We evaluate the spectral determinant for the damped wave equation on an interval of length $T$ with Dirichlet boundary conditions, proving that it does not depend on the damping. This is achieved by analysing the square of the damped wave operator using the general result by Burghelea, Friedlander, and Kappeler on the determinant for a differential operator with matrix coefficients.
We obtain asymptotics for the Airy kernel Fredholm determinant on two intervals. We give explicit formulas for all the terms up to and including the oscillations of order $1$, which are expressed in terms of Jacobi $θ$-functions.
We revisit the damped string equation on a compact interval with a variety of boundary conditions and derive an infinite sequence of trace formulas associated with it, employing methods familiar from supersymmetric quantum mechanics. We also derive completeness and Riesz basis results (with parentheses) for the associated root functions under less smoothness assumptions on the coefficients than usual, using operator theoretic methods (rather than detailed eigenvalue and root function asymptotics) only.
We consider the problem of finding extremal potentials for the functional determinant of a one-dimensional Schrödinger operator defined on a bounded interval with Dirichlet boundary conditions under an $L^q$-norm restriction ($q\geq 1$). This is done by first extending the definition of the functional determinant to the case of $L^q$ potentials and showing the resulting problem to be equivalent to a problem in optimal control, which we believe to be of independent interest. We prove existence, uniqueness and describe some basic properties of solutions to this problem for all $q\geq 1$, providi…
We consider the probability of two large gaps (intervals without eigenvalues) in the bulk scaling limit of the Gaussian Unitary Ensemble of random matrices. We determine the multiplicative constant in the asymptotics. We also provide the full explicit asymptotics (up to decreasing terms) for the transition between one and two large gaps.
We find the probability of two gaps of the form $(sc,sb)\cup (sa,+\infty)$, $c<b<a<0$, for large $s>0$, in the edge scaling limit of the Gaussian Unitary Ensemble of random matrices, including the multiplicative constant in the asymptotics.
Some Wiener--Hopf determinants on [0,s] are calculated explicitly for all s>0. Their symbols are zero on an interval and they are related to the determinant with the sine-kernel appearing in the random matrix theory. The determinants are calculated by taking limits of Toeplitz determinants, which in turn are found from the related systems of polynomials orthogonal on an arc of the unit circle. As is known, the latter polynomials are connected to those orthogonal on an interval of the real axis. This connection is somewhat extended here. The determinants we compute originate from the Bernstein-…
In this article, we study the large $n$ asymptotic expansions of $n\times n$ Toeplitz determinants whose symbols are indicator functions of unions of arc-intervals of the unit circle. In particular, we use an Hermitian matrix model reformulation of the problem to provide a rigorous derivation of the general form of the large $n$ expansion when the symbol is an indicator function of either a single arc-interval or several arc-intervals with a discrete rotational symmetry. Moreover, we prove that the coefficients in the expansions can be reconstructed, up to some constants, from the Eynard-Orant…
We analyze the spectral properties and peculiar behavior of solutions of a damped wave equation on a finite interval with a singular damping of the form $α/x$, $α>0$. We establish the exponential stability of the semigroup for all positive $α$, and determine conditions for the spectrum to consist of a finite number of eigenvalues. As a consequence, we fully characterize the set of initial conditions for which there is extinction of solutions in finite time. Finally, we propose two open problems related to extremal decay rates of solutions.
The principal aim in this paper is to develop an effective and unified approach to the computation of traces of resolvents (and resolvent differences), Fredholm determinants, $ζ$-functions, and $ζ$-function regularized determinants associated with linear operators in a Hilbert space. In particular, we detail the connection between Fredholm and $ζ$-function regularized determinants. Concrete applications of our formalism to general (i.e., three-coefficient) regular Sturm-Liouville operators on compact intervals with various (separated and coupled) boundary conditions, and Schrödinger operators …
Orthogonal polynomial random matrix models of NxN hermitian matrices lead to Fredholm determinants of integral operators with kernel of the form (phi(x) psi(y) - psi(x) phi(y))/x-y. This paper is concerned with the Fredholm determinants of integral operators having kernel of this form and where the underlying set is a union of open intervals. The emphasis is on the determinants thought of as functions of the end-points of these intervals. We show that these Fredholm determinants with kernels of the general form described above are expressible in terms of solutions of systems of PDE's as long a…
The aim of this paper is twofold: On one hand we discuss an abstract approach to symmetrized Fredholm perturbation determinants and an associated trace formula for a pair of operators of positive-type, extending a classical trace formula. On the other hand, we continue a recent systematic study of boundary data maps, that is, 2 \times 2 matrix-valued Dirichlet-to-Neumann and more generally, Robin-to-Robin maps, associated with one-dimensional Schrödinger operators on a compact interval [0,R] with separated boundary conditions at 0 and R. One of the principal new results in this paper reduces a…
