Heat-Trace Asymptotics on Singular and Noncompact Manifolds

Definitions and mathematical statement

Let $P$ be an elliptic operator on a manifold $M$, which may be noncompact or singular. The supplied record asks for criteria ensuring trace-class heat operators and power--logarithm heat-trace asymptotics. It does not specify a class of manifolds, an operator realization, or a precise asymptotic convention, so the following is a formulation of the requested properties rather than a complete conjecture.

For a specified Hilbert space $H$ and a realization of $P$ for which $e^{-tP}$ is defined as a bounded operator, an operator $A$ on $H$ is trace class if \[ \|A\|_1:=\operatorname{Tr}\left((A^*A)^{1/2}\right)<\infty. \] Whenever $e^{-tP}$ is trace class, its heat trace is \[ \Theta_P(t):=\operatorname{Tr}_H(e^{-tP}). \] The requested task is to give hypotheses on the geometry and on the realization of $P$ that ensure trace-class membership for positive times in an appropriate, specified time range and an expansion of the schematic form \[ \Theta_P(t)\sim\sum_{\alpha,k}a_{\alpha,k}\,t^\alpha(\log t)^k, \qquad t\downarrow0. \] Here $\log t$ is the real logarithm for $t>0$, and the coefficients are independent of $t$. The source does not specify the allowable exponents $\alpha$, the range of logarithmic powers $k$, the ordering or local finiteness of the index set, or remainder estimates defining $\sim$. Thus this display cannot yet be read as a precise asymptotic assertion.

A complete statement must also specify the geometric singularities and behavior at infinity allowed for $M$, the measure and any bundle defining $H$, the order and type of ellipticity of $P$, its domain and any boundary or singular-end conditions, and hypotheses making the heat operator well defined. Self-adjointness, positivity, and a particular extension of $P$ are not supplied assumptions. The record also leaves unclear whether trace-class membership is requested for every $t>0$ or only for sufficiently small positive $t$. These choices remain unresolved rather than being imposed in this draft.

Short English statement

Find geometric and operator-theoretic conditions under which an elliptic operator on a possibly noncompact or singular manifold has trace-class heat operators and a heat trace admitting a power--logarithm expansion as $t\downarrow0$. The supplied extraction lacks the hypotheses and asymptotic conventions needed for a complete mathematical statement; its supplied open-problem label requires review.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, with catalog/problems.json as the additional registry specified by the project instructions; permanent ID $30002071$, title Heat-Trace Asymptotics on Singular and Noncompact Manifolds, exact source problem code catalogue entry. This draft uses the supplied complete source record; no registry inspection is claimed.
• Supplied report citation: Analysis and Geometric Singularities (2013), Oberwolfach Reports, pp. 1487--1562, DOI 10.4171/owr/2012/25: https://doi.org/10.4171/owr/2012/25. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No individual proposer is supplied; the proposed year is recorded as $2012$.
• Supplied literature source, labeled "Lesch analytic-torsion paper": https://doi.org/10.2140/apde.2013.6.221.
• Supplied literature source, labeled "original OWR report": https://doi.org/10.4171/owr/2012/25.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the problem open but states that the extraction is a technical setup for analytic-torsion heat asymptotics and stops before a precise criterion is stated; consequently, no complete claim can be assessed beyond the open-problem context. The supplied background asks that time-sensitive status and cited claims be verified independently.
• The record separately labels the statement corrected_verified and asserts that the official report explicitly identifies the trace-class and heat-expansion criteria as open. This is supplied verification metadata, not an independent verification by this draft.