相关论文: On the instability of eigenvalues
In the present work we show that the joint probability distribution of the eigenvalues can be expressed in terms of a differential operator acting on the distribution of some other matrix quantities. Those quantities might be the diagonal…
We study resonances generated by rank one perturbations of selfadjoint operators with eigenvalues embedded in the continuous spectrum. Instability of these eigenvalues is analyzed and almost exponential decay for the associated resonant…
Diagrammatic perturbation theory is a powerful tool for the investigation of interacting many-body systems, the self-energy operator $\Sigma$ encoding all the variety of scattering processes. In the simplest scenario of correlated electrons…
We consider the problem of exact experimental determination of the boundaries of Stability Zones for magneto-conductivity in normal metals in the space of directions of $\, {\bf B} \, $. As can be shown, this problem turns out to be…
We consider the empirical eigenvalue distribution of an $m\times m$ principal submatrix of an $n\times n$ random unitary matrix distributed according to Haar measure. For $n$ and $m$ large with $\frac{m}{n}=\alpha$, the empirical spectral…
Fermi's Golden Rule (FGR) applies in the limit where an initial quantum state is weakly coupled to a {\it continuum} of other final states overlapping its energy. Here we investigate what happens away from this limit, where the set of final…
We study a 2D scalar harmonic wave transmission problem between a classical dielectric and a medium with a real-valued negative permittivity/permeability which models a metal at optical frequency or an ideal negative metamaterial. We…
The (interior) transmission eigenvalue problems are a type of non-elliptic, non-selfadjoint and nonlinear spectral problems that arise in the theory of wave scattering. They connect to the direct and inverse scattering problems in many…
In a previous paper (J. Phys. A 36, 11807 (2003)), we introduced the `asymptotic iteration method' for solving second-order homogeneous linear differential equations. In this paper, we study perturbed problems in quantum mechanics and we…
We investigate almost-degenerate perturbation theory of eigenvalue problems, using spectral projectors, also named density matrices. When several eigenvalues are close to each other, the coefficients of the perturbative series become…
Recently delivered lectures on Self-Referential Mathematics, [2], at the Department of Mathematics and Applied Mathematics, University of Pretoria, are briefly presented. Comments follow on the subject, as well as on Inconsistent…
We establish a set of exact sum rules that relate the interatomic force constants to the frequency-dependent electromagnetic susceptibility of a solid or molecule, thereby generalizing the long-established principles of rototranslational…
Level shift operators describe the second order displacement of eigenvalues under perturbation. They play a central role in resonance theory and ergodic theory of open quantum systems at positive temperatures. We exhibit intrinsic…
We study the perturbative correction to the ground state energy eigenvalue of a 2-dimensional dilute fermi gas with weak short-range two body repulsion. From the structure of the energy shift we infer the presence of an induced two body…
We present first-order perturbation analysis of a simple eigenvalue and the corresponding right and left eigenvectors of a general square matrix, not assumed to be Hermitian or normal. The eigenvalue result is well known to a broad…
The study of a machine learning problem is in many ways is difficult to separate from the study of the loss function being used. One avenue of inquiry has been to look at these loss functions in terms of their properties as scoring rules…
These lectures deal with: (1) a brief review of the theory of flexible random manifolds (with fixed intrinsic metric), connected to the physics of polymerized membranes, and of the effect of extrinsic curvature (crumpling transitions); (2)…
In this paper we consider generalized nonlinear Schr\"odinger equations with external potentials. we compute the forth and the sixed order Fermi Golden Rules (FGR), conjectured in our previous papers, which is used in a study of the…
We introduce the concept of shape partition of a tensor and formulate a general tensor eigenvalue problem that includes all previously studied eigenvalue problems as special cases. We formulate irreducibility and symmetry properties of a…
Eigenvalues inequalities involving (log) convex/concav functions and Hermitian matrices, positive unital maps are considered. Simple proofs of Bhatia-Kittaneh inequality and Naimark dilation theorem are given.