Related papers: The characteristic polynomial of the next-nearest-…

Given a non-hermitean matrix M, the structure of its minimal polynomial encodes whether M is diagonalizable or not. This note will explain how to determine the minimal polynomial of a matrix without going through its characteristic…

We show that the exact energy eigenvalues and eigenfunctions of the Schrodinger equation for charged particles moving in certain class of non-central potentials can be easily calculated analytically in a simple and elegant manner by using…

The evolution of entanglement for two identical two-level atoms coupled to a resonant thermal field is studied for two different families of input states. Entanglement enhancement is predicted for a well defined region of the parameter…

The behaviour of a system of two two-level atoms, both identical as well as non-identical, under the influence of a continuously varying dipole-dipole coupling parameter for a wide range of other parameters such as the atomic level…

Rydberg-atom ensembles are switched from a weakly- into a strongly-interacting regime via adiabatic transformation of the atoms from an approximately non-polar into a highly dipolar quantum state. The resultant electric dipole-dipole forces…

Using Chebyshev polynomialsof both kinds, we construct rational fractions which are convergents of the smallest root of $x^2-\alpha x+1$ for $\alpha=3,4,5,\dots$.Some of the underlying identities suggest an identity involving…

The multidimensional chain rule formula for analytic functions and its generalisation to higher derivatives perfectly work in the algebraic setting in characteristic zero. In positive characteristic one runs into problems due to…

In this note we consider a pair of particles moving on the positive half-line with the pairing generated by a hard-wall potential. This model was first introduced in [arXiv:1604.06693] and later applied to investigate condensation of pairs…

By applying a Gr\"{o}bner-Shirshov basis of the symmetric group $S_{n}$, we give two formulas for Schubert polynomials, either of which involves only nonnegative monomials. We also prove some combinatorial properties of Schubert…

We explore the block nature of the matrix representation of multiplex networks, introducing a new formalism to deal with its spectral properties as a function of the inter-layer coupling parameter. This approach allows us to derive…

We consider the problem of learning local quantum Hamiltonians given copies of their Gibbs state at a known inverse temperature, following Haah et al. [2108.04842] and Bakshi et al. [arXiv:2310.02243]. Our main technical contribution is a…

We prove the bivariate Cayley-Hamilton theorem, a powerful generalization of the classical Cayley-Hamilton theorem. The bivariate Cayley-Hamilton theorem has three direct corollaries that are usually proved independently: The classical…

In this paper, we investigate some properties of Chebyshev polynomials arising from non-linear differential equations. From our investigation, we derive some new and interesting identities on Chebyshev polynomials.

We analyze the characteristic polynomial associated to an ellipsoid and another quadric in the context of the contact detection problem. We obtain a necessary and sufficient condition for an efficient method to detect contact. This…

We calculate the exceptional points of the eigenvalues of several parameter-dependent Hamiltonian operators of mathematical and physical interest. We show that the calculation is greatly facilitated by the application of the discriminant to…

Chebyshev interpolation polynomials exhibit the exponential approximation property to analytic functions on a cube. Based on the Chebyshev interpolation polynomial approximation, we propose iterative polynomial approximation algorithms to…

We introduce a q-analogue of the classical Zeta polynomial of finite partially ordered sets, as a polynomial in one variable x with coefficients depending on the indeterminate q. We prove some properties of this polynomial invariant,…

Ionization-excitation of an atom induced by the absorption of a single photon in the presence of a neighbouring atom is studied. The latter is, first, resonantly photoexcited and, afterwards, transfers the excitation energy radiationlessly…

Let $k$ be a field of characteristic two. We prove that a non constant monic polynomial $f\in k[X]$ of degree $n$ is the minimal/characteristic polynomial of a symmetric matrix with entries in $k$ if and only if it is not the product of…

The coherent nonlinear process where a single photon simultaneously excites two or more two-level systems (qubits) in a single-mode resonator has recently been theoretically predicted. Here we explore the case where the two qubits are…