An Optimal L-bound on the Krein Spectral Shift Function
نویسندگان
چکیده
and |ξA,B(λ)| ≤ n if A −B is rank n (2) are well known; see, for example, [5] or [6]. The Krein spectral shift function can also be defined for unbounded self-adjoint operators A,B and enjoys the same properties as long as their difference is trace class. The results of this paper extend to general unbounded operators A and B (as long as their difference is trace class) but for simplicity, we will suppose that A and B are bounded. For applications
منابع مشابه
Krein Spectral Shift Function
A b s t r a c t . Let ~A,B be the Krein spectral shift function for a pair of operators A, B, with C = A B trace class. We establish the bound f F(I~A,B()~)I ) d,~ <_ f F ( 1 5 1 c l , o ( ) , ) l ) d A = ~ [F(j) F ( j 1 ) ] # j ( C ) , j= l where F is any non-negative convex function on [0, oo) with F(O) = 0 and #j (C) are the singular values of C. The choice F(t) = t p, p > 1, improves a rece...
متن کاملReference Potential Approach to the Quantum-mechanical Inverse Problem: Ii. Solution of Krein Equation
A reference potential approach to the one-dimensional quantummechanical inverse problem is developed. All spectral characteristics of the system, including its discrete energy spectrum, the full energy dependence of the phase shift, and the Jost function, are expected to be known. The technically most complicated task in ascertaining the potential, solution of a relevant integral equation, has ...
متن کاملEstimates for the spectral shift function of the polyharmonic operator
The Lifshits–Krein spectral shift function is considered for the pair of operators H0 = (−4)l, l > 0 and H = H0 + V in L2(R), d ≥ 1; here V is a multiplication operator. The estimates for this spectral shift function ξ(λ;H,H0) are obtained in terms of the spectral parameter λ > 0 and the integral norms of V . These estimates are in a good agreement with the ones predicted by the classical phase...
متن کامل. SP ] 1 6 Ja n 20 09 HIGHER ORDER SPECTRAL SHIFT , II . UNBOUNDED CASE
Abstract. 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...
متن کاملar X iv : m at h / 07 03 44 2 v 1 [ m at h . O A ] 1 4 M ar 2 00 7 OPERATOR INTEGRALS , SPECTRAL SHIFT AND SPECTRAL FLOW
We present a new and simple approach to the theory of multiple operator in-tegrals that applies to unbounded operators affiliated with general von Neumann algebras. For semifinite von Neumann algebras we give applications to the Fréchet differentiation of operator functions that sharpen existing results, and establish the Birman-Solomyak representation of the spectral shift function of M. G. Kr...
متن کامل