Spectral averaging for trace compatible operators

In this note the notions of trace compatible operators and infinitesimal spectral flow are introduced. We define the spectral shift function as the integral of infinitesimal spectral flow. It is proved that the spectral shift function thus defined is absolutely continuous and Kreĭn's formula is established. Some examples of trace compatible affine spaces of operators are given.

     

Spectral flow is the integral of one forms on the Banach manifold of self adjoint Fredholm operators

One may trace the idea that spectral flow should be given as the integral of a one form back to the 1974 Vancouver ICM address of I.M. Singer. Our main theorem gives analytic formulae for the spectral flow along a norm differentiable path of self adjoint bounded Breuer-Fredholm operators in a semifinite von Neumann algebra. These formulae have a geometric interpretation which derives from the proof. Namely we define a family of Banach submanifolds of all bounded self adjoint Breuer-Fredholm operators and on each submanifold define global one forms whose integral on a norm differentiable path contained in the submanifold calculates the spectral flow along this path. We emphasise that our methods do not give a single globally defined one form on the self adjoint Breuer-Fredholms whose integral along all paths is spectral flow rather, as the choice of the plural ‘forms’ in the title suggests, we need a family of such one forms in order to confirm Singer's idea. The original context for this result concerned paths of unbounded self adjoint Fredholm operators. We therefore prove analogous formulae for spectral flow in the unbounded case as well. The proof is a synthesis of key contributions by previous authors, whom we acknowledge in detail in the introduction, combined with an additional important recent advance in the differential calculus of functions of non-commuting operators.     

On a formula for the spectral flow and its applications

We consider a continuous path of bounded symmetric Fredholm bilinear forms with arbitrary endpoints on a real Hilbert space, and we prove a formula that gives the spectral flow of the path in terms of the spectral flow of the restriction to a finite codimensional closed subspace. We also discuss the case of restrictions to a continuous path of finite codimensional closed subspaces. As an application of the formula, we introduce the notion of spectral flow for a periodic semi‐Riemannian geodesic, and we compute its value in terms of the Maslov index (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximation of Navier–Stokes incompressible flow using a spectral element method with a local discretization in spectral space

A spectral element technique is examined, which builds upon a local discretization within the spectral space. To approximate a given system of equations the domain is subdivided into nonoverlapping quadrilateral elements, and within each element a discretization is found in the spectral space. The difference is that the test functions are divided into the higher-order polynomials, which have zero boundaries and lower-order polynomials, which are nonzero on one boundary. The method is examined for Navier–Stokes incompressible flow for fluid flow within a driven cavity and for flow over a backstep. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 587–599, 1997     

Uniqueness of spectral flow

To each path of self-adjoint Fredholm operators acting on a real separable Hilbert space H with invertible ends, there is associated an integer called spectral flow. The purpose of this brief note is to show that spectral flow is uniquely characterized by four elementary properties: normalization, continuity, additivity over direct sums, and its value as the difference of the Morse indices of the ends when H is finite dimensional. The proof of uniqueness relies of the invariance of spectral flow of the path under cogredient transformations of the path.     

Spectral flow and iteration of closed semi-Riemannian geodesics

We introduce the notion of spectral flow along a periodic semi-Riemannian geodesic, as a suitable substitute of the Morse index in the Riemannian case. We study the growth of the spectral flow along a closed geodesic under iteration, determining its asymptotic behavior. M. A. J. is sponsored by Fapesp; P. P. is partially sponsored by CNPq.     

Semi-classical behavior of the spectral function

We study the semi-classical behavior of the spectral function of the Schrödinger operator with short range potential. We prove that the spectral function is a semi-classical Fourier integral operator quantizing the forward and backward Hamiltonian flow relations of the system. Under a certain geometric condition we explicitly compute the phase in an oscillatory integral representation of the spectral function.

     

Asymptotic Optimality of Isoperimetric Constants

In this paper, we investigate the existence of L ²(π)-spectral gaps for π-irreducible, positive recurrent Markov chains with a general state space Ω. We obtain necessary and sufficient conditions for the existence of L ²(π)-spectral gaps in terms of a sequence of isoperimetric constants. For reversible Markov chains, it turns out that the spectral gap can be understood in terms of convergence of an induced probability flow to the uniform flow. These results are used to recover classical results concerning uniform ergodicity and the spectral gap property as well as other new results. As an application of our result, we present a rather short proof for the fact that geometric ergodicity implies the spectral gap property. Moreover, the main result of this paper suggests that sharp upper bounds for the spectral gap should be expected when evaluating the isoperimetric flow for certain sets. We provide several examples where the obtained upper bounds are exact.     

Prediction of the evolution of the dispersed phase in bubbly flow problems

For modeling multi-phase where the dispersed phase plays a major role in determining the flow structure and inter phase transfer quantities, the size distribution of the bubbles has to be considered. This can be done by extension of the mass balance equation to a population balance equation. In this work, a least squares spectral method is tested for predicting the evolution of the dispersed phase in a vertical two-phase bubbly flow. The least squares spectral method consists in minimizing the L² norm of the residual over the simulation domain. The results are compared with experimental data obtained for two different initial bubble distributions.     

New Zonal,Spectral Solutions for the Navier‐Stokes Layer and Their Applications

New zonal, spectral forms for the axial, lateral and vertical velocity's components, density function and absolute temperature inside of compressible three‐dimensional Navier‐Stokes layer (NSL) over flattened, flying configurations (FC), are here proposed. The inviscid flow over the FC, obtained after the solidification of the NSL, is here used as outer flow. If the spectral forms of the velocity's components are introduced in the partial differential equations of the NSL and the collocation method is used, the spectral coefficients are obtained by the iterative solving of an equivalent quadratical algebraic system with slightly variable coefficients.     

