arXiv subjects

M. A. Rodriguez

Publications and source records attributed to M. A. Rodriguez.

32 records · Page 2Linked to original sources

arXiv2004

Solvable scalar and spin models with near-neighbors interactions

We construct new solvable rational and trigonometric spin models with near-neighbors interactions by an extension of the Dunkl operator formalism. In the trigonometric case we obtain a finite number of energy levels in the center of mass frame, while the rational models are shown to possess an equally spaced infinite algebraic spectrum. For the trigonometric and one of the rational models, the corresponding eigenfunctions are explicitly computed. We also study the scalar reductions of the models, some of which had already appeared in the literature, and compute their algebraic eigenfunctions in closed form. In the rational cases, for which only partial results were available, we give concise expressions of the eigenfunctions in terms of generalized Laguerre and Jacobi polynomials.

hep-th↗

arXiv2002

On the Sutherland Spin Model of B_N Type and its Associated Spin Chain

The B_N hyperbolic Sutherland spin model is expressed in terms of a suitable set of commuting Dunkl operators. This fact is exploited to derive a complete family of commuting integrals of motion of the model, thus establishing its integrability. The Dunkl operators are shown to possess a common flag of invariant finite-dimensional linear spaces of smooth scalar functions. This implies that the Hamiltonian of the model preserves a corresponding flag of smooth spin functions. The discrete spectrum of the restriction of the Hamiltonian to this spin flag is explicitly computed by triangularization. The integrability of the hyperbolic Sutherland spin chain of B_N type associated with the dynamical model is proved using Polychronakos's "freezing trick".

hep-th↗

arXiv2002

Anomalous roughening of Hele-Shaw flows with quenched disorder

The kinetic roughening of a stable oil--air interface, moving in a Hele--Shaw cell which contains a quenched columnar disorder (tracks) has been studied. A capillary effect is responsible for the dynamic evolution of the resulting rough interface, which exhibits anomalous scaling. The three independent exponents needed to characterize the anomalous scaling are determined experimentally. The anomalous scaling is explained in terms of the initial acceleration and subsequent deceleration of the interface tips in the tracks coupled by mass conservation. A phenomenological model that reproduces the measured global and local exponents has been introduced.

cond-mat.dis-nn↗

arXiv2001

Interface roughening in Hele--Shaw flows with quenched disorder: experimental and theoretical results

We study the forced fluid invasion of an air-filled model porous medium at constant flow rate, in 1+1 dimensions, both experimentally and theoretically. We focus on the non-local character of the interface dynamics, due to liquid conservation, and its effect on the scaling properties of the interface upon roughening. Specifically, we study the limit of large flow rates and weak capillary forces. Our theory predicts a roughening behaviour characterized at short times by a growth exponent $β_1 = 5/6$, a roughness exponent $α_1=5/2$, and a dynamic exponent $z_1=3$, and by $β_2=1/2$, $α_2=1/2$, and $z_2=1$ at long times, before saturation. This theoretical prediction is in good agreement with the experiments at long times.The ensemble of experiments, theory, and simulations provides evidence for a new universality class of interface roughening in 1+1 dimensions.

cond-mat↗

arXiv2001

New spin Calogero-Sutherland models related to B_N-type Dunkl operators

We construct several new families of exactly and quasi-exactly solvable BC_N-type Calogero-Sutherland models with internal degrees of freedom. Our approach is based on the introduction of two new families of Dunkl operators of B_N type which, together with the original B_N-type Dunkl operators, are shown to preserve certain polynomial subspaces of finite dimension. We prove that a wide class of quadratic combinations involving these three sets of Dunkl operators always yields a spin Calogero-Sutherland model, which is (quasi-)exactly solvable by construction. We show that all the spin Calogero-Sutherland models obtainable within this framework can be expressed in a unified way in terms of a Weierstrass P function with suitable half-periods. This provides a natural spin counterpart of the well-known general formula for a scalar completely integrable potential of BC_N type due to Olshanetsky and Perelomov. As an illustration of our method, we exactly compute several energy levels and their corresponding wavefunctions of an elliptic quasi-exactly solvable potential for two and three particles of spin 1/2.

