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Register a new userThe spectral shift function and spectral flow
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This paper extends Kreins 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.
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It is a well-known result of T.,Kato that given a continuous path of square matrices of a fixed dimension, the eigenvalues of the path can be chosen continuously. In this paper, we give an infinite-dimensional analogue of this result, which naturally arises in the context of the so-called unitary spectral flow. This provides a new approach to spectral flow, which seems to be missing from the literature. It is the purpose of the present paper to fill in this gap.
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 Kreins formula is established. Some examples of trace compatible affine spaces of operators are given.
The main object of the paper is a symmetric system $J y-B(t)y=lD(t) y$ defined on an interval $cI=[a,b) $ with the regular endpoint $a$. Let $f(cd,l)$ be a matrix solution of this system of an arbitrary dimension and let $(Vf)(s)=intlimits_cI f^*(t,s)D(t)f(t),dt$ be the Fourier transform of the function $f(cd)in L_D^2(cI)$. We define a pseudospectral function of the system as a matrix-valued distribution function $s(cd)$ of the dimension $n_s$ such that $V$ is a partial isometry from $L_D^2(cI)$ to $L^2(s;bC^{n_s})$ with the minimally possible kernel. Moreover, we find the minimally possible value of $n_s$ and parameterize all spectral and pseudospectral functions of every possible dimensions $n_s$ by means of a Nevanlinna boundary parameter. The obtained results develop the results by Arov and Dym; A.~Sakhnovich, L.~Sakhnovich and Roitberg; Langer and Textorius.
This note is to indicate the new sphere of applicability of the method developed by Mlak as well as by the author. Restoring those ideas is summoned by current developments concerning $K$-spectral sets on numerical ranges.
It was recently shown that the nodal deficiency of an eigenfunction is encoded in the spectrum of the Dirichlet-to-Neumann operators for the eigenfunctions positive and negative nodal domains. While originally derived using symplectic methods, this result can also be understood through the spectral flow for a family of boundary conditions imposed on the nodal set, or, equivalently, a family of operators with delta function potentials supported on the nodal set. In this paper we explicitly describe this flow for a Schrodinger operator with separable potential on a rectangular domain, and determine a mechanism by which lower energy eigenfunctions do or do not contribute to the nodal deficiency.
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