Concrete Hilbert spaces for quantum systems with infinitely many degrees of freedom

We demonstrate a method to describe quantum systems with infinitely many degrees of freedom in concrete Hilbert spaces, using the electromagnetic radiation field as a well-known example of such a system. Since our method is not only applicable to the case of countably many but even to the case of uncountably many degrees of freedom, there is no need for a finite quantization volume in radiation theory.     

A Nekhoroshev-type theorem for Hamiltonian systems with infinitely many degrees of freedom

We study the propagation of lattice vibrations in models of disordered, classical anharmonic crystals. Using classical perturbation theory with an optimally chosen remainder term (i.e. a Nekhoroshev-type scheme), we are able to show that vibrations corresponding to localized initial conditions do essentially not propagate through the crystal up to times larger than any inverse power of the strength of the anharmonic couplings.     

On wavefunctions of discrete spectrum states of integrable quantum systems with N degrees of freedom

We propose a method for constructing the wavefunctions of excited states of the discrete spectrum for an N-dimensional Schrödinger equation in which the structure of a potential is determined by systems of roots of the Lie algebras.     

Minimum-uncertainty states of systems with many degrees of freedom

The minimum-uncertainty states for systems with many degrees of freedom are investigated. The limiting situation, relevant to the electromagnetic field, is discussed and it is pointed out that the states that minimize Δp Δq do not tend to the coherent states. These latter, including the vacuum state, are minimum-uncertainty states for the transverse electric and magnetic fields. In an appendix a direct method to obtain the wave functional for the vacuum is presented.     

Resonance transport in near-integrable systems with many degrees of freedom

When a small perturbation is added to the Hamiltonian function of an integrable multi-dimensional nonlinear system, classical transport processes on the action space can be dramatically altered when the phase point is inside a nonlinear resonance. This phenomenon, called “streaming” here, can result in the rapid migration of the phase point along the resonance, and a corresponding enhanced transport through the action space. The magnitude of the enhanced transport depends on the direction of the resonant libration induced by the perturbation.     

The classical mechanics of one-dimensional systems of infinitely many particles

We apply the existence theorem for solutions of the equations of motion for infinite systems to study the time evolution of measures on the set of locally finite configurations of particles. The set of allowed initial configurations and the time evolution mappings are shown to be measurable. It is shown that infinite volume limit states of thermodynamic ensembles at low activity or for positive potentials are concentrated on the set of allowed initial configurations and are invariant under the time evolution. The total entropy per unit volume is shown to be constant in time for a large class of states, if the potential satisfies a stability condition.On leave from: Department of Mathematics, University of California, Berkeley, California.     

The classical mechanics of one-dimensional systems of infinitely many particles

We prove a global existence and uniqueness theorem for solutions of the classical equations of motion for a one-dimensional system of infinitely many particles interacting by finite-range two-body forces which satisfy a Lipschitz condition.     

Intrinsic stochasticity with many degrees of freedom

We study a simple model equation describing a system with an infinity of degrees of freedom which displays an intrinsically chaotic behavior. Some concepts of fully developed turbulence are discussed in relation to this model. We also develop an approach based on Lyapunov exponent measurements. Numerical results on the distribution of Lyapunov numbers and the power spectrum of the associated Lyapunov vectors are presented and briefly discussed.     

New integrable classical system with two degrees of freedom

It is shown that a classical system with hamiltonian $\text{H = (p}{}^2{}_1\text{+ p}{}^2{}_2\text{)/2 + λ(q}{}_1\text{q}{}_2\text{)}{}^{\mathrm{<mtext>-</mtext><mtext>2</mtext><mtext>3</mtext>}}$ possesses a constant of the motion $\text{K = p}{}_1\text{p}{}_2\text{(p}{}_1\text{q}{}_2\text{? p}{}_2\text{q}{}_1\text{) + 2λ(p}{}_2\text{q}{}_2\text{? p}{}_1\text{q}{}_1\text{)(q}{}_1\text{q}{}_2\text{)}{}^{\mathrm{<mtext>-</mtext><mtext>2</mtext><mtext>3</mtext>}}$; the quantum system with the same hamiltonian has no constants of the motion which are polynomials in the momenta of order not higher than three, except H.     

Influence of internal degrees of freedom on the structure of one-dimensional systems

Summary The influence of internal degrees of freedom on the behaviour of one-dimensional systems is discussed. For systems with half-filled bands the coupling to internalviz. lattice coordinates decides whether Peierls distortion is caused by intramonomer coordinates or by a lattice coordinate. Thereby the various intramonomer degrees of freedom act cooperatively. We show that there is a small regime of parameters where both kinds of distortion exist simultaneously. For increasing temperature we find that distortions can also move from the lattice coordinate to the intramonomer coordinate.     

Transient response of unidirectional vibration isolators with many degrees of freedom

Following a procedure developed by the authors for the investigation of the dynamic behaviour of an n-degree-of-freedom linear elastic system with damping, the corresponding transient response is dealt with. The procedure is based on a class of polynomials depending on the mechanical properties of the system and the Laplace transformation is applied. As an example, a four-degree-of-freedom is examined.     

From classical mechanics with doubled degrees of freedom to quantum field theory for nonconservative systems

The 2×2$2\times 2$-matrix structure of Green?s functions is a common feature for the real-time formalisms of quantum field theory under thermal situations, such as the closed time path formalism and Thermo Field Dynamics (TFD). It has been believed to originate from quantum nature. Recently, Galley has proposed the Hamilton?s principle with initial data for nonconservative classical systems, doubling each degree of freedom [1]. We show that the Galley?s Hamilton formalism can be extended to quantum field and that the resulting theory is naturally identical with nonequilibrium TFD.     

Excited-state quantum phase transitions in systems with two degrees of freedom: II. Finite-size effects

This article extends our previous analysis Stránský et al. (2014) of Excited-State Quantum Phase Transitions (ESQPTs) in systems of dimension two. We focus on the oscillatory component of the quantum state density in connection with ESQPT structures accompanying a first-order ground-state transition. It is shown that a separable (integrable) system can develop rather strong finite-size precursors of ESQPT expressed as singularities in the oscillatory component of the state density. The singularities originate in effectively 1-dimensional dynamics and in some cases appear in multiple replicas with increasing excitation energy. Using a specific model example, we demonstrate that these precursors are rather resistant to proliferation of chaotic dynamics.     

Filters with infinitely many components

With the use of a suitable assumption about the structure of the class of experimental filters, it is shown that the sequence of alternating replicas of two filters is their greatest lower bound, as Jauch suggests. A generalization of his suggestion yields the greatest lower bound of a denumerable set of filters. The criteria of admissibility of filters are briefly discussed.