Gabriel Álvarez

Bender-Wu branch points in the cubic oscillator

The analytic continuation of the resonances of the cubic anharmonic oscillator to complex values of the coupling constant is studied with semiclassical and numerical methods. Bender-Wu branch points, at which level crossing occurs, are calculated and labelled by a process of analytic continuation. The different resonances are the values that a single analytic function takes on different sheets of a Riemann surface whose topology is described.

Summability of the perturbative expansion for a zero-dimensional disordered spin model

We show analytically that the perturbative expansion for the free energy of the zero dimensional (quenched) disordered Ising model is Borel-summable in a certain range of parameters, provided that the summation is carried out in two steps: first, in the strength of the original coupling of the Ising model and subsequently in the variance of the quenched disorder. This result is illustrated by some high-precision calculations of the free energy obtained by a straightforward numerical implementation of our sequential summation method.

Critical Role of Two-Dimensional Island-Mediated Growth on the Formation of Semiconductor Heterointerfaces

We experimentally demonstrate a sigmoidal variation of the composition profile across semiconductor heterointerfaces. The wide range of material systems (III-arsenides, III-antimonides, III-V quaternary compounds, III-nitrides) exhibiting such a profile suggests a universal behavior. We show that sigmoidal profiles emerge from a simple model of cooperative growth mediated by two-dimensional island formation, wherein cooperative effects are described by a specific functional dependence of the sticking coefficient on the surface coverage. Experimental results confirm that, except in the very early stages, island growth prevails over nucleation as the mechanism governing the interface development and ultimately determines the sigmoidal shape of the chemical profile in these two-dimensional-grown layers. In agreement with our experimental findings, the model also predicts a minimum value of the interfacial width, with the minimum attainable value depending on the chemical identity of the species.

Langer–Cherry derivation of the multi-instanton expansion for the symmetric double well

The multi-instanton expansion for the eigenvalues of the symmetric double well is derived using a Langer–Cherry uniform asymptotic expansion of the solution of the corresponding Schrodinger equation. The Langer–Cherry expansion is anchored to either one of the minima of the potential, and by construction has the correct asymptotic behavior at large distance, while the quantization condition amounts to imposing the even or odd parity of the wave function. This method leads to an efficient algorithm for the calculation to virtually any desired order of all the exponentially small series of the multi-instanton expansion, and with trivial modifications can also be used for nonsymmetric double wells.

Exponentially small corrections in the asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator

The asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator is studied in a region of the coupling constant plane in which there is a sequence of exponentially small subseries beyond the standard Rayleigh-Schrodinger perturbation theory (RSPT) power series. We give a simple algorithm for the calculation of these subseries (to any desired order) in terms of the RSPT coefficients expressed as polynomials in the quantum number, and illustrate our results with numerical Borel-Pade summations of the expansion up to third exponentially small order.

Determination of S-curves with applications to the theory of non-Hermitian orthogonal polynomials

This paper deals with the determination of the S-curves in the theory of non-hermitian orthogonal polynomials with respect to exponential weights along suitable paths in the complex plane. It is known that the corresponding complex equilibrium potential can be written as a combination of Abelian integrals on a suitable Riemann surface whose branch points can be taken as the main parameters of the problem. Equations for these branch points can be written in terms of periods of Abelian differentials and are known in several equivalent forms. We select one of these forms and use a combination of analytic an numerical methods to investigate the phase structure of asymptotic zero densities of orthogonal polynomials and of asymptotic eigenvalue densities of random matrix models. As an application we give a complete description of the phases and critical processes of the standard cubic model.

Uniform asymptotic and JWKB expansions for anharmonic oscillators

We show explicitly the relation between the uniform asymptotic and the Jeffreys-Wentzel-Kramers-Brillouin (JWKB) wavefunctions, and between the matching of uniform asymptotic expansions and the complete JWKB connection formulae written in terms of Stokes multipliers and loop integrals. As an application we give a unified derivation of the asymptotic behaviour of the imaginary part of the resonances in anharmonic oscillators and, via dispersion relations, the corresponding asymptotic behaviour of the Rayleigh-Schrodinger perturbation theory coefficients.

Dispersive hyperasymptotics and the anharmonic oscillator

Hyperasymptotic summation of steepest-descent asymptotic expansions of integrals is extended to functions that satisfy a dispersion relation. We apply the method to energy eigenvalues of the anharmonic oscillator, for which there is no known integral representation, but for which there is a dispersion relation. Hyperasymptotic summation exploits the rich analytic structure underlying the asymptotics and is a practical alternative to Borel summation of the Rayleigh–Schrodinger perturbation series.

Anharmonic oscillator discontinuity formulae up to second-exponentially-small order

The eigenvalues of the quartic anharmonic oscillator as functions of the anharmonicity constant satisfy a once-subtracted dispersion relation. In turn, this dispersion relation is driven by the purely imaginary discontinuity of the eigenvalues across the negative real axis. In this paper we calculate explicitly the asymptotic expansion of this discontinuity up to second-exponentially-small order.

Phase transitions in multi-cut matrix models and matched solutions of Whitham hierarchies

We present a method to study phase transitions in the large N limit of matrix models using matched solutions of Whitham hierarchies. The endpoints of the eigenvalue spectrum as functions of the temperature are characterized both as solutions of hodograph equations and as solutions of a system of ordinary differential equations. In particular we show that the free energy of the matrix model is the quasiclassical �-function of the associated hierarchy, and that critical processes in which the number of cuts changes in one unit are third-order phase transitions described by C 1 matched solutions of Whitham hierarchies. The method is illustrated with the Bleher-Eynard model for the merging of two cuts. We show that this model involves also a birth of a cut.