J. S. Dehesa

On orthogonal polynomials with perturbed recurrence relations

Abstract Orthogonal polynomials may be fully characterized by the following recurrence relation: Pn(x) = (x − βn-1)Pn-1(x)-γn-1Pn-2(x), with P0(x)=1, P1(x) = x - β0 and γn ≠ 0. Here we study how the structure and the spectrum of these polynomials get modified by a local perturbation in the β and γ parameters of a co-recursive (βk → βk + μ), co-dilated (γk → λγk and co-modified (βk → βk + μ; γk → λγk) nature for an arbitrary (but fixed) kth element (1 ⩽ k). Specifically, Stieltjes functions, differential equations and distributions of zeros as well as representations of the new perturbed polynomials in terms of the old unperturbed ones are given. This type of problems is strongly related to the boundary value problems of finite-difference equations and to the quantum mechanical study of physical many-body systems (atoms, molecules, nuclei and solid state systems).

Information-theoretic measures for Morse and Pöschl–Teller potentials

The spreading of the quantum-mechanical probability cloud for the ground state of the Morse and modified Pöschl–Teller potentials, which controls the chemical and physical properties of some molecular systems, is studied in position and momentum space by means of global (Shannons information entropy, variance) and local (Fishers information) information-theoretic measures. We establish a general relation between variance and Fishers information, proving that, in the case of a real-valued and symmetric wavefunction, the well-known Cramer–Rao and Heisenberg uncertainty inequalities are equivalent. Finally, we discuss the asymptotics of all three information measures, showing that the ground state of these potentials saturates all the uncertainty relations in an appropriate limit of the parameter.

Uncertainty relation for Fisher information of D-dimensional single-particle systems with central potentials

An uncertainty Fisher information relation in quantum mechanics is derived for multidimensional single-particle systems with central potentials. It is based on the concept of Fisher information in the two complementary position and momentum spaces, which is a gradient functional of the corresponding probability distributions. The lower bound of the product of position and momentum Fisher informations is shown to depend on the orbital and magnetic quantum numbers of the physical state and the space dimensionality. Applications to various elementary systems is discussed.

Fisher information of D-dimensional hydrogenic systems in position and momentum spaces

The spreading of the quantum-mechanical probability distribution density of D-dimensional hydrogenic orbitals is quantitatively determined by means of the local information-theoretic quantity of Fisher in both position and momentum spaces. The Fisher information is found in closed form in terms of the quantum numbers of the orbital.

Spatial entropy of central potentials and strong asymptotics of orthogonal polynomials

The Boltzmann–Shannon information entropy of quantum‐mechanical systems in central potentials can be expressed in terms of the entropy Sn of the classical orthogonal polynomials. Here, an asymptotic formula for the entropy of general orthogonal polynomials on finite intervals is obtained. It is shown that this entropy is intimately related to the relative entropy I (ρ0,ρ) of the equilibrium measure ρ0(x) and the weight function ρ(x) of the polynomials. To do so, the theory of strong asymptotics of orthogonal polynomials on compact sets is used.

Quantum entanglement in helium

We compute the entanglement of the ground state and several singlet and triplet excited states of the helium atom using high-quality, state-of-the-art wavefunctions. The behaviour of the entanglement of the helium eigenstates is similar to that observed in some exactly soluble two-electron systems. In particular, the amount of entanglement exhibited by the eigenstates tends to increase with energy.

Separability criteria and entanglement measures for pure states of N identical fermions

The study of the entanglement properties of systems of N fermions has attracted considerable interest during the last few years. Various separability criteria for pure states of N identical fermions have been recently discussed but, except for the case of two-fermions systems, these criteria are difficult to implement and are of limited value from the practical point of view. Here we advance simple necessary and sufficient separability criteria for pure states of N identical fermions. We found that to be identified as separable, a state has to comply with one single identity involving either the purity or the von Neumann entropy of the single-particle reduced density matrix. These criteria, based on the verification of only one identity, are drastically simpler than the criteria discussed in the recent literature. We also derive two inequalities verified, respectively, by the purity and the entropy of the single-particle, reduced density matrix, which lead to natural entanglement measures for N-fermion pure states. Our present considerations are related to some classical results from the Hartree-Fock theory, which are here discussed from a different point of view in order to clarify some important points concerning the separability of fermionic pure states.

Quantum entanglement in two-electron atomic models

We explore the main entanglement properties exhibited by the eigenfunctions of two exactly soluble two-electron models, the Crandall atom and the Hooke atom, and compare them with the entanglement features of helium-like systems. We compute the amount of entanglement associated with the wavefunctions corresponding to the fundamental and first few excited states of these models. We investigate the dependence of the entanglement on the parameters of the models and on the quantum numbers of the eigenstates. It is found that the amount of entanglement of the system tends to increase with energy in both models. In addition, we study the entanglement of a few states of helium-like systems, which we compute using high-quality Kinoshita-like eigenfunctions. The dependence of the entanglement of helium-like atoms on the nuclear charge and on energy is found to be consistent with the trends observed in the previous two model systems.

Entropy of orthogonal polynomials with Freud weights and information entropies of the harmonic oscillator potential

The information entropy of the harmonic oscillator potential V(x)=1/2λx2 in both position and momentum spaces can be expressed in terms of the so‐called ‘‘entropy of Hermite polynomials,’’ i.e., the quantity Sn(H):= −∫−∞+∞H2n(x)log H2n(x) e−x2dx. These polynomials are instances of the polynomials orthogonal with respect to the Freud weights w(x)=exp(−‖x‖m), m≳0. Here, a very precise and general result of the entropy of Freud polynomials recently established by Aptekarev et al. [J. Math. Phys. 35, 4423–4428 (1994)], specialized to the Hermite kernel (case m=2), leads to an important refined asymptotic expression for the information entropies of very excited states (i.e., for large n) in both position and momentum spaces, to be denoted by Sρ and Sγ, respectively. Briefly, it is shown that, for large values of n, Sρ+1/2logλ≂log(π√2n/e)+o(1) and Sγ−1/2log λ≂log(π√2n/e)+o(1), so that Sρ+Sγ≂log(2π2n/e2)+o(1) in agreement with the generalized indetermination relation of Byalinicki‐Birula and Mycielski [Commun. M...

General linearization formulae for products of continuous hypergeometric-type polynomials

The linearization of products of wavefunctions of exactly solvable potentials often reduces to the generalized linearization problem for hypergeometric polynomials (HPs) of a continuous variable, which consists of the expansion of the product of two arbitrary HPs in series of an orthogonal HP set. Here, this problem is algebraically solved directly in terms of the coefficients of the second-order differential equations satisfied by the involved polynomials. General expressions for the expansion coefficients are given in integral form, and they are applied to derive the connection formulae relating the three classical families of hypergeometric polynomials orthogonal on the real axis (Hermite, Laguerre and Jacobi), as well as several generalized linearization formulae involving these families. The connection and linearization coefficients are generally expressed as finite sums of terminating hypergeometric functions, which often reduce to a single function of the same type; when possible, these functions are evaluated in closed form. In some cases, sign properties of the coefficients such as positivity or non-negativity conditions are derived as a by-product from their resulting explicit representations.