On the spin fluctuation approach to the anderson and Kondo hamiltonians

The free energy expression of the full Anderson model is derived in a similar way as has been done before for the Kondo model. Use is made of the “asymptotic time approximation” first invented to study the x-ray threshold singularity. Again the procedure leads to a classical Coulomb gas on a ring. The magnetic field is included and plays the role of an electric field for the Coulomb gas. Further it turns out that the “symmetric” Anderson model (2ε _d =?U) is identical to the antiferromagnetic Kondo model. The method and the results suggest the construction of a “polaron” model which in the approximation used is equivalent to the Kondo model as well as the Anderson model. From this a new picture of the “Kondo effect” in terms of spin fluctuations is developed.     

On Wigner's semicircle law for eigenvalues of non-random hamiltonians

It is shown that physical many-body systems with hamiltonians which belong to a large class of non-random matrices of rational Jacobi type have as level density Wigner's famous semicircle law.     

Maslov-WKB theory for supersymmetric hamiltonians

Maslov's asymptotic analysis of Schrödinger operators is extended to the case of supersymmetric hamiltonians. It is shown how this leads to a modified quantization condition, a special case of which has been found previously by Comtet et al.     

Effective hamiltonians

Non-linear equations, generalizing those determining Bloch's transformation to an effective hamiltonian, are set up in a very simple way. It is shown how they immediately determine the results of Van Vleck, Bloch and Soliverez. Treating the Foldy-Wouthuysen transformation as a special case of the Van Vleck transformation, sixth-order expressions are obtained by means of the non-linear equations. The transformations of Soliverez, Roussy and Kvasnicka are shown to be strictly identical—because they all have a simple property which is proven to be unique for that of Soliverez. Kvasnicka relates his transformation to that of Primas'. We shall see that it is also very closely related to that of Van Vleck's. A certain non-hermitian effective hamiltonian also given by Roussy is proven to be identical to that of Bloch's. Primas' level shift transformation gives an effective hamiltonian for each level in zeroth order. This effective hamiltonian is seen to be identical to that of Soliverez's up to and including third order—and to all orders if one makes a simple change in ‘normalization’. Computer treatment of finite matrices is discussed. The sixth-order Van Vleck hamiltonian recently given by Jørgensen and Pedersen in terms of commutators is derived in a more simple way.     

On a class of solvable Pauli-Schrödinger hamiltonians

To every scalar potential for which the bound state problem of the Schrödinger equation can be (at least partially) solved it is possible to associate a new potential containing Pauli matrices and for which the associated Pauli-Schrödinger equation can also be solved to the same extent. The new system can be regarded as a finite break of the supersymmetry that the original system and its supersymmetric partner have.     

Continuity and relative hamiltonians

Let ( _n ) _n 1 be a norm convergent sequence of normal states on a von Neumann algebraA with _n . Let (k _n ) _n1 be a strongly convergent sequence of self-adjoint elements ofA withk _n k. It is shown that the sequence of perturbed states converges in norm to . A related result holds forC ^*-algebras. A counter-example is provided to show that it is not sufficient to assume weak convergence of ( _n ) _n 1 even whenk _n=k for alln. However, conditions are given which, together with weak convergence, are sufficient. Relative entropy methods are used, and a relative entropy inequality is proved.     

Mean-field approximation and the collective transformation for Dyson boson hamiltonians

Hartree and TDA approximations are derived for non-hermitian boson hamiltonians such as the one obtained with the Dyson boson expansion. It is shown how to construct a collective transformation from the structure of Hartree and TDA bosons. This approach is applied to monopole pairing hamiltonian in the tin region.     

