Koplienko–Neidhardt trace formula for pairs of contraction operators and pairs of maximal dissipative operators

We present a generalization of Koplienko–Neidhardt trace formula for pairs of Hilbert space operators (T , V ) with T contractive and V unitary such that T – V is a Hilbert–Schmidt operator. We extend the result to pairs of contractions and then, via Cayley transform, to pairs of maximal dissipative operators. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

von Neumann’s trace inequality for Hilbert–Schmidt operators

von Neumann’s inequality in matrix theory refers to the fact that the Frobenius scalar product of two matrices is less than or equal to the scalar product of the respective singular values. Moreover, equality can only happen if the two matrices share a joint set of singular vectors, and this latter part is hard to find in the literature. We extend these facts to the separable Hilbert space setting, and provide a self-contained proof of the “latter part”.     

Spectral shift function and resonances for slowly varying perturbations of periodic Schrödinger operators

We study the spectral shift function s(λ,h) and the resonances of the operator P(h)=-Δ+V(x)+W(hx). Here V is a periodic potential, W a decreasing perturbation and h a small positive constant. We give a representation of the derivative of s(λ,h) related to the resonances of P(h), and we obtain a Weyl-type asymptotics of s(λ,h). We establish an upper bound O(h^-n+1) for the number of the resonances of P(h) lying in a disk of radius h.     

Higher order spectral shift

We construct higher order spectral shift functions, extending the perturbation theory results of M.G. Krein [M.G. Krein, On a trace formula in perturbation theory, Mat. Sb. 33 (1953) 597-626 (in Russian)] and L.S. Koplienko [L.S. Koplienko, Trace formula for perturbations of nonnuclear type, Sibirsk. Mat. Zh. 25 (1984) 62-71 (in Russian); translation in: Trace formula for nontrace-class perturbations, Siberian Math. J. 25 (1984) 735-743] on representations for the remainders of the first and second order Taylor-type approximations of operator functions. The higher order spectral shift functions represent the remainders of higher order Taylor-type approximations; they can be expressed recursively via the lower order (in particular, Krein's and Koplienko's) ones. We also obtain higher order spectral averaging formulas generalizing the Birman-Solomyak spectral averaging formula. The results are obtained in the semi-finite von Neumann algebra setting, with the perturbation taken in the Hilbert-Schmidt class of the algebra.     

Time delay in scattering by potentials and by magnetic fields with two supports at large separation

We study the asymptotic behavior of the time delay (defined as the trace of the Eisenbud-Wigner time delay operator) for scattering by potential and by magnetic field with two compact supports as the separation of supports goes to infinity. The emphasis is placed on analyzing how different the asymptotic formulae are in potential and magnetic scattering. The difference is proper to scattering in two dimensions.     

Eigenvalues of the reference operator and semiclassical resonances

We prove that the estimate of the number of the eigenvalues in intervals , of the reference operator L^#(h) related to a self-adjoint operator L(h) is equivalent to the estimate of the integral over [λ−δ,λ+δ] of the sum of harmonic measures associated to the resonances of L(h) lying in a complex neighborhood Ω of λ>0 and the number of the positive eigenvalues of L(h) in [λ−δ,λ+δ]. We apply this result to obtain a Breit-Wigner approximation of the derivative of the spectral shift function near critical energy levels.     

三种不同意义下的最佳求积公式之间的关系

详细讨论了函数类KWr[a,b]上Sard和N iko lsk ii意义下以及基于给定信息的最佳求积公式三者之间的关系,并且提供了一种由基于给定信息的最佳求积公式得到其它两种求积公式的方法.     

Künneth formulae and cross products for the symplectic Floer cohomology

For a symplectic monotone manifold (P,ω) and φSymp₀(P,ω), we define a -graded symplectic Floer cohomology (a local invariant) over integral coefficients. There is a spectral sequence which arises from a filtration on the -graded symplectic Floer cochain complex. The spectral sequence converges to the -graded symplectic Floer cohomology (a global invariant). We show that there are cross products on the -graded symplectic Floer cohomology and on the spectral sequence, hence on the usual -graded symplectic Floer cohomology. The Künneth formula for the -graded symplectic Floer cohomology is proved and similar results for the spectral sequence are obtained.     

Hilbert重级数定理的一个新推广

通过引入一个适当的对数函数建立了一种新的Hilbert型不等式.利用Euler-Maclaurin求和公式对权函数进行了估计.证明了常数因子π2r＋1Er是最佳的,其中Er是Euler数.作为应用,给出了一些互相等价的不等式.     

Contributions of non-extremum critical points to the semi-classical trace formula

We study the semi-classical trace formula at a critical energy level for a h-pseudo-differential operator on whose principal symbol has a totally degenerate critical point for that energy. We compute the contribution to the trace formula of isolated non-extremum critical points under a condition of “real principal type”. The new contribution to the trace formula is valid for all time in a compact subset of but the result is modest since we have restrictions on the dimension.     

A trace class operator inequality

In this note, we generalize the inequality about the trace of positive semidefinite matrix trk(AB)?k(trA)k(trB) to Hilbert space, and obtain a relevant inequality about positive trace class operator.     

Operator-norm inequalities and interpolated trace inequalities

In this article, we investigate some operator-norm inequalities related to some conjectures posed by Hayajneh and Kittaneh that are related to questions of Bourin regarding a special type of inequalities referred to as subadditivity inequalities. While some inequalities are meant to answer these conjectures, other inequalities present reverse-type inequalities for these conjectures. Then, we present some new trace inequalities related to Heinz means inequality and use these inequalities to prove some variants of the aforementioned conjectures.     

Geometric estimates for the trace formula

In order to study the asymptotic distribution of geometric or spectral data associated with quotients of a reductive group by a lattice, one needs a trace formula for test functions on that group with noncompact support. Arthur has proved a trace formula for compactly supported test functions on reductive groups of arbitrary rank. We show that the coarse geometric expansion in his formula converges for rapidly decreasing functions.      

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.

     

Gaussian extended cubature formulae for polyharmonic functions

The purpose of this paper is to show certain links between univariate interpolation by algebraic polynomials and the representation of polyharmonic functions. This allows us to construct cubature formulae for multivariate functions having highest order of precision with respect to the class of polyharmonic functions. We obtain a Gauss type cubature formula that uses values of linear functionals (integrals over hyperspheres) and is exact for all -harmonic functions, and consequently, for all algebraic polynomials of variables of degree .

     

Necessary and sufficient conditions for absolute summability of the trace formulas for certain one dimensional Schrödinger operators

For the general one dimensional Schrödinger operator with real we study some analytic aspects related to order-one trace formulas originally due to Buslaev-Faddeev, Faddeev-Zakharov, and Gesztesy-Holden-Simon-Zhao. We show that the condition guarantees the existence of the trace formulas of order one only with certain resolvent regularizations of the integrals involved. Our principle results are simple necessary and sufficient conditions on absolute summability of the formulas under consideration. These conditions are expressed in terms of Fourier transforms related to .

     

The Spectral Shift Function for Certain Block Operator Matrices

Let L₀ be a 2 × 2 diagonal self‐adjoint block operator matrix with entries A and D. If operators B and B* are added to the off diagonal zeros, certain parts of the spectrum of L₀ move to the right and other parts move to the left. In this paper it is shown that, correspondingly, if B is a trace class operator M. G. Krein's spectral shift function is of constant sign on certain intervals.