We revisit the computation of (2-modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels. The latter occur, for instance, in the form of Green's functions associated with closed ordinary differential operators on arbitrary intervals on the real line. Our approach determines the (2-modified) Fredholm determinants in terms of solutions of closely associated Volterra integral equations, and as a result offers a natural way to compute such determinants. We illustrate our approach by identifying classical objects such as the Jost function for half-line Schrö…
We study the determinant $\det(I-K_{\textnormal{PII}})$ of an integrable Fredholm operator $K_{\textnormal{PII}}$ acting on the interval $(-s,s)$ whose kernel is constructed out of the $Ψ$-function associated with the Hastings-McLeod solution of the second Painlevé equation. This Fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random Hermitian matrix chosen from the Unitary Ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density. Using the Riemann-Hilbert method, we evaluate the large $s$-asymptotics …
The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and $ζ$-functions to efficiently compute values of spectral $ζ$-functions at positive integers associated to regular (three-coefficient) self-adjoint Sturm--Liouville differential expressions $τ$. Depending on the underlying boundary conditions, we express the $ζ$-function values in terms of a fundamental system of solutions of $τy = z y$ and their expansions about the spectral point $z=0$. Furthermore, we give the full analytic continuation of the $ζ$-function through …
The level spacing distributions which arise when one rescales the Laguerre or Jacobi ensembles of hermitian matrices is studied. These distributions are expressible in terms of a Fredholm determinant of an integral operator whose kernel is expressible in terms of Bessel functions of order $α$. We derive a system of partial differential equations associated with the logarithmic derivative of this Fredholm determinant when the underlying domain is a union of intervals. In the case of a single interval this Fredholm determinant is a Painleve tau function.
We consider a damped/driven nonlinear Schrödinger equation in an $n$-cube $K^{n}\subset\mathbb{R}^n$, $n$ is arbitrary, under Dirichlet boundary conditions \[ u_t-νΔu+i|u|^2u=\sqrtνη(t,x),\quad x\in K^{n},\quad u|_{\partial K^{n}}=0, \quad ν>0, \] where $η(t,x)$ is a random force that is white in time and smooth in space. It is known that the Sobolev norms of solutions satisfy $ \| u(t)\|_m^2 \le Cν^{-m}, $ uniformly in $t\ge0$ and $ν>0$. In this work we prove that for small $ν>0$ and any initial data, with large probability the Sobolev norms $\|u(t,\cdot)\|_m$ of the solutions with $m>2$ beco…
In the hard edge scaling limit of the Jacobi unitary ensemble generated by the weight $x^α(1-x)^β,~x\in[0,1],~α,β>0$, the probability that all eigenvalues of Hermitian matrices from this ensemble lie in the interval $[t,1]$ is given by the Fredholm determinant of the Bessel kernel. We derive the constant in the asymptotics of this Bessel-kernel determinant. A specialization of the results gives the constant in the asymptotics of the probability that the interval $(-a,a),a>0,$ is free of eigenvalues in the Jacobi unitary ensemble with the symmetric weight $(1-x^2)^β, x\in[-1,1]$.
We consider Fredholm determinants of the form identity minus product of spectral projections corresponding to isolated parts of the spectrum of a pair of self-adjoint operators. We show an identity relating such determinants to an integral over the spectral shift function in the case of a rank-one perturbation. More precisely, we prove $$ -\ln \left(\det \big(\mathbf{1} -\mathbf{1} _{I}(A) \mathbf{1}_{\mathbb R\backslash I}(B)\mathbf{1}_{I}(A)\big) \right) = \int_I \text{d} x \int_{\mathbb R\backslash I} \text{d} y\, \frac{ξ(x)ξ(y)}{(y-x)^2}, $$ where $\mathbf{1}_J (\cdot)$ denotes the spectra…
The linearized of the Poisson-Nernst-Planck (PNP) equation under closed ends around a neutral state is studied. It is reduced to a damped heat equation under non-local boundary conditions, which leads to a stochastic interpretation of the linearized equation as a Brownian particle which jump and is reflected, at Poisson distributed time, to one of the end points of the channel, with a probability which is proportional to its distance from this end point. An explicit expansion of the heat kernel reveals the eigenvalues and eigenstates of both the PNP equation and its adjoint. For this, we take …
We study the Fredholm determinant of an integrable operator acting on the interval $(0,s)$ whose kernel is constructed out of a hierarchy of higher order analogues to the Painlevé III equation. This Fredholm determinant describes the critical behavior of the eigenvalue gap probability at the hard edge of unitary invariant random matrix ensembles perturbed by poles of order $k$ in the double scaling regime. Using the Riemann-Hilbert method, we obtain the large $s$ asymptotics of the Fredholm determinant. Moreover, we derive a Painlevé type formula of the Fredholm determinant, which is expressed…
We obtain uniform asymptotics for polynomials orthogonal on a fixed and varying arc of the unit circle with a positive analytic weight function. We also complete the proof of the large $s$ asymptotic expansion for the Fredholm determinant with the kernel $\sin z/(πz)$ on the interval $[0,s]$, verifying a conjecture of Dyson for the constant term in the expansion. In the Gaussian Unitary Ensemble of random matrices, this determinant describes the probability for an interval of length $s$ in the bulk scaling limit to be free from the eigenvalues.
