Trace expansions for pseudodifferential boundary problems for Dirac-type operators and more general systems
The author proves asymptotic expansions of heat traces \[ (\ast)\qquad \text{Tr}(\varphi e^{-t\Delta_i})\sim \sum_{-n\leq k<0} a_{i,k}t^{k/2}+ \sum_{k=0}^\infty (a_{i,k}\log t+ a'_{i,k})t^{k/2} \text{ for} t\to 0 \] \[ \Delta_1=D^\ast_BD_B,\qquad \Delta_2=D_BD_B^\ast \] for general realizations \(D_B\) of first-order differential operators \(D\) (Dirac-type operators) on a manifold \(X\) with pseudodifferential boundary conditions \(B|_{\partial X}=0.\) The expansion \((\ast)\) and the index formula \[ \text{index} \;D_B=a'_{1,0}-a'_{2,0},\;\text{when} \varphi=1\;\text{and} X\;\text{is\;compact} \] are obtained also for any first-order elliptic differential operator \(D\) and any well-posed for it pseudodifferential operator \(B\) for compact \(X.\) The expansion \((\ast)\) is generalized to suitable noncompact situations.
scientific article; zbMATH DE number 1928881
scientific article