We prove that a first order linear differential operator G with unbounded operator coefficients is Fredholm on spaces of functions on the real line with values in a reflexive Banach space if and only if the corresponding strongly continuous evolution family has exponential dichotomies on both semiaxises and a pair of the ranges of the dichotomy projections is Fredholm, and that the Fredholm index of G is equal to the Fredholm index of the pair. The operator G is the generator of the evolution semigroup associated with the evolution family. In the case when the evolution family is the propagator of a well-posed differential equation u'(t)=A(t)u(t) with, generally, unbounded operators A(t), the operator G is a closure of the operator -d/dt+A(t). Thus, this paper provides a complete infinite dimensional generalization of well-known finite dimensional results by K. Palmer, and by A. Ben-Artzi and I. Gohberg.
We consider the $3-D$ Dirac operator $\mathfrak{D}_{\boldsymbol{A},Φ,Q_{\sin }}$ with variable regular magnetic and electrostatic potentials $ \boldsymbol{A}$,$Φ$ and with singular potentials $Q_{\sin }$ with support on a smooth unbounded surface $Σ\subset \mathbb{R}^{3}$ which divides $\mathbb{R}^{3}$ on two open domains $Ω_{\pm }$. We associate with the formal Dirac operator $\mathfrak{D}_{\boldsymbol{A},Φ,Q_{\sin }} $ an unbounded operator $\mathcal{D}_{\boldsymbol{A},Φ,Q_{\sin }}$ in $ L^{2}(\mathbb{R}^{3},\mathbb{C}^{4})$ generated by the regular part of $ \mathfrak{D}_{\boldsymbol{A},Φ,Q…
In this article we give a comprehensive treatment of a `Clifford module flow' along paths in the skew-adjoint Fredholm operators on a real Hilbert space that takes values in KO${}_{*}(\mathbb{R})$ via the Clifford index of Atiyah-Bott-Shapiro. We develop its properties for both bounded and unbounded skew-adjoint operators including an axiomatic characterization. Our constructions and approach are motivated by the principle that \[ \text{spectral flow} = \text{Fredholm index}. \] That is, we show how the KO--valued spectral flow relates to a KO-valued index by proving a Robbin-Salamon type resu…
As Fredholm determinants are more and more frequent in the context of stochastic integrability, we unveil the existence of a common framework in many integrable systems where they appear. This consists in a quasi-universal hierarchy of equations, partly unifying an integro-differential generalization of the Painlevé II hierarchy, the finite-time solutions of the Kardar-Parisi-Zhang equation, multi-critical fermions at finite temperature and a notable solution to the Zakharov-Shabat system associated to the largest real eigenvalue in the real Ginibre ensemble. As a byproduct, we obtain the expl…
The Evans function is a well known tool for locating spectra of differential operators in one spatial dimension. In this paper we construct a multidimensional analogue as the modified Fredholm determinant of a ratio of Dirichlet-to-Robin operators on the boundary. This gives a tool for studying the eigenvalue counting functions of second-order elliptic operators that need not be self-adjoint. To do this we use local representation theory for meromorphic operator-valued pencils, and relate the algebraic multiplicities of eigenvalues of elliptic operators to those of the Robin-to-Robin and Robin…
Based on operators borrowed from scattering theory, several concrete realizations of index theorems are proposed. The corresponding operators belong to some C*-algebras of pseudo-differential operators with coefficients which either have limits at plus and minus infinity, or which are periodic or asymptotically periodic, or which are uniformly almost periodic. These various situations can be deduced from a single partial isometry which depends on several parameters. All computations are explicitly performed.
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…
We survey the notion of the spectral shift function of a pair of self-adjoint operators and recent progress on its connection with the Witten index. We also describe a proof of Krein's Trace Theorem that does not use complex analysis [53] and develop its extension to general $σ$-finite von Neumann algebras $\mathcal{M}$ of type II and unbounded perturbations from the predual of $\mathcal{M}$. We also discuss the connection between the theory of the spectral shift function and index theory for certain model operators. We start by introducing various definitions of the Witten index, (an extensio…
An odd Fredholm module for a given invertible operator on a Hilbert space is specified by an unbounded so-called Dirac operator with compact resolvent and bounded commutator with the given invertible. Associated to this is an index pairing in terms of a Fredholm operator with Noether index. Here it is shown by a spectral flow argument how this index can be calculated as the signature of a finite dimensional matrix called the spectral localizer.
