Anna Skripka
Multiple scalar integral representations for traces of operator derivatives are obtained and applied in the proof of existence of the higher order spectral shift functions.
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We construct higher order spectral shift functions, which represent the remainders of Taylor-type approximations for the value of a function at a perturbed self-adjoint operator by derivatives of the function at an initial unbounded operator. In the particular cases of the zero and the first order approximations, the corresponding spectral shift functions have been constructed by M. G. Krein and L. S. Koplienko, respectively. The higher order spectral shift functions obtained in this paper can be expressed recursively via the lower order ones, in particular, Krein's and Koplienko's spectral sh…
We construct higher order spectral shift functions, extending the perturbation theory results of M. G. Krein and L. S. Koplienko 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…
Spectral Shift Functions of Lamé Operators
本文利用分布傅里叶变换得到的格林函数，计算了Weierstrass形式的Lamé算子和Brioschi-Halphen算子的Kreın谱位移函数，推进了该开放领域的研究。
We study the spectrum of one dimensional integral operators in bounded real intervals of length $2L$, for value of $L$ large. The integral operators are obtained by linearizing a non local evolution equation for a non conserved order parameter describing the phases of a fluid. We prove a Perron-Frobenius theorem showing that there is an isolated, simple minimal eigenvalue strictly positive for $L$ finite, going to zero exponentially fast in $L$. We lower bound, uniformly on $L$, the spectral gap by applying a generalization of the Cheeger's inequality. These results are usefulfor deriving spec…
It is well known that the sum of negative (positive) eigenvalues of some finite Hermitian matrix $V$ is concave (convex) with respect to $V$. Using the theory of the spectral shift function we generalize this property to self-adjoint operators on a separable Hilbert space with an arbitrary spectrum. More precisely, we prove that the spectral shift function integrated with respect to the spectral parameter from $-\infty$ to $λ$ (from $λ$ to $+\infty$) is concave (convex) with respect to trace class perturbations. The case of relative trace class perturbations is also considered.
We consider the Schrödinger operator on a star shaped graph with $n$ edges joined at a single vertex. We derive an expression for the trace of the difference of the perturbed and unperturbed resolvent in terms of a Wronskian. This leads to representations for the perturbation determinant and the spectral shift function, and to an analog of Levinson's formula.
In this article we continue our analysis of Schroedinger operators with a random potential using scattering theory. In particular the theory of Krein's spectral shift function leads to an alternative construction of the density of states in arbitrary dimensions. For arbitrary dimension we show existence of the spectral shift density, which is defined as the bulk limit of the spectral shift function per unit interaction volume. This density equals the difference of the density of states for the free and the interaction theory. This extends the results previously obtained by the authors in one d…
In this paper we study the asymptotic expansion of the spectral shift function for the slowly varying perturbations of periodic Schrödinger operators. We give a weak and pointwise asymptotics expansions in powers of $h$ of the derivative of the spectral shift function corresponding to the pair $\big(P(h)=P_0+ϕ(hx),P_0=-Δ+V(x)\big),$ where $ϕ(x)\in {\mathcal C}^\infty(\mathbb R^n,\mathbb R)$ is a decreasing function, ${\mathcal O}(|x|^{-δ})$ for some $δ>n$ and $h$ is a small positive parameter. Here the potential $V$ is real, smooth and periodic with respect to a lattice $Γ$ in ${\mathbb R}^n$.…
This paper is a continuation of my previous work on absolutely continuous and singular spectral shift functions, where it was in particular proved that the singular part of the spectral shift function is an a.e. integer-valued function. It was also shown that the singular spectral shift function is a locally constant function of the coupling constant $r,$ with possible jumps only at resonance points. Main result of this paper asserts that the jump of the singular spectral shift function at a resonance point is equal to the so-called resonance index, --- a new (to the best of my knowledge) noti…
In this paper it is shown that in case of trace class perturbations the singular part of Pushnitski $μ$-invariant does not depend on the angle variable. This gives an alternative proof of integer-valuedness of the singular part of the spectral shift function. As a consequence, the Birman-Krein formula for trace class perturbations follows.
The spectral shift function of a pair of self-adjoint operators is expressed via an abstract operator valued Titchmarsh--Weyl $m$-function. This general result is applied to different self-adjoint realizations of second-order elliptic partial differential operators on smooth domains with compact boundaries, Schrödinger operators with compactly supported potentials, and finally, Schrödinger operators with singular potentials supported on hypersurfaces. In these applications the spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-N…
We obtain a representation formula for the derivative of the spectral shift function $ξ(λ; B, ε)$ related to the operators $H_0(B,ε) = (D_x - By)^2 + D_y^2 + εx$ and $H(B, ε) = H_0(B, ε) + V(x,y), \: B > 0, ε> 0$. We establish a limiting absorption principle for $H(B, ε)$ and an estimate ${\mathcal O}(ε^{n-2})$ for $ξ'(λ; B, ε)$, provided $λ\notin σ(Q)$, where $Q = (D_x - By)^2 + D_y^2 + V(x,y).$
A general integral formula for the spectral flow of a path of unbounded selfadjoint Fredholm operators subject to certain summability conditions is derived from the interpretation of the spectral flow as a winding number.
