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Register a new userSpectral Theory as Influenced by Fritz Gesztesy
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We survey a selection of Fritzs principal contributions to the field of spectral theory and, in particular, to Schroedinger operators.
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The spectral flow is a classical notion of functional analysis and differential geometry which was given different interpretations as Fredholm index, Witten index, and Maslov index. The classical theory treats spectral flow outside the essential spectrum. Inside essential spectrum, the spectral shift function could be considered as a proper analogue of spectral flow, but unlike the spectral flow, the spectral shift function is not an integer-valued function. In this paper it is shown that the notion of spectral flow admits a natural integer-valued extension for a.e. value of the spectral parameter inside essential spectrum too and appropriate theory is developed. The definition of spectral flow inside essential spectrum given in this paper applies to the classical spectral flow and thus gives one more new alternative definition of it.
We consider a compact perturbation $H_0 = S + K_0^* K_0$ of a self-adjoint operator $S$ with an eigenvalue $lambda^circ$ below its essential spectrum and the corresponding eigenfunction $f$. The perturbation is assumed to be along the eigenfunction $f$, namely $K_0f=0$. The eigenvalue $lambda^circ$ belongs to the spectra of both $H_0$ and $S$. Let $S$ have $sigma$ more eigenvalues below $lambda^circ$ than $H_0$; $sigma$ is known as the spectral shift at $lambda^circ$. We now allow the perturbation to vary in a suitable operator space and study the continuation of the eigenvalue $lambda^circ$ in the spectrum of $H(K)=S + K^* K$. We show that the eigenvalue as a function of $K$ has a critical point at $K=K_0$ and the Morse index of this critical point is the spectral shift $sigma$. A version of this theorem also holds for some non-positive perturbations.
We present a brief survey of the spectral theory and dynamics of infinite volume asymptotically hyperbolic manifolds. Beginning with their geometry and examples, we proceed to their spectral and scattering theories, dynamics, and the physical description of their quantum and classical mechanics. We conclude with a discussion of recent results, ideas, and conjectures.
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.
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.
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