Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSpectral inverse problems for compact Hankel operators
172
0
0.0
(
0
)
Authors
Patrick Gerard
Ask ChatGPT about the research
No Arabic abstract
Given two arbitrary sequences $(lambda_j)_{jge 1}$ and $(mu_j)_{jge 1}$ of real numbers satisfying $$|lambda_1|>|mu_1|>|lambda_2|>|mu_2|>...>| lambda_j| >| mu_j| to 0 ,$$ we prove that there exists a unique sequence $c=(c_n)_{ninZ_+}$, real valued, such that the Hankel operators $Gamma_c$ and $Gamma_{tilde c}$ of symbols $c=(c_{n})_{nge 0}$ and $tilde c=(c_{n+1})_{nge 0}$ respectively, are selfadjoint compact operators on $ell^2(Z_+)$ and have the sequences $(lambda_j)_{jge 1}$ and $(mu_j)_{jge 1}$ respectively as non zero eigenvalues. Moreover, we give an explicit formula for $c$ and we describe the kernel of $Gamma_c$ and of $Gamma_{tilde c}$ in terms of the sequences $(lambda_j)_{jge 1}$ and $(mu_j)_{jge 1}$. More generally, given two arbitrary sequences $(rho_j)_{jge 1}$ and $(sigma_j)_{jge 1}$ of positive numbers satisfying $$rho_1>sigma_1>rho_2>sigma_2>...> rho_j> sigma_j to 0 ,$$ we describe the set of sequences $c=(c_n)_{ninZ_+}$ of complex numbers such that the Hankel operators $Gamma_c$ and $Gamma_{tilde c}$ are compact on $ell ^2(Z_+)$ and have sequences $(rho_j)_{jge 1}$ and $(sigma_j)_{jge 1}$ respectively as non zero singular values.
rate research
Read More
We consider inverse boundary value problems for general real principal type differential operators. The first results state that the Cauchy data set uniquely determines the scattering relation of the operator and bicharacteristic ray transforms of lower order coefficients. We also give two different boundary determination methods for general operators, and prove global uniqueness results for determining coefficients in nonlinear real principal type equations. The article presents a unified approach for treating inverse boundary problems for transport and wave equations, and highlights the role of propagation of singularities in the solution of related inverse problems.
In this paper we explore the theory of fractional powers of non-negative (and not necessarily self-adjoint) operators and its amazing relationship with the Chebyshev polynomials of the second kind to obtain results of existence, regularity and behavior asymptotic of solutions for linear abstract evolution equations of $n$-th order in time, where $ngeqslant3$. We also prove generalizations of classical results on structural damping for linear systems of differential equations.
Soliton theory and the theory of Hankel (and Toeplitz) operators have stayed essentially hermetic to each other. This paper is concerned with linking together these two very active and extremely large theories. On the prototypical example of the Cauchy problem for the Korteweg-de Vries (KdV) equation we demonstrate the power of the language of Hankel operators in which symbols are conveniently represented in terms of the scattering data for the Schrodinger operator associated with the initial data for the KdV equation. This approach yields short-cuts to already known results as well as to a variety of new ones (e.g. wellposedness beyond standard assumptions on the initial data) which are achieved by employing some subtle results for Hankel operators.
In this paper we consider second order parabolic partial differential equations subject to the Dirichlet boundary condition on smooth domains. We establish weighted $L_{q}$-maximal regularity in weighted Triebel-Lizorkin spaces for such parabolic problems with inhomogeneous boundary data. The weights that we consider are power weights in time and space, and yield flexibility in the optimal regularity of the initial-boundary data, allow to avoid compatibility conditions at the boundary and provide a smoothing effect. In particular, we can treat rough inhomogeneous boundary data.
In this article we present three robust instability mechanisms for linear and nonlinear inverse problems. All of these are based on strong compression properties (in the sense of singular value or entropy number bounds) which we deduce through either strong global smoothing, only weak global smoothing or microlocal smoothing for the corresponding forward operators, respectively. As applications we for instance present new instability arguments for unique continuation, for the backward heat equation and for linear and nonlinear Calderon type problems in general geometries, possibly in the presence of rough coefficients. Our instability mechanisms could also be of interest in the context of control theory, providing estimates on the cost of (approximate) controllability in rather general settings.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
