arXiv subjects

Pablo Román

Publications and source records attributed to Pablo Román.

24 records · Page 2Linked to original sources

arXiv2016

Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures

A matrix-valued measure $Θ$ reduces to measures of smaller size if there exists a constant invertible matrix $M$ such that $MΘM^*$ is block diagonal. Equivalently, the real vector space ${\mathscr A}$ of all matrices $T$ such that $TΘ(X)=Θ(X) T^*$ for any Borel set $X$ is non-trivial. If the subspace $A_h$ of self-adjoints elements in the commutant algebra $A$ of $Θ$ is non-trivial, then $Θ$ is reducible via a unitary matrix. In this paper we prove that ${\mathscr A}$ is $*$-invariant if and only if $A_h={\mathscr A}$, i.e., every reduction of $Θ$ can be performed via a unitary matrix. The motivation for this paper comes from families of matrix-valued polynomials related to the group ${\rm SU}(2)\times {\rm SU}(2)$ and its quantum analogue. In both cases the commutant algebra $A=A_h\oplus iA_h$ is of dimension two and the matrix-valued measures reduce unitarily into a $2\times 2$ block diagonal matrix. Here we show that there is no further non-unitary reduction.

math.CA↗

arXiv2015

Asymptotics of orthogonal polynomials generated by a Geronimus perturbation of the Laguerre measure

This paper deals with monic orthogonal polynomials generated by a Geronimus canonical spectral transformation of the Laguerre classical measure: \[ \frac{1}{x-c}x^{α}e^{-x}dx+Nδ(x-c), \] for $x\in[0,\infty)$, $α>-1$, a free parameter $N\in \mathbb{R}_{+}$ and a shift $c<0$. We analyze the asymptotic behavior (both strong and relative to classical Laguerre polynomials) of these orthogonal polynomials as $n$ tends to infinity.

math.CA↗

arXiv2014

Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One

We present a method to obtain infinitely many examples of pairs $(W,D)$ consisting of a matrix weight $W$ in one variable and a symmetric second-order differential operator $D$. The method is based on a uniform construction of matrix valued polynomials starting from compact Gelfand pairs $(G,K)$ of rank one and a suitable irreducible $K$-representation. The heart of the construction is the existence of a suitable base change $Ψ_{0}$. We analyze the base change and derive several properties. The most important one is that $Ψ_{0}$ satisfies a first-order differential equation which enables us to compute the radial part of the Casimir operator of the group $G$ as soon as we have an explicit expression for $Ψ_{0}$. The weight $W$ is also determined by $Ψ_{0}$. We provide an algorithm to calculate $Ψ_{0}$ explicitly. For the pair $(\mathrm{USp}(2n),\mathrm{USp}(2n-2)\times\mathrm{USp}(2))$ we have implemented the algorithm in GAP so that individual pairs $(W,D)$ can be calculated explicitly. Finally we classify the Gelfand pairs $(G,K)$ and the $K$-representations that yield pairs $(W,D)$ of size $2\times2$ and we provide explicit expressions for most of these cases.

math.RT↗

arXiv2014

Asymptotic behavior and zero distribution of polynomials orthogonal with respect to Bessel functions

We consider polynomials $P_n$ orthogonal with respect to the weight $J_ν$ on $[0,\infty)$, where $J_ν$ is the Bessel function of order $ν$. Asheim and Huybrechs considered these polynomials in connection with complex Gaussian quadrature for oscillatory integrals. They observed that the zeros are complex and accumulate as $n \to \infty$ near the vertical line $\textrm{Re}\, z = \frac{νπ}{2}$. We prove this fact for the case $0 \leq ν\leq 1/2$ from strong asymptotic formulas that we derive for the polynomials $P_n$ in the complex plane. Our main tool is the Riemann-Hilbert problem for orthogonal polynomials, suitably modified to cover the present situation, and the Deift-Zhou steepest descent method. A major part of the work is devoted to the construction of a local parametrix at the origin, for which we give an existence proof that only works for $ν\leq 1/2$.

math.CA↗

arXiv2011

Non-intersecting squared Bessel paths with one positive starting and ending point

We consider a model of $n$ non-intersecting squared Bessel processes with one starting point $a>0$ at time t=0 and one ending point $b>0$ at time $t=T$. After proper scaling, the paths fill out a region in the $tx$-plane. Depending on the value of the product $ab$ the region may come to the hard edge at 0, or not. We formulate a vector equilibrium problem for this model, which is defined for three measures, with upper constraints on the first and third measures and an external field on the second measure. It is shown that the limiting mean distribution of the paths at time $t$ is given by the second component of the vector that minimizes this vector equilibrium problem. The proof is based on a steepest descent analysis for a $4 \times 4$ matrix valued Riemann-Hilbert problem which characterizes the correlation kernel of the paths at time $t$. We also discuss the precise locations of the phase transitions.

math-ph↗

arXiv2011

Asymptotic zero distribution of multiple orthogonal polynomials associated with Macdonald functions

We study the asymptotic zero distribution of type II multiple orthogonal polynomials associated with two Macdonald functions (modified Bessel functions of the second kind). Based on the four-term recurrence relation, it is shown that, after proper scaling, the sequence of normalized zero counting measures converges weakly to the first component of a vector of two measures which satisfies a vector equilibrium problem with two external fields. We also give the explicit formula for the equilibrium vector in terms of solutions of an algebraic equation.

math.CA↗