116 research outputs found
Hardy-Carleman Type Inequalities for Dirac Operators
General Hardy-Carleman type inequalities for Dirac operators are proved. New
inequalities are derived involving particular traditionally used weight
functions. In particular, a version of the Agmon inequality and Treve type
inequalities are established. The case of a Dirac particle in a (potential)
magnetic field is also considered. The methods used are direct and based on
quadratic form techniques
On a Watson-like Uniqueness Theorem and Gevrey Expansions
We present a maximal class of analytic functions, elements of which are in
one-to-one correspondence with their asymptotic expansions. In recent decades
it has been realized (B. Malgrange, J. Ecalle, J.-P. Ramis, Y. Sibuya et al.),
that the formal power series solutions of a wide range of systems of ordinary
(even non-linear) analytic differential equations are in fact the Gevrey
expansions for the regular solutions. Watson's uniqueness theorem belongs to
the foundations of this new theory. This paper contains a discussion of an
extension of Watson's uniqueness theorem for classes of functions which admit a
Gevrey expansion in angular regions of the complex plane with opening less than
or equal to (\frac \pi k,) where (k) is the order of the Gevrey expansion. We
present conditions which ensure uniqueness and which suggest an extension of
Watson's representation theorem. These results may be applied for solutions of
certain classes of differential equations to obtain the best accuracy estimate
for the deviation of a solution from a finite sum of the corresponding Gevrey
expansion.Comment: 18 pages, 4 figure
Zeta function regularization in Casimir effect calculations and J.S. Dowker's contribution
A summary of relevant contributions, ordered in time, to the subject of
operator zeta functions and their application to physical issues is provided.
The description ends with the seminal contributions of Stephen Hawking and
Stuart Dowker and collaborators, considered by many authors as the actual
starting point of the introduction of zeta function regularization methods in
theoretical physics, in particular, for quantum vacuum fluctuation and Casimir
effect calculations. After recalling a number of the strengths of this powerful
and elegant method, some of its limitations are discussed. Finally, recent
results of the so called operator regularization procedure are presented.Comment: 16 pages, dedicated to J.S. Dowker, version to appear in
International Journal of Modern Physics
The stability for the Cauchy problem for elliptic equations
We discuss the ill-posed Cauchy problem for elliptic equations, which is
pervasive in inverse boundary value problems modeled by elliptic equations. We
provide essentially optimal stability results, in wide generality and under
substantially minimal assumptions. As a general scheme in our arguments, we
show that all such stability results can be derived by the use of a single
building brick, the three-spheres inequality.Comment: 57 pages, review articl
The Boundary Convergence of the Steady Zero-Temperature-Driven Hard Spheres
We study the fundamental problem of two gas species in two dimensional
velocity space whose molecules collide as hard circles in the presence of a
flat boundary and with dependence on only one space dimension. The case of
three-dimensional velocity space is a generalization. More speciffically the
linear problem arising when the second gas dominates as a flow with constant
velocity (and hence zero temperature) is considered. The boundary condition
adopted consists of prescribing the outgoing velocity distribution at the wall.
It is discovered that the presence of the boundary under general assumptions on
the outgoing distribution ensures the convergence of a series of path integrals
and thus a convenient representation for the solution is obtained.Comment: 13 pages, 2 figure
Scaling Theory of Conduction Through a Normal-Superconductor Microbridge
The length dependence is computed of the resistance of a disordered
normal-metal wire attached to a superconductor. The scaling of the transmission
eigenvalue distribution with length is obtained exactly in the metallic limit,
by a transformation onto the isobaric flow of a two-dimensional ideal fluid.
The resistance has a minimum for lengths near l/Gamma, with l the mean free
path and Gamma the transmittance of the superconductor interface.Comment: 8 pages, REVTeX-3.0, 3 postscript figures appended as self-extracting
archive, INLO-PUB-94031
Self-dual noncommutative \phi^4-theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory
We study quartic matrix models with partition function Z[E,J]=\int dM
\exp(trace(JM-EM^2-(\lambda/4)M^4)). The integral is over the space of
Hermitean NxN-matrices, the external matrix E encodes the dynamics, \lambda>0
is a scalar coupling constant and the matrix J is used to generate correlation
functions. For E not a multiple of the identity matrix, we prove a universal
algebraic recursion formula which gives all higher correlation functions in
terms of the 2-point function and the distinct eigenvalues of E. The 2-point
function itself satisfies a closed non-linear equation which must be solved
case by case for given E. These results imply that if the 2-point function of a
quartic matrix model is renormalisable by mass and wavefunction
renormalisation, then the entire model is renormalisable and has vanishing
\beta-function.
As main application we prove that Euclidean \phi^4-quantum field theory on
four-dimensional Moyal space with harmonic propagation, taken at its
self-duality point and in the infinite volume limit, is exactly solvable and
non-trivial. This model is a quartic matrix model, where E has for N->\infty
the same spectrum as the Laplace operator in 4 dimensions. Using the theory of
singular integral equations of Carleman type we compute (for N->\infty and
after renormalisation of E,\lambda) the free energy density
(1/volume)\log(Z[E,J]/Z[E,0]) exactly in terms of the solution of a non-linear
integral equation. Existence of a solution is proved via the Schauder fixed
point theorem.
The derivation of the non-linear integral equation relies on an assumption
which we verified numerically for coupling constants 0<\lambda\leq (1/\pi).Comment: LaTeX, 64 pages, xypic figures. v4: We prove that recursion formulae
and vanishing of \beta-function hold for general quartic matrix models. v3:
We add the existence proof for a solution of the non-linear integral
equation. A rescaling of matrix indices was necessary. v2: We provide
Schwinger-Dyson equations for all correlation functions and prove an
algebraic recursion formula for their solutio
Celebrating Cercignani's conjecture for the Boltzmann equation
Cercignani's conjecture assumes a linear inequality between the entropy and
entropy production functionals for Boltzmann's nonlinear integral operator in
rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities
and spectral gap inequalities, this issue has been at the core of the renewal
of the mathematical theory of convergence to thermodynamical equilibrium for
rarefied gases over the past decade. In this review paper, we survey the
various positive and negative results which were obtained since the conjecture
was proposed in the 1980s.Comment: This paper is dedicated to the memory of the late Carlo Cercignani,
powerful mind and great scientist, one of the founders of the modern theory
of the Boltzmann equation. 24 pages. V2: correction of some typos and one
ref. adde
Asymptotic Improvement of Resummation and Perturbative Predictions in Quantum Field Theory
The improvement of resummation algorithms for divergent perturbative
expansions in quantum field theory by asymptotic information about perturbative
coefficients is investigated. Various asymptotically optimized resummation
prescriptions are considered. The improvement of perturbative predictions
beyond the reexpansion of rational approximants is discussed.Comment: 21 pages, LaTeX, 3 tables; title shortened; typographical errors
corrected; minor changes of style; 2 references adde
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Spectral bounds for the Neumann-Poincaré operator on planar domains with corners
The boundary double layer potential, or the Neumann-Poincaré operator, is studied on the Sobolev space of order 1/2 along the boundary, coinciding with the space of charges giving rise to double layer potentials with finite energy in the whole space. Poincaré’s program of studying the spectrum of the boundary double layer potential is developed in complete generality on closed Lipschitz hypersurfaces in euclidean space. Furthermore, the Neumann-Poincaré operator is realized as a singular integral transform bearing similarities to the Beurling-Ahlfors transform in 2 dimensions. As an application, in the case of planar curves with corners, bounds for the spectrum of the Neumann-Poincaré operator are derived from recent results in quasi-conformal mapping theory
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