[Paper Review] An explicit Laurent expansion for regularised integrals of holomorphic symbols

Sylvie Paycha, Simon Scott|arXiv (Cornell University)|Jun 10, 2005

Holomorphic and Operator Theory7 references11 citations

TL;DR

This paper provides explicit formulae for all coefficients in the Laurent expansion of the Kontsevich-Vishik canonical trace for holomorphic families of classical pseudodifferential operators on a closed manifold. By generalizing the known identification of the Wodzicki residue with the pole at zero, it extends this relationship to all higher-order terms in the expansion, offering a complete algebraic characterization of the canonical trace's singular behavior.

ABSTRACT

Abstract. For a holomorphic family of classical pseudodifferential operators on a closed manifold we give exact formulae for all coefficients in the Laurent expansion of its Kontsevich-Vishik canonical trace. This generalizes a known result identifying the Wodzicki residue with the pole at zero to all higher order terms.

Motivation & Objective

• To derive exact formulae for all coefficients in the Laurent expansion of the Kontsevich-Vishik canonical trace for holomorphic families of classical pseudodifferential operators.
• To extend the known result identifying the Wodzicki residue as the residue at the pole at zero to all higher-order singular terms in the expansion.
• To provide a complete algebraic description of the canonical trace's singular structure beyond the leading-order pole.
• To establish a systematic method for computing all coefficients in the Laurent series of the regularized trace.

Proposed method

• The authors analyze the holomorphic family of classical pseudodifferential operators on a closed manifold using the theory of regularized traces.
• They apply the Kontsevich-Vishik canonical trace to the family and expand it as a Laurent series around the pole at zero.
• The method relies on the structure of the symbol expansion and the homogeneity properties of classical pseudodifferential operators.
• Key components include the use of the Wodzicki residue and its generalization to higher-order terms via the canonical trace's singular part.
• The derivation uses analytic continuation and residue calculus techniques applied to the regularized integral of the symbol.
• The final formulae express each coefficient in the Laurent expansion in terms of intrinsic geometric and analytic invariants of the operator family.

Experimental results

Research questions

• RQ1 How can all coefficients in the Laurent expansion of the Kontsevich-Vishik canonical trace be explicitly computed for holomorphic families of classical pseudodifferential operators?
• RQ2 What is the generalization of the Wodzicki residue beyond the leading-order pole to higher-order singular terms?
• RQ3 Can a uniform formula be derived for all coefficients in the Laurent series of the regularized trace?
• RQ4 What role do the symbol's homogeneity and operator class play in determining the structure of the Laurent coefficients?
• RQ5 How do the coefficients relate to geometric invariants of the underlying manifold and operator family?

Key findings

• The paper provides explicit formulae for all coefficients in the Laurent expansion of the canonical trace, not just the residue at the pole.
• The Wodzicki residue is identified as the coefficient of the first-order pole, generalizing a known result to all higher-order terms.
• Each coefficient in the Laurent series is expressed as a regularized integral of a specific homogeneous component of the operator's symbol.
• The structure of the coefficients is shown to be governed by the homogeneity and principal type of the pseudodifferential operator family.
• The method yields a complete algebraic description of the canonical trace's singular behavior, including both the residue and higher-order terms.
• The results establish a systematic framework for computing the full Laurent expansion of the trace for any holomorphic family of classical pseudodifferential operators.