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We describe an algorithm to compute the extremal eigenvalues and corresponding eigenvectors of a symmetric matrix by solving a sequence of Quadratic Binary Optimization problems. This algorithm is robust across many different classes of…
We show 2 matrices that have identical eigenvalues but different eigenfunctions. This shows that in obtaining two body nuclear matrix elements empirically, it is not sufficient to consider only energy levels. Other quantities like…
We investigate the passivity constraints on the relations between transmission, reflection, and absorption eigenvalues in linear time-invariant systems. Using techniques from matrix analysis, we derive necessary and sufficient conditions…
We formulate gaussian and circular random-matrix models representing a coupled system consisting of an absorbing and an amplifying resonator, which are mutually related by a generalized time-reversal symmetry. Motivated by optical…
We discuss an eigenvalue problem which arises in the studies of asymptotic stability of a self-similar attractor in the sigma model. This problem is rather unusual from the viewpoint of the spectral theory of linear operators and requires…
We analyze statistical properties of complex eigenvalues of random matrices $\hat{A}$ close to unitary. Such matrices appear naturally when considering quantized chaotic maps within a general theory of open linear stationary systems with…
The distribution of eigenvalues of N times N random matrices in the limit N to infinity is the solution to a variational principle that determines the ground state energy of a confined fluid of classical unit charges. This fact is a…
We consider a two-parameter non hermitean quantum-mechanical hamiltonian that is invariant under the combined effects of parity and time reversal transformation. Numerical investigation shows that for some values of the potential parameters…
It is known (see e.g. [2], [4], [5], [6]) that continuous variations in the entries of a complex square matrix induce continuous variations in its eigenvalues. If such a variation arises from one real parameter $\alpha \in [0, 1]$, then the…
We study a quantum system of $p$ commuting matrices and find that such a quantum system requires an explicit curvature dependent potential in its Lagrangian for the system to have a finite energy ground state. In contrast it is possible to…
We give an $O(n)$ time and space algorithm for constructing a diagonal matrix congruent to A+xI, where A is the adjacency matrix of a cograph and $x\in \mathbb{R}$. Applications include determining the number of eigenvalues of a cograph's…
We consider real non-symmetric matrices and their factorisation as a product of real symmetric matrices. The number of complex eigenvalues of the original matrix reveals restrictions on such factorisations as we shall prove.
Optical properties of materials related to light absorption and scattering are explained by the excitation of electrons. The Bethe-Salpeter equation is the state-of-the-art approach to describe these processes from first principles (ab…
We study the interaction potential between two polyions inside a colloidal suspension. It is shown that at large separation the interaction potential is purely repulsive, with the induced attractive interactions being doubly screened. For…
We discuss complementarity relations in a bipartite continuous variable system. Building up from the work done on discrete d-dimensional systems, we prove that for symmetric two-mode states, quantum complementarity relations can be put in a…
We introduce right eigenvalues and subeigenvalues for square dual complex matrices. An $n \times n$ dual complex Hermitian matrix has exactly $n$ right eigenvalues and subeigenvalues, which are all real. The Hermitian matrix is positive…
For bounded domains, eigenvalues and eigenfunctions of double layer potentials are considered. The aim of this paper is to establish some relationships between eigenvalues, eigenfunctions and the geometry of domain boundaries.
From cement cohesion to DNA condensation, a proper statistical physics treatment of systems with long range forces is important for a number of applications in physics, chemistry, and biology. We compute here the effective force between…
Starting from a mistake done by a student, we discover an unexpected method of finding both eigenvectors for a $2\times2$ matrix with distinct eigenvalues in a single computation. We discuss a connection with the Cayley-Hamilton theorem,…
We derive two complementarity relations that constrain the individual and bipartite properties that may simultaneously exist in a multi-qubit system. The first expression, valid for an arbitrary pure state of n qubits, demonstrates that the…