On the vortex flow in bounded domains

We consider the motion ofN vortices in bounded domains in IR². We prove that the set of initial positions which lead to a collapse of two or more vortices has Lebesgue measure zero. We extend this result to the stochastic motion of the vortices, where the stochasticity comes from a Wiener-noise term, which is added to the deterministic equation of motion.On leave of the Fachbereich Mathematik, RUB, 4630 Bochum, Federal Republic of Germany. Supported by a DFG-fellowshipPartially supported by Italian CNR     

First-passage times for random walks in bounded domains

We present a novel computational method of first-passage times between a starting site and a target site of regular bounded lattices. We derive accurate expressions for all the moments of this first-passage time, validated by numerical simulations. Their range of validity is discussed. We also consider the case of a starting site and two targets. In addition, we present the extension to continuous Brownian motion. These results are of great relevance to any system involving diffusion in confined media.     

Positive operators and automorphism groups of bounded symmetric domains

The spectrum of self-adjoint operators arising from unitary representations of semi-simple Lie groups is investigated. A series of irreducible unitary representations in which certain generators of non-compact one-parameter subgroups are realized by positive operators is described. These representations occur only for groups of automorphisms of bounded symmetric domains.     

Zero constant formula for first-passage observables in bounded domains

In this Letter, we develop an analytical approach which provides an explicit determination of mean first-passage times (MFPTs) for random walks in bounded domains for a wide class of transport processes. In particular, we derive for the first time explicit expressions of MFPTs for emblematic models of transport in complex media, such as diffusion on deterministic and random fractals. This approach relies on a scale-invariance hypothesis and a large volume expansion of the MFPT, which actually proves to be very accurate even for small system sizes as shown by numerical simulations. This explicit determination of MFPTs can be straightforwardly generalized to further useful first-passage observables such as occupation times and splitting probabilities.     

Number of degrees of freedom of the radiated field over multiple bounded domains

The problem of determining the number of degrees of freedom (NDF) of the field radiated by an electric current supported over a bounded rectilinear domain and observed over multiple bounded domains parallel to the source is addressed. The analysis is achieved by means of the singular value decomposition of the radiation operator so that the NDF is identified as the number of 'significant' singular values. The aim is to analyze whether the multidomain observation allows to increase the available NDF. By analytical arguments, we show that collecting data over multiple domains shapes the singular value behavior but it still presents a steep decay in correspondence to an index dictated by the observation domain that subtends the largest observation angular sector.     

On the unstable discrete spectrum of the linearized 2-D Euler equations in bounded domains

We investigate the behavior of the unstable discrete spectrum of the linearized 2-D Euler equation when the domain is smoothly perturbed. It is shown that when a self-adjoint Schrödinger-type operator undergoes a codimension-1 bifurcation it translates into a bifurcation in the linearized Euler equation associated with an instability either appearing or disappearing.We give sufficient conditions in order to observe smooth quadratic growth of the unstable eigencurves of the linearized Euler equation. The critical exponent is explicitly given as a function of the null-vector involved into the codimension-1 bifurcation using first and second-order moments of a Laplace transform.This analysis provides an explanation for the successive symmetry-breaking bifurcations observed in models of the mid-latitude oceans. An explicit example is also given.     

General Quantization

In the Darwinian evolution of physical theories, stability (genericity) has survival value. To convert a singular physical theory based on Lie algebras of several levels into a generic quantum theory with the same levels and nearly the same predictions and symmetries in a limited correspondence domain, it suffices to simplify the algebra of each level by a small homotopy (general quantization). This extends and unifies special relativization, general relativization, and canonical quantization. For exercise I general-quantize the scalar meson field in Minkowski space-time. The predictions of the resulting theory are finite, including its zero-point energy.This updates part of a talk given at Glafka 2005, Athens. quant-ph/0601002     

Action Quantization,Energy Quantization,and Time Parametrization

The additional information within a Hamilton–Jacobi representation of quantum mechanics is extra, in general, to the Schrödinger representation. This additional information specifies the microstate of \(\psi \) that is incorporated into the quantum reduced action, W. Non-physical solutions of the quantum stationary Hamilton–Jacobi equation for energies that are not Hamiltonian eigenvalues are examined to establish Lipschitz continuity of the quantum reduced action and conjugate momentum. Milne quantization renders the eigenvalue J. Eigenvalues J and E mutually imply each other. Jacobi’s theorem generates a microstate-dependent time parametrization \(t-\tau =\partial _E W\) even where energy, E, and action variable, J, are quantized eigenvalues. Substantiating examples are examined in a Hamilton–Jacobi representation including the linear harmonic oscillator numerically and the square well in closed form. Two byproducts are developed. First, the monotonic behavior of W is shown to ease numerical and analytic computations. Second, a Hamilton–Jacobi representation, quantum trajectories, is shown to develop the standard energy quantization formulas of wave mechanics.     

