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Register a new userNon-trivial singular spectral shift functions exist
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Authors
Nurulla Azamov
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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 $H.$
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Given a self-adjoint operator H, a self-adjoint trace class operator V and a fixed Hilbert-Schmidt operator F with trivial kernel and co-kernel, using limiting absorption principle an explicit set of full Lebesgue measure is defined such that for all points of this set the wave and the scattering matrices can be defined and constructed unambiguously. Many well-known properties of the wave and scattering matrices and operators are proved, including the stationary formula for the scattering matrix. This new abstract scattering theory allows to prove that for any trace class perturbations of arbitrary self-adjoint operators the singular part of the spectral shift function is an almost everywhere integer-valued function.
In this paper it is shown that in case of trace class perturbations the singular part of Pushnitski $mu$-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.
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) notion introduced in this paper. Resonance index can be described as follows. For a fixed $lambda$ the resonance points $r_0$ of a path $H_r$ of self-adjoint operators are real poles of a certain meromorphic function associated with the triple $(lambda+i0; H_0,V).$ When $lambda+i0$ is shifted to $lambda+iy$ with small $y>0,$ that pole get off the real axis in the coupling constant complex plane and, in general, splits into some $N_+$ poles in the upper half-plane and some $N_-$ poles in the lower half-plane (counting multiplicities). Resonance index of the triple $(lambda; H_{r_0},V)$ is the difference $N_+-N_-.$ Based on the theorem just described, a non-trivial example of singular spectral shift function is given.
Let $H_0 = -Delta + V_0(x)$ be a Schroedinger operator on $L_2(mathbb{R}^ u),$ $ u=1,2,$ or 3, where $V_0(x)$ is a bounded measurable real-valued function on $mathbb{R}^ u.$ Let $V$ be an operator of multiplication by a bounded integrable real-valued function $V(x)$ and put $H_r = H_0+rV$ for real $r.$ We show that the associated spectral shift function (SSF) $xi$ admits a natural decomposition into the sum of absolutely continuous $xi^{(a)}$ and singular $xi^{(s)}$ SSFs. This is a special case of an analogous result for resolvent comparable pairs of self-adjoint operators, which generalises the known case of a trace class perturbation while also simplifying its proof. We present two proofs -- one short and one long -- which we consider to have value of their own. The long proof along the way reframes some classical results from the perturbation theory of self-adjoint operators, including the existence and completeness of the wave operators and the Birman-Krein formula relating the scattering matrix and the SSF. The two proofs demonstrate the equality of the singular SSF with two a priori different but intrinsically integer-valued functions: the total resonance index and the singular $mu$-invariant.
We derive two main results: First, assume that $A$, $B$, $A_n$, $B_n$ are self-adjoint operators in the Hilbert space $mathcal{H}$, and suppose that $A_n$ converges to $A$ and $B_n$ to $B$ in strong resolvent sense as $n to infty$. Fix $m in mathbb{N}$, $m$ odd, $p in [1,infty)$, and assume that $T:= big[( A + iI_{mathcal{H}})^{-m} - ( B + iI_{mathcal{H}})^{-m}big] in mathcal{B}_p(mathcal{H})$, $T_n := big[( A_n + iI_{mathcal{H}})^{-m} - ( B_n + iI_{mathcal{H}})^{-m}big] in mathcal{B}_p(mathcal{H})$, and $lim_{n rightarrow infty} |T_n - T|_{mathcal{B}_p(mathcal{H})} =0$. Then for any function $f$ in the class $mathfrak F_{k}(mathbb{R}) supset C_0^{infty}(mathbb{R})$ (cf. (1.1)), $$ lim_{n rightarrow infty} big| [f(A_n) - f(B_n)] - [f(A)- f(B)]big|_{mathcal{B}_p(mathcal{H})}=0. $$ Our second result concerns the continuity of spectral shift functions $xi(cdot; B,B_0)$ with respect to the operator parameter $B$. For $T$ self-adjoint in $mathcal{H}$ we denote by $Gamma_m(T)$, $m in mathbb{N}$ odd, the set of all self-adjoint operators $S$ in $mathcal{H}$ satisfying $big[(S - z I_{mathcal{H}})^{-m} - (T - z I_{mathcal{H}})^{-m}big] in mathcal{B}_1(mathcal{H})$, $z in mathbb{C}backslash mathbb{R}$. Employing a suitable topology on $Gamma_m(T)$ (cf. (1.9), we prove the following: Suppose that $B_1in Gamma_m(B_0)$ and let ${B_{tau}}_{tauin [0,1]}subset Gamma_m(B_0)$ denote a path from $B_0$ to $B_1$ in $Gamma_m(B_0)$ depending continuously on $tauin [0,1]$ with respect to the topology on $Gamma_m(B_0)$. If $f in L^{infty}(mathbb{R})$, then $$ lim_{tauto 0^+} |xi(, cdot , ; B_{tau}, A_0) f - xi(, cdot , ; B_0, A_0) f|_{L^1(mathbb{R}; (| u|^{m+1} + 1)^{-1}d u)} = 0. $$
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