We study the asymptotic behavior as L \to \infty of the finite-volume spectral shift function for a positive, compactly-supported perturbation of a Schrödinger operator in d-dimensional Euclidean space, restricted to a cube of side length L with Dirichlet boundary conditions. The size of the support of the perturbation is fixed and independent of L. We prove that the Cesàro mean of finite-volume spectral shift functions remains pointwise bounded along certain sequences L_n \to \infty for Lebesgue-almost every energy. In deriving this result, we give a short proof of the vague convergence of th…
Positivity of spectral shift functions and infinite-dimensional BMV conjecture
本文解决了半界算子迹类扰动下的BMV猜想，并确认了Schatten-von Neumann扰动下高阶谱移函数的正性。
This paper extends Krein's spectral shift function theory to the setting of semifinite spectral triples. We define the spectral shift function under these hypotheses via Birman-Solomyak spectral averaging formula and show that it computes spectral flow.
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 $x=R$.
A noncommutative integral on spectrally truncated spectral triples, and a link with quantum ergodicity
本研究针对谱截断谱三元组提出非交换积分的简单近似，推导得到非交换几何的Szegő极限公式，建立其与态密度的关联，提出紧谱三元组测地流遍历性的定义，揭示量子遍历性与Connes积分公式的紧密联系。
We study the manner in which a sequence of spectral shift functions $ξ(\cdot;H_j,H_{0,j})$ associated with abstract pairs of self-adjoint operators $(H_j, H_{0,j})$ in Hilbert spaces $\cH_j$, $j\in\bbN$, converge to a limiting spectral shift function $ξ(\cdot;H,H_0)$ associated with a pair $(H,H_0)$ in the limiting Hilbert space $\cH$ as $j\to\infty$ (mimicking the infinite volume limit in concrete applications to multi-dimensional Schrödinger operators). Our techniques rely on a Fredholm determinant approach combined with certain measure theoretic facts. In particular, we show that prior vagu…
We discuss the well known ``Fredholm index=spectral flow'' theorem and show that it can be interpreted as a limit case of an identity involving two spectral shift functions.
We generalize the $L^p$ spectral cluster bounds of Sogge for the Laplace-Beltrami operator on compact Riemannian manifolds to systems of orthonormal functions. The optimality of these new bounds is also discussed. These spectral cluster bounds follow from Schatten-type bounds on oscillatory integral operators.
We introduce the concept of a spectral shift operator and use it to derive Krein's spectral shift function for pairs of self-adjoint operators. Our principal tools are operator-valued Herglotz functions and their logarithms. Applications to Krein's trace formula and to the Birman-Solomyak spectral averaging formula are discussed.
A nonstandard approach to the direct integral version of the Spectral Theorem
本文通过非标准方法证明了无界自伴算子的谱定理直接积分版本，并给出了谱测度版本的新证明。
We discuss convergence properties of the spectral shift functions associated with a pair of Schrodinger operators with Dirichlet boundary conditions at the end points of a finite interval (0, r) as the length of interval approaches infinity.
We derive strong estimates for Schatten norms of operator derivatives along paths of contractions and apply them to prove existence of higher order spectral shift functions for pairs of contractions.
The new representation formula for the spectral shift function due to F.Gesztesy and K.A.Makarov is considered. This formula is extended to the case of relatively trace class perturbations.
We show that spectral problems for periodic operators on lattices with embedded defects of lower dimensions can be solved with the help of matrix-valued integral continued fractions. While these continued fractions are usual in the approximation theory, they are less known in the context of spectral problems. We show that the spectral points can be expressed as zeroes of determinants of the continued fractions. They are also useful in the study of inverse problems (one-to-one correspondence between spectral data and defects). Finally, the explicit formula for the resolvent in terms of the cont…
The recently introduced concept of a spectral shift operator is applied in several instances. Explicit applications include Krein's trace formula for pairs of self-adjoint operators, the Birman-Solomyak spectral averaging formula and its operator-valued extension, and an abstract approach to trace formulas based on perturbation theory and the theory of self-adjoint extensions of symmetric operators.
Sharp Spectral Gap Estimates on Manifolds under Integral Ricci Curvature Bounds
本文在积分曲率边界下证明了紧流形上的尖锐谱间隙估计，推广了 Kröger (1992) 以及 Bakry 和 Qian (2000) 的结果，并证实了 Ramos 等人 (2020) 中的猜想。
We discuss applications of the M. G. Kre\uın theory of the spectral shift function to the multi-dimensional Schrödinger operator as well as specific properties of this function, for example, its high-energy asymptotics. Trace identities for the Schrödinger operator are derived.