It is shown how the bilinear differential equations satisfied by Fredholm determinants of integral operators appearing as spectral distribution functions for random matrices may be deduced from the associated systems of nonautonomous Hamiltonian equations satisfied by auxiliary canonical phase space variables introduced by Tracy and Widom. The essential step is to recast the latter as isomonodromic deformation equations for families of rational covariant derivative operators on the Riemann sphere and interpret the Fredholm determinants as isomonodromic $τ$-functions.
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 generalize to quantum weighted projective spaces in any dimension previous results of us on K-theory and K-homology of quantum projective spaces `tout court'. For a class of such spaces, we explicitly construct families of Fredholm modules, both bounded and unbounded (that is spectral triples), and prove that they are linearly independent in the K-homology of the corresponding C*-algebra. We also show that the quantum weighted projective spaces are base spaces of quantum principal circle bundles whose total spaces are quantum lens spaces. We construct finitely generated projective modules a…
A natural formulation of the theory of quantum measurements in continuous time is based on quantum stochastic differential equations (Hudson-Parthasarathy equations). However, such a theory was developed only in the case of Hudson-Parthasarathy equations with bounded coefficients. By using some results on Hudson-Parthasarathy equations with unbounded coefficients, we are able to extend the theory of quantum continuous measurements to cases in which unbounded operators on the system space are involved. A significant example of a quantum optical system (the degenerate parametric oscillator) is s…
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…
Let $\{A(t)\}_{t \in \mathbb{R}}$ be a path of self-adjoint Fredholm operators in a Hilbert space $\mathcal{H}$, joining endpoints $A_\pm$ as $t \to \pm \infty$. Computing the index of the operator $D_A= (d/d t) + A$ acting in $L^2(\mathbb{R}; \mathcal{H})$, where $A = \int_{\mathbb{R}}^{\oplus} dt \, A(t)$, and its relation to spectral flow along this path, has a long history. While most of the latter focuses on the case where $A(t)$ all have purely discrete spectrum, we now particularly study situations permitting essential spectra. Introducing $H_1={D_A}^* D_A$ and $H_2=D_A {D_A}^*$, we con…
We show how many classes of partial differential systems with local and nonlocal nonlinearities are linearisable in the sense that they are realisable as Fredholm Grassmannian flows. In other words, time-evolutionary solutions to such systems can be constructed from solutions to the corresponding underlying linear partial differential system, by solving a linear Fredholm equation. For example, it is well-known that solutions to classical integrable partial differential systems can be generated by solving a corresponding linear partial differential system for the scattering data and then solvin…
Continuing a line of investigation initiated in [11] exploring the connections between Jost and Evans functions and (modified) Fredholm determinants of Birman-Schwinger type integral operators, we here examine the stability index, or sign of the first nonvanishing derivative at frequency zero of the characteristic determinant, an object that has found considerable use in the study by Evans function techniques of stability of standing and traveling wave solutions of partial differential equations (PDE) in one dimension. This leads us to the derivation of general perturbation expansions for anal…
We consider resonances for third order ordinary differential operator with compactly supported coefficients on the real line. Resonance are defined as zeros of a Fredholm determinant on a non-physical sheet of three sheeted Riemann surface. We determine upper bounds of the number of resonances in complex discs at large radius. We express the trace formula in terms of resonances only.
Given a Schrödinger differential expression on an exterior Lipschitz domain we prove strict inequalities between the eigenvalues of the corresponding selfadjoint operators subject to Dirichlet and Neumann or Dirichlet and mixed boundary conditions, respectively. Moreover, we prove a strict inequality between the eigenvalues of two different elliptic differential operators on the same domain with Dirichlet boundary conditions.
