Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTrace Formulas for a Class of non-Fredholm Operators: A Review
89
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We review previous work on spectral flow in connection with certain self-adjoint model operators ${A(t)}_{tin mathbb{R}}$ on a Hilbert space $mathcal{H}$, joining endpoints $A_pm$, and the index of the operator $D_{A}^{}= (d/d t) + A$ acting in $L^2(mathbb{R}; mathcal{H})$, where $A$ denotes the operator of multiplication $(A f)(t) = A(t)f(t)$. In this article we review what is known when these operators have some essential spectrum and describe some new results in terms of associated spectral shift functions. We are especially interested in extensions to non-Fredholm situations, replacing the Fredholm index by the Witten index, and use a particular $(1+1)$-dimensional model setup to illustrate our approach based on spectral shift functions.
rate research
Read More
We examine quantum normal typicality and ergodicity properties for quantum systems whose dynamics are generated by Hamiltonians which have residual degeneracy in their spectrum and resonance in their energy gaps. Such systems can be considered atypical in the sense that degeneracy, which is usually a sign of symmetry, is naturally broken in typical systems due to stochastic perturbations. In particular, we prove a version of von Neumanns quantum ergodic theorem, where a modified condition needs to hold in order to have normal typicality and ergodicity. As a result, we show that degeneracy of spectrum does not considerably modify the condition of the theorem, whereas the existence of resonance is more dominant for obstructing ergodicity.
A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group. In this paper, we firstly introduce the motivation for study of the generalized Maxwell operators and bosonic Laplacians (also known as the higher spin Laplace operators). Then, with the help of connections between Rarita-Schwinger type operators and bosonic Laplacians, we solve Poissons equation for bosonic Laplacians. A representation formula for bounded solutions to Poissons equation in Euclidean space is also provided. In the end, we provide Greens formulas for bosonic Laplacians in scalar-valued and Clifford-valued cases, respectively. These formulas reveal that bosonic Laplacians are self-adjoint with respect to a given $L^2$ inner product on certain compact supported function spaces.
We study the inverse electrostatic and elasticity problems associated with Poisson and Navier equations. The uniqueness of solutions of these problems is proved for piecewise constant electric charge and internal stress distributions having a checkered structure: they are constant on rectangular blocks. Such distributions appear naturally in practical applications. We also discuss computational challenges arising in the numerical implementation of our method.
We study the relaxation to equilibrium for a class linear one-dimensional Fokker-Planck equations characterized by a particular subcritical confinement potential. An interesting feature of this class of Fokker-Planck equations is that, for any given probability density $e(x)$, the diffusion coefficient can be built to have $e(x)$ as steady state. This representation of the equilibrium density can be fruitfully used to obtain one-dimensional Wirtinger-type inequalities and to recover, for a sufficiently regular density $e(x) $, a polynomial rate of convergence to equilibrium.Numerical results then confirm the theoretical analysis, and allow to conjecture that convergence to equilibrium with positive rate still holds for steady states characterized by a very slow polynomial decay at infinity.
Continuing a line of investigation initiated in [11] exploring the connections between Jost and Evans functions and (modified) Fredholm determinants of Birman-Schwinger type integral operators, we here examine the stability index, or sign of the first nonvanishing derivative at frequency zero of the characteristic determinant, an object that has found considerable use in the study by Evans function techniques of stability of standing and traveling wave solutions of partial differential equations (PDE) in one dimension. This leads us to the derivation of general perturbation expansions for analytically-varying modified Fredholm determinants of abstract operators. Our main conclusion, similarly in the analysis of the determinant itself, is that the derivative of the characteristic Fredholm determinant may be efficiently computed from first principles for integral operators with semi-separable integral kernels, which include in particular the general one-dimensional case, and for sums thereof, which latter possibility appears to offer applications in the multi-dimensional case. A second main result is to show that the multi-dimensional characteristic Fredholm determinant is the renormalized limit of a sequence of Evans functions defined in [23] on successive Galerkin subspaces, giving a natural extension of the one-dimensional results of [11] and answering a question of [27] whether this sequence might possibly converge (in general, no, but with renormalization, yes). Convergence is useful in practice for numerical error control and acceleration.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