We study the joint probability generating function for $k$ occupancy numbers on disjoint intervals in the Bessel point process. This generating function can be expressed as a Fredholm determinant. We obtain an expression for it in terms of a system of coupled Painlevé V equations, which are derived from a Lax pair of a Riemann-Hilbert problem. This generalizes a result of Tracy and Widom [24], which corresponds to the case $k = 1$. We also provide some examples and applications. In particular, several relevant quantities can be expressed in terms of the generating function, like the gap probab…
We extend the formalism of integrable operators a' la Its-Izergin-Korepin-Slavnov to matrix-valued convolution operators on a semi-infinite interval and to matrix integral operators with a kernel of the form E_1^T(x) E_2(y)/(x+y) thus proving that their resolvent operators can be expressed in terms of solutions of some specific Riemann-Hilbert problems. We also describe some applications, mainly to a noncommutative version of Painleve' II (recently introduced by Retakh and Rubtsov), a related noncommutative equation of Painleve' type. We construct a particular family of solutions of the noncom…
It was proved by Akemann, Ipsen and Kieburg that squared singular values of products of $M$ complex Ginibre random matrices form a determinantal point process whose correlation kernel is expressible in terms of Meijer's $G$-functions. Kuijlaars and Zhang recently showed that at the edge of the spectrum, this correlation kernel has a remarkable scaling limit $K_M(x,y)$ which can be understood as a generalization of the classical Bessel kernel of Random Matrix Theory. In this paper we investigate the Fredholm determinant of the operator with the kernel $K_M(x,y)χ_J(y)$, where $J$ is a disjoint u…
We study the determinant $\det(I-γK_s), 0<γ<1$, of the integrable Fredholm operator $K_s$ acting on the interval $(-1,1)$ with kernel $K_s(λ, μ)= \frac{\sin s(λ- μ)}{π(λ-μ)}$. This determinant arises in the analysis of a log-gas of interacting particles in the bulk-scaling limit, at inverse temperature $β=2$, in the presence of an external potential $v=-\frac{1}{2}\ln(1-γ)$ supported on an interval of length $\frac{2s}π$. We evaluate, in particular, the double scaling limit of $\det(I-γK_s)$ as $s\rightarrow\infty$ and $γ\uparrow 1$, in the region $0\leqκ=\frac{v}{s}=-\frac{1}{2s}\ln(1-γ)\leq …
On generalised d'Alembert-type integral representations for damped wave equations on the quarter-plane
National and Kapodistrian University of Athens（雅典国立卡波蒂斯特里安大学）；University of Cyprus（塞浦路斯大学）；Universita degli Studi di Brescia（布雷西亚大学）；Istituto Nazionale di Fisica Nucleare, Milano Bicocca（意大利国家核物理研究所，米兰比科卡）；Henan Normal University（河南师范大学）；Laboratoire de Mathématiques Appliquées du Havre (LMAH), Université Le Havre Normandie（勒阿弗尔应用数学实验室，勒阿弗尔诺曼底大学）
The probabilities for gaps in the eigenvalue spectrum of the finite dimension $ N \times N $ random matrix Hermite and Jacobi unitary ensembles on some single and disconnected double intervals are found. These are cases where a reflection symmetry exists and the probability factors into two other related probabilities, defined on single intervals. Our investigation uses the system of partial differential equations arising from the Fredholm determinant expression for the gap probability and the differential-recurrence equations satisfied by Hermite and Jacobi orthogonal polynomials. In our stud…
We study the manner in which spectral shift functions associated with self-adjoint one-dimensional Schrödinger operators on the finite interval $(0,R)$ converge in the infinite volume limit $R\to\infty$ to the half-line spectral shift function. Relying on a Fredholm determinant approach combined with certain measure theoretic facts, we show that prior vague convergence results in the literature in the special case of Dirichlet boundary conditions extend to the notion of weak convergence and arbitrary separated self-adjoint boundary conditions at $x=0$ and $x=R$.
We consider the generating function of the sine point process on $m$ consecutive intervals. It can be written as a Fredholm determinant with discontinuities, or equivalently as the convergent series \begin{equation*} \sum_{k_{1},...,k_{m} \geq 0} \mathbb{P}\Bigg(\bigcap_{j=1}^{m} \#\{\mbox{points in j-th interval}\}=k_{j}\Bigg)\prod_{j=1}^{m} s_{j}^{k_{j}}, \end{equation*} where $s_{1},\ldots,s_{m} \in [0,1]$. In particular, we can deduce from it joint probabilities of the counting function of the process. In this work, we obtain large gap asymptotics for the generating function, which are asy…