In this paper I prove existence of an irreducible pair of operators $H$ and $H+V,$ where $H$ is a self-adjoint operator and $V$ is a self-adjoint trace-class operator, such that the singular spectral shift function of the pair is non-zero on the absolutely continuous spectrum of the operator $H.$
Sharp endpoint multilinear estimates for oscillatory integrals and spectral clusters
本文证明了任意分离频率尺度下Carleson–Sjölin振荡积分算子的尖锐k线性L^p估计，并解决了三维闭Riemann流形上Burq–Gérard–Tzvetkov的对数自由端点双线性谱簇估计问题。
In this note it is shown that for trace-class perturbations of self-adjoint operators the singular part of the spectral shift function is additive.
We study convergence of the spectral shift function for the finite interval restrictions of a pair of full-line Schrödinger operators to an interval of the form $(-\ell,\ell)$ with coupled boundary conditions at the endpoints as $\ell\to \infty$ in the case when the finite interval restrictions are relatively prime to those with Dirichlet boundary conditions. Using a Krein-type resolvent identity we show that the spectral shift function for the finite interval restrictions converges weakly to that for the pair of full-line Schrödinger operators as the length of the interval tends to infinity.
We introduce the concept of Stieltjes integral of an operator-valued function with respect to the spectral measure associated with a normal operator. We give sufficient conditions for the existence of this integral and find bounds on its norm. The results obtained are applied to the Sylvester and Riccati operator equations. Assuming that the entry C is a normal operator, the spectrum of the entry A is separated from the spectrum of C, and D is a bounded operator, we obtain a representation for the strong solution X to the Sylvester equation XA-CX=D in the form of an operator Stieltjes integral…
Let $H$ signify the free non-negative Laplacian on $\mathbb{R}^2$ and $H_Y$ the non-negative Dirichlet Laplacian on the complement $Y$ of a nonpolar compact subset $K$ in the plane. We derive the low-energy expansion for the Krein spectral shift function (scattering phase) for the obstacle scattering system $\{\,H_Y,\,H\,\}$ including detailed expressions for the first three coefficients. We use this to investigate the large time behaviour of the expected volume of the pinned Wiener sausage associated to $K$.
The Krein spectral shift and rank one perturbations of spectra
利用最近关于Cauchy积分边界表现的结果，研究了自伴算子秩一扰动问题的Krein谱移.作为应用，我们证明了自伴算子的秩一扰动是纯点的当且仅当该算子的谱是可数的。我们还研究了纯点算子对酉等价的秩一扰动，并给出了各种例子秩一扰动的奇异谱。
Real spectral shift functions for pairs of contractions and pairs of dissipative operators
本文研究了收缩算子对和耗散算子对的实谱移函数条件，推导了其存在性，并应用于耗散Schrödinger算子。
In this paper we develop the method of double operator integrals to prove trace formulae for functions of contractions, dissipative operators, unitary operators and self-adjoint operators. To establish the absolute continuity of spectral shift, we use the Sz.-Nagy theorem on the absolute continuity of the spectrum of the minimal unitary dilation of a completely nonunitary contraction. We also give a construction of an intermediate contraction for a pair of contractions with trace class difference.
We consider the 3D Pauli operator with nonconstant magnetic field B of constant direction, perturbed by a symmetric matrix-valued electric potential V whose coefficients decay fast enough at infinity. We investigate the low-energy asymptotics of the corresponding spectral shift function. As a corollary, for generic negative V, we obtain a generalized Levinson formula, relating the low-energy asymptotics of the eigenvalue counting function and of the scattering phase of the perturbed operator.
Real-valued spectral shift functions for contractions and dissipative operators
本文针对收缩算子对与耗散算子对，公布了可积实值谱移函数存在性的充分条件，补充了此前迹类差收缩算子可积谱移函数存在性相关研究的实值性结论。
Spectral convergence of empirical integral operators with discontinuous kernels
研究经验积分算子在样本量趋于无穷时的谱行为，证明在放宽正性和连续性假设下，经验算子收敛于连续算子并给出明确收敛速率。
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 semi-finite 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 …
We develop an analog of classical oscillation theory for Sturm-Liouville operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing zeros of solutions of one operator by weighted zeros of Wronskians of solutions of two different operators. In particular, we show that a Sturm-type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the finiteness of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein's spec…
For singular numbers of integral operators of the form $u(x)\mapsto \int F_1(X)K(X,Y,X-Y)F_2(Y)u(Y)μ(dY),$ with measure $μ$ singular with respect to the Lebesgue measure in $\mathbb{R}^\mathbf{N}$, order sharp estimates for the counting function are established. The kernel $K(X,Y,Z)$ is supposed to be smooth in $X,Y$ and in $Z\ne 0$ and to admit an asymptotic expansion in homogeneous functions in $Z$ variable as $Z\to 0.$ The order in estimates is determined by the leading homogeneity order in the kernel and geometric properties of the measure $μ$ and involves integral norms of the weight func…
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