Linear relativistic hamiltonians

A 2(2J + 1)-component relativistic Hamiltonian H that describes free particles of mass m and spin J is said to be linear if it has the form $\text{H = h}\text{x}\text{?}\text{·}\text{p}\text{+ gm}$, where $\text{x}\text{\_.}\text{= i[H,}\text{x}\text{]}{}_{\mathrm{?}}$, h is a numerical factor, and g commutes with x and p. All such Hamiltonians are found, provided that the metric is either the unit matrix or ?₃ and provided that the theory is invariant under the discrete symmetries. If the operator Γ in the generator $\text{K}\text{=}\text{1}\text{2}\text{[}\text{x}\text{, H]}{}_{\mathrm{+}}\text{+ Γ}$ of Lorentz boosts is required to be local, there are only two possibilities; either Γ = 0, which generalizes the Dirac spin-$\text{1}\text{2}$ theory, or $\text{Γ = ??}{}_3\text{(}\text{1}\text{2m}\text{)}\text{S × p}$, which generalizes the Sakata-Taketani spin-0 and spin-1 theories. The relationship to linear manifestly covariant equations and its significance is discussed.     

On the averaged quantum dynamics by white-noise hamiltonians with and without dissipation

Exact results are derived on the averaged dynamics of a class of random quantum-dynamical systems in continuous space. Each member of the class is characterized by a Hamiltonian which is the sum of two parts. While one part is deterministic, time-independent and quadratic, the Weyl-Wigner symbol of the other part is a homogeneous Gaussian random field which is delta correlated in time, but smoothly correlated in position and momentum. The averaged dynamics of the resulting white-noise system is shown to be a monotone mixing increasing quantum-dynamical semigroup. Its generator is computed explicitly. Typically, in the course of time the mean energy of such a system grows linearly to infinity. In the second part of the paper an extended model is studied, which, in addition, accounts for dissipation by coupling the white-noise system linearly to a quantum-mechanical harmonic heat bath. It is demonstrated that, under suitable assumptions on the spectral density of the heat bath, the mean energy then saturates for long times.     

On the magneto-optical activity of singlet-triplet transitions in near-symmetric tops

Expressions are derived for the magnetic rotational strengths associated with singlet-triplet transitions in molecules which are Hund's coupling case (b) near-symmetric tops. The analysis closely parallels that developed by Hougen in his treatment of dipole intensities in these systems.     

Hidden hamiltonians of first-order equations

There are equations, like the KDV equation, of which the solutions behave like conservative systems although the equation is of first order in time. It is shown how equations of this kind can originate by a direct-product like process of fusion of two canonical conjugate variables. Conversely, for a class of dynamically well-behaved first-order equations a splitting of the independent variable into two conjugate parts and a corresponding hamiltonian functional can be found. It is shown how the action principle and the Noether theorem transform during this fusion or splitting process. A number of examples are discussed. It is shown how a KDV approximation can be derived directly from the hamiltonian of a second-order system without using the second-order wave equations.     

Generic properties of quantum Hall hamiltonians for finite systems

We discuss various qualitative aspects related to the Hall conductance of finite systems. We consider general, multiparticle hamiltonians with no symmetries. The identification of the Hall conductance with a first Chern character, its geometrical content and a theorem developed by Wigner and von Neumann on level crossing play key roles in the analysis.     

Energy scaling for model hamiltonians in molecule-surface scattering

It is shown that scattering by model potentials for gas-surface interactions scales exactly with energy. The semiclassical sudden approximation preserves this property. Some applications are discussed in the framework of the sudden approximation.     

Note on the dynamical invariants of time-dependent hamiltonians

With the help of a suitable example we demonstrate the importance of the choice of the dynamical algebra in finding the invariants of time-dependent hamiltonians. Several cases of time-dependent oscillators are discussed briefly illustrating the algebraic method of finding the invariants.     

Collective hamiltonians with Kac-Moody algebraic conditions

We describe the general framework for constructing collective-theory hamiltonians whose hermiticity requirements imply a Kac-Moody algebra of constraints on the associated jacobian. We give explicit examples for the algebras sl(2)_k and sl(3)_k. The reduction to W_n-constraints, relevant to n-matrix models, is described for the jacobians.     

Exact generator coordinate wave functions for some simple hamiltonians

We obtain the superconformal transformation laws of theN=4 supersymmetric Yang-Mills theory and explicitly demonstrate the closure of the algebra.     

On the validity of the non-perturbative solutions for multiphoton processes at high intensities

It is shown that a unifying sufficient condition for the validity of two classes of non-perturbative solutions to multiphoton transitions to be found in the literature, is the requirement of spectral degeneracy of the target Hamiltonian.