arXiv subjects

J. Ostensson

Publications and source records attributed to J. Ostensson.

3 recordsLinked to original sources

arXiv2011

A trace formula for differential operators of arbitrary order

An operator $H=H_{0}+V$ where $H_{0}=i^{-N} \partial^{N}$ ($N$ is arbitrary) and $V$ is a differential operator of order $N-1$ with coefficients decaying sufficiently rapidly at infinity is considered in the space $L^2(\Bbb R)$. The goal of the paper is to find an expression for the trace of the difference of the resolvents $(H-z)^{-1}$ and $(H_{0}-z)^{-1}$ in terms of the Wronskian of appropriate solutions to the differential equation $Hu=zu$. This also leads to a representation for the perturbation determinant of the pair $H_{0}, H$.

math.SP↗

arXiv2008

Asymptotics for a special solution of the thirty fourth Painleve equation

In a previous paper we studied the double scaling limit of unitary random matrix ensembles of the form Z_{n,N}^{-1} |\det M|^{2α} e^{-N \Tr V(M)} dM with α> -1/2. The factor |\det M|^{2α} induces critical eigenvalue behavior near the origin. Under the assumption that the limiting mean eigenvalue density associated with V is regular, and that the origin is a right endpoint of its support, we computed the limiting eigenvalue correlation kernel in the double scaling limit as n, N \to \infty such that n^{2/3}(n/N-1) = O(1) by using the Deift-Zhou steepest descent method for the Riemann-Hilbert problem for polynomials on the line orthogonal with respect to the weight |x|^{2α} e^{-NV(x)}. Our main attention was on the construction of a local parametrix near the origin by means of the ψ-functions associated with a distinguished solution u_α of the Painleve XXXIV equation. This solution is related to a particular solution of the Painleve II equation, which however is different from the usual Hastings-McLeod solution. In this paper we compute the asymptotic behavior of u_α(s) as s \to \pm \infty. We conjecture that this asymptotics characterizes u_α and we present supporting arguments based on the asymptotic analysis of a one-parameter family of solutions of the Painleve XXXIV equation which includes u_α. We identify this family as the family of tronquee solutions of the thirty fourth Painleve equation.

math.CA↗

arXiv2007

Critical edge behavior in unitary random matrix ensembles and the thirty fourth Painleve transcendent

We describe a new universality class for unitary invariant random matrix ensembles. It arises in the double scaling limit of ensembles of random $n \times n$ Hermitian matrices $Z_{n,N}^{-1} |\det M|^{2α} e^{-N \Tr V(M)} dM$ with $α> -1/2$, where the factor $|\det M|^{2α}$ induces critical eigenvalue behavior near the origin. Under the assumption that the limiting mean eigenvalue density associated with $V$ is regular, and that the origin is a right endpoint of its support, we compute the limiting eigenvalue correlation kernel in the double scaling limit as $n, N \to \infty$ such that $n^{2/3}(n/N-1) = O(1)$. We use the Deift-Zhou steepest descent method for the Riemann-Hilbert problem for polynomials on the line orthogonal with respect to the weight $|x|^{2α} e^{-NV(x)}$. Our main attention is on the construction of a local parametrix near the origin by means of the $ψ$-functions associated with a distinguished solution of the Painleve XXXIV equation. This solution is related to a particular solution of the Painleve II equation, which however is different from the usual Hastings-McLeod solution.

math.CA↗