiltonian of the periodic Anderson model in the regime of strong electron correlations. With the virtual transfers into high-energy double states taken into account, this Hamiltonian involves exchange interactions between spin moments of the f ions and the s-f-exchange coupling between the two subsystems. For a noncollinear problem geometry, we introduce a unitary transformation that allows reducing an eighth-order equation for the spectrum of heavy fermions to two fourth-order equations. We show that the quasimomentum dependence of the heavy-fermion energy changes qualitatively under the transition from the antiferromagnetic phase to a phase with a considerable canting angle emerging in an external magnetic field.     

Multiplicity of ground states in quantum field models: applications of asymptotic fields

Ground states of Hamiltonian H of quantum field models are investigated. The infimum of the spectrum of H is in the edge of its essential spectrum. By means of the asymptotic field theory, we give a necessary and sufficient condition for that the expectation value of the number operator of ground states is finite, from which we give an upper bound of the multiplicity of ground states of H. Typical examples are massless GSB models and the Pauli-Fierz model with spin 1/2.