## Operator theory, singular integral equations, and PDEs

#### Event Title

Witten Index and spectral shift function

#### Location

Cupples I Room 115

#### Start Date

7-18-2016 4:00 PM

#### End Date

18-7-2016 4:20 PM

#### Description

Let $D$ be a selfadjoint unbounded operator on a Hilbert space and let $\{B(t)\}$ be a one parameter norm continuous family of self-adjoint bounded operators that converges in norm to asymptotes $B_\pm$. Then setting $A(t)=D+B(t)$ one can consider the operator $\mathbf{D_A}^{}=d/dt+A(t)$ on the Hilbert space $L_2(\mathbb{R},H)$. We present a connection between the theory of spectral shift function for the pair of the asymptotes $(A_+,A_-)$ and index theory for the operator $\mathbf{D_A}^{}$. Under the condition that the operator $B_+$ is a $p$-relative trace-class perturbation of $A_-$ and some additional smoothness assumption we prove a heat kernel formula for all $t>0$, $$\mathrm{tr}\Big(e^{-t\mathbf{D_A}^{}\mathbf{D_A}^{*}}-e^{-t\mathbf{D_A}^{*}\mathbf{D_A}^{}}\Big)=-\Big(\frac{t}{\pi}\Big)^{1/2}\int_0^1\mathrm{tr}\Big(e^{-tA_s^2}(A_+-A_-)\Big)ds,$$ where $A_s, s\in[0,1]$ is a straight path joining $A_-$ and $A_+$. Using this heat kernel formula we obtain the description of the Witten index of the operator $\mathbf{D_A}^{}$ in terms of the spectral shift function for the pair $(A_+,A_-)$. {\bf Theorem.} \textit{If\, $0$ is a right and a left Lebesgue point of the spectral shift function $\xi(\cdot;A_+,A_-)$ for the pair $(A_+,A_-)$ (denoted by $\xi_L(0_+; A_+,A_-)$ and $\xi_L(0_-; A_+, A_-)$, respectively), then the Witten index $W_s(\mathbf{D_A})$ of the operator $\mathbf{D_A}$ exists and equals $$W_s(\mathbf{D_A})=\frac12\big(\xi(0+;A_+,A_-)+\xi(0-;A_+,A_-)\big).$$} We note that our assumptions include the cases studied earlier. In particular, we impose no assumption on the spectra of $A_\pm$ and we can treat differential operators in any dimension. As a corollary of this theorem we have the following result. {\bf Corollary.} \textit{Assume that the asymptotes $A_\pm$ are boundedly invertible. Then the operator $\mathbf{D_A}$ is Fredholm and for the Fredholm index $\mathrm{index}(\mathbf{D_A})$ of the operator $\mathbf{D_A}$ we have $$\mathrm{index}(\mathbf{D_A})=\xi(0;A_+,A_-)=\mathrm{sf}\{A(t)\}_{t=-\infty}^{+\infty},$$ where $\mathrm{sf}\{A(t)\}_{t=-\infty}^{+\infty}$ denotes the spectral flow along the path $\{A(t)\}_{t=-\infty}^{+\infty}.$}

#### Share

COinS

Jul 18th, 4:00 PM Jul 18th, 4:20 PM

Witten Index and spectral shift function

Cupples I Room 115

Let $D$ be a selfadjoint unbounded operator on a Hilbert space and let $\{B(t)\}$ be a one parameter norm continuous family of self-adjoint bounded operators that converges in norm to asymptotes $B_\pm$. Then setting $A(t)=D+B(t)$ one can consider the operator $\mathbf{D_A}^{}=d/dt+A(t)$ on the Hilbert space $L_2(\mathbb{R},H)$. We present a connection between the theory of spectral shift function for the pair of the asymptotes $(A_+,A_-)$ and index theory for the operator $\mathbf{D_A}^{}$. Under the condition that the operator $B_+$ is a $p$-relative trace-class perturbation of $A_-$ and some additional smoothness assumption we prove a heat kernel formula for all $t>0$, $$\mathrm{tr}\Big(e^{-t\mathbf{D_A}^{}\mathbf{D_A}^{*}}-e^{-t\mathbf{D_A}^{*}\mathbf{D_A}^{}}\Big)=-\Big(\frac{t}{\pi}\Big)^{1/2}\int_0^1\mathrm{tr}\Big(e^{-tA_s^2}(A_+-A_-)\Big)ds,$$ where $A_s, s\in[0,1]$ is a straight path joining $A_-$ and $A_+$. Using this heat kernel formula we obtain the description of the Witten index of the operator $\mathbf{D_A}^{}$ in terms of the spectral shift function for the pair $(A_+,A_-)$. {\bf Theorem.} \textit{If\, $0$ is a right and a left Lebesgue point of the spectral shift function $\xi(\cdot;A_+,A_-)$ for the pair $(A_+,A_-)$ (denoted by $\xi_L(0_+; A_+,A_-)$ and $\xi_L(0_-; A_+, A_-)$, respectively), then the Witten index $W_s(\mathbf{D_A})$ of the operator $\mathbf{D_A}$ exists and equals $$W_s(\mathbf{D_A})=\frac12\big(\xi(0+;A_+,A_-)+\xi(0-;A_+,A_-)\big).$$} We note that our assumptions include the cases studied earlier. In particular, we impose no assumption on the spectra of $A_\pm$ and we can treat differential operators in any dimension. As a corollary of this theorem we have the following result. {\bf Corollary.} \textit{Assume that the asymptotes $A_\pm$ are boundedly invertible. Then the operator $\mathbf{D_A}$ is Fredholm and for the Fredholm index $\mathrm{index}(\mathbf{D_A})$ of the operator $\mathbf{D_A}$ we have $$\mathrm{index}(\mathbf{D_A})=\xi(0;A_+,A_-)=\mathrm{sf}\{A(t)\}_{t=-\infty}^{+\infty},$$ where $\mathrm{sf}\{A(t)\}_{t=-\infty}^{+\infty}$ denotes the spectral flow along the path $\{A(t)\}_{t=-\infty}^{+\infty}.$}