Study of spectral characteristics of a homogeneous turbulent flow

A spectral representation of kinetic energy for a vortex cascade of instability in a compressible inviscid shear flow is considered, and the Rayleigh-Taylor instability is studied. A comparative analysis is given to the spectral decompositions of kinetic energy for both problems. The classical Kolmogorov −5/3 power law is proved to hold for developed turbulent flows.     

MASLOV-TYPE INDEX THEORY FOR SYMPLECTIC PATHS AND SPECTRAL FLOW (Ⅰ)

51.IntroductionThespectralfiowforaoneparameterfamilyoflinearselfadjointFredholmoperatorsisintroducedbyAtiyah-Patodi-Singerl2]intheirstudyofindextheoryonmanifoldswithboundary.Sincethenothersignificantapplicationshavebeenfound.In[17],J.RobbinandD.SalamonstudiedindetailthespectralflowforthecurvesoflinearselfadjointFredhomoperatorswithinvertibleoperatorsattheendpointsandprovedanindextheorem.In[7]and[8]thenotionofthespectralflowwasgeneralizedtothehigherdimensionalcasebyX.DaiandW.Zhang.Inthisp…     

Quasi-hyperbolic semigroups

We introduce the notion of quasi-hyperbolic operators and C₀-semigroups. Examples include the push-forward operator associated with a quasi-Anosov diffeomorphism or flow. A quasi-hyperbolic operator can be characterised by a simple spectral property or as the restriction of a hyperbolic operator to an invariant subspace. There is a corresponding spectral property for the generator of a C₀-semigroup, and it characterises quasi-hyperbolicity on Hilbert spaces but not on other Banach spaces. We exhibit some weaker properties which are implied by the spectral property.     

Spectral representations of sum- and max-stable processes

To each max-stable process with α-Fréchet margins, α ∈ (0,2), a symmetric α-stable process can be associated in a natural way. Using this correspondence, we deduce known and new results on spectral representations of max-stable processes from their α-stable counterparts. We investigate the connection between the ergodic properties of a stationary max-stable process and the recurrence properties of the non-singular flow generating its spectral representation. In particular, we show that a stationary max-stable process is ergodic iff the flow generating its spectral representation has vanishing positive recurrent component. We prove that a stationary max-stable process is ergodic (mixing) iff the associated SαS process is ergodic (mixing). We construct non-singular flows generating the max-stable processes of Brown and Resnick.     

Spectral Flow in Fredholm Modules, Eta Invariants and the JLO Cocycle

We give a comprehensive account of an analytic approach to spectral flow along paths of self-adjoint Breuer–Fredholm operators in a type I or II von Neumann algebra N. The framework is that of odd unbounded-summable Breuer–Fredholm modules for a unital Banach *-algebra, A. In the type II case spectral flow is real-valued, has no topological definition as an intersection number and our formulae encompass all that is known. We borrow Ezra Getzlers idea (suggested by I. M. Singer) of considering spectral flow (and eta invariants) as the integral of a closed one-form on an affine space. Applications in both the types I and II cases include a general formula for the relative index of two projections, representing truncated eta functions as integrals of one forms and expressing spectral flow in terms of the JLO cocycle to give the pairing of the K-homology and K-theory of A.     

Transparent boundary conditions as dissipative closures

We study the systematic development of subgrid closures via transparent boundary conditions in spectral space in one of the simplest possible settings: passive tracer transport in a doubly–periodic shear flow that has small support in the spectral domain. The resulting closure schemes have small dissipation rate errors and are only minimally over–dissipative for unsteady flows. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Macroscopic current induced boundary conditions for Schrödinger-type operators

We describe an embedding of a quantum mechanically described structure into a macroscopic flow. The open quantum system is partly driven by an adjacent macroscopic flow acting on the boundary of the bounded spatial domain designated to quantum mechanics. This leads to an essentially non-selfadjoint Schrödinger-type operator, the spectral properties of which will be investigated.     

Bifurcation of critical points for continuous families of C 2 functionals of Fredholm type

Given a continuous family of C ² functionals of Fredholm type, we show that the nonvanishing of the spectral flow for the family of Hessians along a known (trivial) branch of critical points not only entails bifurcation of nontrivial critical points but also allows to estimate the number of bifurcation points along the branch. We use this result for several parameter bifurcation, estimating the number of connected components of the complement of the set of bifurcation points in the parameter space and apply our results to bifurcation of periodic orbits of Hamiltonian systems. By means of a comparison principle for the spectral flow, we obtain lower bounds for the number of bifurcation points of periodic orbits on a given interval in terms of the coefficients of the linearization.     

Instability in Parallel Flows Revisited

An example of an unstable inviscid plane parallel shear flow with classical boundary conditions is presented. The complete unstable spectrum is exhibited using techniques of continued fractions for the shear flow with profile U ( y )=cos  m y . For such flows spectral instability implies nonlinear instability. A three-dimensional generalization is discussed.     

Spectral gap on Riemannian path space over static and evolving manifolds

In this article, we continue the discussion of Fang–Wu (2015) to estimate the spectral gap of the Ornstein–Uhlenbeck operator on path space over a Riemannian manifold of pinched Ricci curvature. Along with explicit estimates we study the short-time asymptotics of the spectral gap. The results are then extended to the path space of Riemannian manifolds evolving under a geometric flow. Our paper is strongly motivated by Naber's recent work (2015) on characterizing bounded Ricci curvature through stochastic analysis on path space.