Trace-class properties of semi-groups associated with operator valued differential operators and their Witten index

Abstract

We consider the operator <i>D</i> = &part;<i> + A(X)</i> in L²(&#8477, <i>H</i>), where &part; is the closure of the derivative along &#8477 in L²(&#8477, <i>H</i>), the operator <i>A(X)</i> is the fibre-wise multiplication in L²(&#8477, <i>H</i>) by self-adjoint operators <i>A(x)</i>, x &isin; &#8477, in a separable Hilbert space <i>H</i>, and <i>A_&plusmn;</i> = lim_{<i>x &rarr; &plusmn;&infin;</i>} <i>A(x)</i> are limiting operators in an appropriate sense. <br /> We aim to investigate the index and related trace formulae of <i>D</i> purely in terms of the operator family <i>A(&middot;)</i>. <br /> This type of problem originated from the seminal papers "Spectral asymmetry and Riemannian geometry" by Atiyah, Patodi and Singer (1973-1976). Here, the authors showed in particular that the Fredholm index of <i>D</i> is the spectral flow through <i>0</i> of the family <i>A(&middot;)</i>, which was further discussed by Callias (1978) and Robbin and Salamon (1995). <br /> Pushnitski (2008) and Gesztesy, Latushkin, Makarov, Sukochev and Tomilov (2011) generalized these results, and dropped the assumption of discrete spectra from the works of the previous authors, and replaced it by a (relative) trace-class condition. The methodology in both works is centered around resolvents of the involved operators. <br /> The goal of this dissertation is to investigate the problem consequently from the viewpoint of semi-groups. <br /> As main results, we determine the L²(&#8477, <i>H</i>)-trace of the operator <i>e^{-tDD^*} - e^{-tD^*D}</i> in L²(&#8477, <i>H</i>) in terms of the <i>H</i>-trace of operators only involving <i>A₊</i> and <i>A_-</i> for <i>t > 0</i>. We also generalize the "index = spectral flow"-theorem to a formula for the Witten index of <i>D</i> and give a functional equation for the spectral shift functions of the pairs <i>(DD^*, D^*D)</i> and <i>(A₊, A_-)</i>.