Retrospective on Quantization

Quantization is still a central problem of modern physics. One example of an unsolved problem is the quantization of Nambu mechanics. After a brief comment on the role of Harrison cohomology, this review concentrates on the central problem of quantization of QCD and, more generally, quark confinement seen as a problem of quantization. Several suggestions are made, some of them rather extravagant.     

Quantization of Co-Isotropic Subgroups

The purpose of this Letter is to show that the concept of co-isotropic subgroups of a Poisson–Lie group can be given a natural analogue in the context of quantum homogeneous spaces, explaining some of the special features of this theory. We will give examples related to some previously known and some new quantum homogeneous spaces of the two-dimensional Euclidean quantum groups.     

Quantization of nonintegrable maps

Using Heisenberg's matrix formulation of quantum mechanics, a method is given for quantizing volume-preserving polynomial mappings. The energy levels of the linear map are obtained exactly and those of the cubic, nonintegrable map are obtained approximately and numerically.     

Quantization of gauge theories

A path-integral procedure for quantizing gauge theories is proposed (on a heuristic level). The Hilbert space of physical states is constructed. Each physical state is represented by an infinite set of gauge equivalent configurations. All physical transition amplitudes are defined. In this approach, the “natural” value of parameter θ is zero.     

Geometric Quantization of Vector Bundles and the Correspondence with Deformation Quantization

I repeat my definition for quantization of a vector bundle. For the cases of the Toeplitz and geometric quantizations of a compact K?hler manifold, I give a construction for quantizing any smooth vector bundle, which depends functorially on a choice of connection on the bundle. Using this, the classification of formal deformation quantizations, and the formal, algebraic index theorem, I give a simple proof as to which formal deformation quantization (modulo isomorphism) is derived from a given geometric quantization. Received: 16 November 1998 / Accepted: 29 June 2000     

Quantization of chaotic systems

Starting from the semiclassical dynamical zeta function for chaotic Hamiltonian systems we use a combination of the cycle expansion method and a functional equation to obtain highly excited semiclassical eigenvalues. The power of this method is demonstrated for the anisotropic Kepler problem, a strongly chaotic system with good symbolic dynamics. An application of the transfer matrix approach of Bogomolny is presented leading to a significant reduction of the classical input and to comparable accuracy for the calculated eigenvalues.     

Quantization on Curves

Deformation quantization on varieties with singularities offers perspectives that are not found on manifolds. The Harrison component of Hochschild cohomology, vanishing on smooth manifolds, reflects information about singularities. The Harrison 2-cochains are symmetric and are interpreted in terms of abelian *-products. This paper begins a study of abelian quantization on plane curves over , being algebraic varieties of the form , where R is a polynomial in two variables; that is, abelian deformations of the coordinate algebra ). To understand the connection between the singularities of a variety and cohomology we determine the algebraic Hochschild (co)homology and its Barr–Gerstenhaber–Schack decomposition. Homology is the same for all plane curves , but the cohomology depends on the local algebra of the singularity of R at the origin. The Appendix, by Maxim Kontsevich, explains in modern mathematical language a way to calculate Hochschild and Harrison cohomology groups for algebras of functions on singular planar curves etc. based on Koszul resolutions.      

Quantization of Brownian Motion

Following our work on the quantization of nonconservative systems using fractional calculus, the canonical quantization of a system with Brownian motion is carried out according to the Dirac method. A suitable Lagrangian corresponding to the Langevin equation is set up. Further, a Hamiltonian is constructed and is transformed to Schrödinger's equation which is solved.     

Quantization of Galilean systems

The Kostant quantization method is applied to the Galilean group.     

Quantization of bag-like solitons

The method of collective coordinates is used to quantize bag-like solitons formed by scalar and spinor fields. This method leads to approximate wave functions for quarks in the bag that are and spinor fields. This method leads to approximate wave functions for quarks in the bag that are orthogonal to the translation modes. Solutions are given for the MIT bag limit of the fields.