hep-th↗

arXiv2001

A_N-type Dunkl operators and new spin Calogero-Sutherland models

A new family of A_N-type Dunkl operators preserving a polynomial subspace of finite dimension is constructed. Using a general quadratic combination of these operators and the usual Dunkl operators, several new families of exactly and quasi-exactly solvable quantum spin Calogero-Sutherland models are obtained. These include, in particular, three families of quasi-exactly solvable elliptic spin Hamiltonians.

hep-th↗

arXiv2000

The Lame equation in parametric resonance after inflation

We show that the most general inflaton potential in Minkowski spacetime for which the differential equation for the Fourier modes of the matter fields reduces to Lamé's equation is of the form $V(ϕ)= λϕ^4/4+Kϕ^2/2+μ/(2ϕ^2)+V_0$. As an application, we study the preheating phase after inflation for the above potential with K=0 and arbitrary $λ,μ>0$. For certain values of the coupling constant between the inflaton and the matter fields, we calculate the instability intervals and the characteristic exponents in closed form.

hep-ph↗

arXiv2000

Exact solutions of an elliptic Calogero--Sutherland model

A model describing N particles on a line interacting pairwise via an elliptic function potential in the presence of an external field is partially solved in the quantum case in a totally algebraic way. As an example, the ground state and the lowest excitations are calculated explicitly for N=2.

hep-th↗

arXiv2000

New Algebraic Quantum Many-body Problems

We develop a systematic procedure for constructing quantum many-body problems whose spectrum can be partially or totally computed by purely algebraic means. The exactly-solvable models include rational and hyperbolic potentials related to root systems, in some cases with an additional external field. The quasi-exactly solvable models can be considered as deformations of the previous ones which share their algebraic character.

nlin.SI↗

arXiv1999

A New Algebraization of the Lame Equation

We develop a new way of writing the Lame Hamiltonian in Lie-algebraic form. This yields, in a natural way, an explicit formula for both the Lame polynomials and the classical non-meromorphic Lame functions in terms of Chebyshev polynomials and of a certain family of weakly orthogonal polynomials

math-ph↗

arXiv1998

Pseudo-orthogonal groups and integrable dynamical systems in two dimensions

Integrable systems in low dimensions, constructed through the symmetry reduction method, are studied using phase portrait and variable separation techniques. In particular, invariant quantities and explicit periodic solutions are determined. Widely applied models in Physics are shown to appear as particular cases of the method.

solv-int↗

arXiv1994

Scaling Relations and Exponents in the Growth of Rough Interfaces Through Random Media

The growth of a rough interface through a random media is modelled by a continuous stochastic equation with a quenched noise. By use of the Novikov theorem we can transform the dependence of the noise on the interface height into an effective temporal correlation for different regimes of the evolution of the interface. The exponents characterizing the roughness of the interface can thus be computed by simple scaling arguments showing a good agreement with recent experiments and numerical simulations.

cond-mat↗

arXiv1994

Analysis of self--averaging properties in the transport of particles through random media

We investigate self-averaging properties in the transport of particles through random media. We show rigorously that in the subdiffusive anomalous regime transport coefficients are not self--averaging quantities. These quantities are exactly calculated in the case of directed random walks. In the case of general symmetric random walks a perturbative analysis around the Effective Medium Approximation (EMA) is performed.

cond-mat↗

arXiv1994

The Conformal Group SU(2,2) and Integrable Systems on a Lorentzian Hyperboloid

Eleven different types of "maximally superintegrable" Hamiltonian systems on the real hyperboloid $(s^0)^2-(s^1)^2+(s^2)^2-(s^3)^2=1$ are obtained. All of them correspond to a free Hamiltonian system on the homogeneous space $SU(2,2)/U(2,1)$, but to reductions by different maximal abelian subgroups of $SU(2,2)$. Each of the obtained systems allows 5 functionally independent integrals of motion, from which it is possible to form two or more triplets in involution (each of them includes the hamiltonian). The corresponding classical and quantum equations of motion can be solved by separation of variables on the $O(2,2)$ space.

hep-th↗