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**Logarithmic terms in trace expansions of Atiyah-Patodi-Singer problems.**
*(English)*
Zbl 1048.35095

Author’s abstract: For Dirac-type operator \(D\) on a manifold \(X\) with a spectral boundary condition (defined by a pseudodifferential projection), the associated heat operator trace has an expansion in integer and half-integer powers and log-powers of \(t\); the interest in the expansion coefficients goes back to the work of Atiyah, Patodi and Singer. In the product case considered by APS, it is known that all the log-coefficients vanish when \(\dim X\) is odd, whereas the log-coefficients at integer powers vanish when \(\dim X\) is even. We investigate here whether this partial vanishing of logarithms holds more generally. One type of result, shown for general \(D\) with well-posed boundary conditions, is that a perturbation of \(D\) by a tangential differential operator vanishing to order \(k\) on the boundary leaves the first \(k\) log-power terms invariant (and the nonlocal power terms of the same degree are only locally perturbed). Another type of result is that for perturbations of the APS product case by tangential operators commuting with the tangential part of \(D\), all the logarithmic terms vanish when \(\dim X\) is odd (whereas they can all be expected to be nonzero when \(\dim X\) is even). The treatment is based on earlier joint work with R. Seeley and a recent systematic parameter-dependent pseudodifferential boundary operator calculus, applied to the resolvent.

Reviewer: C. Bouzar (Oran)

### MSC:

35Q40 | PDEs in connection with quantum mechanics |

35S15 | Boundary value problems for PDEs with pseudodifferential operators |

58J28 | Eta-invariants, Chern-Simons invariants |

81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |