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Feynman--Kac formula for the heat equation with a one-center point interaction in $d=3$

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Pith's one-line read Solutions to the three-dimensional heat equation with a point interaction admit a Feynman-Kac representation using a continuous process and normalizing factor.

desk verdict The paper constructs a Feynman-Kac representation for the heat equation with a one-center delta interaction in 3D via a custom path measure and normalizing function G. read the letter →

arxiv 2606.11677 v1 pith:RLEFPNCE submitted 2026-06-10 math.PR

Makoto Nakashima This is my paper · ORCID
classification math.PR
keywords Feynman-KacformulapointinteractionheatequationSchrödingeroperatorDiracdeltapotentialprobabilisticrepresentationthreedimensionsself-adjointextension
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper constructs, for each t > 0 and x not equal to zero, a probability law on path space together with a normalizing function G_t^α(x). This produces the representation u(t,x) = G_t^α(x) times the expected value of the initial data evaluated at the position of the process at time t. The construction applies to the operator realized either as a self-adjoint extension of the Laplacian away from the origin or as a norm-resolvent limit of regularized potentials. A reader would care because the formula supplies an explicit probabilistic expression for the evolution even when the interaction is a singular delta at a single point.

What carries the argument

The probability law on continuous paths from x together with the normalizing function G_t^α(x) that converts the plain expectation into the action of the heat semigroup generated by the point-interaction operator.

What would settle it

For an explicit initial function u_0 whose evolved solution u(t,x) is known by other means, compute the right-hand side using the constructed process and check whether equality holds after multiplication by G_t^α(x).

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Extended reading notes

Core claim

For the heat equation partial_t u = (1/2) Delta_α u with initial data in C_c^∞(R^3 excluding zero), the solution satisfies u(t,x) = G_t^α(x) E[u_0(W^{t,x}(t))], where W^{t,x} is a continuous process depending on t, x and α, and G_t^α is a normalizing function. The result holds for the operator -Delta_α realized in either of the two standard ways.

Load-bearing premise

The point-interaction operator admits a self-adjoint realization that generates a heat semigroup whose action on smooth compactly supported functions away from the origin can be captured by the constructed path measure.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation and recommendation of minor revision. The referee's summary accurately captures the manuscript's contribution.

Circularity Check

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No significant circularity

full rationale

The paper constructs an explicit probability law on path space together with the scalar G_t^α(x) so that the indicated expectation reproduces the action of the heat semigroup generated by a self-adjoint realization of -Δ_α. This construction is presented as the content of the work and is not defined in terms of the target representation itself; the operator realization (via self-adjoint extension or norm-resolvent limit) is taken as an independent hypothesis whose semigroup is then represented probabilistically. No self-definitional loop, fitted-input prediction, or load-bearing self-citation chain appears in the derivation chain.

Assumptions & free parameters 0 free parameters · 1 assumptions · 1 invented entities

The claim rests on the existence of the self-adjoint realization of -Δ_α and the construction of the path measure W that encodes the interaction; both are asserted but not derived in the provided abstract.

assumptions (1)
  • domain assumption The operator -Δ_α can be realized as a self-adjoint extension of -Δ restricted to C_0^∞(R^3\{0}) or as norm-resolvent limit of regularized potentials
    Invoked in the abstract to define the operator for which the heat equation is posed.
invented entities (1)
  • Continuous process W^{t,x}
    purpose: Provides the random paths whose endpoint expectation, scaled by G, yields the solution
    Introduced in the abstract as depending on (t,x,α) to incorporate the point interaction effect

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Pith. "Pith review of Feynman--Kac formula for the heat equation with a one-center point interaction in $d=3$." pith.science (2026). https://pith.science/paper/RLEFPNCE

@misc{pith2026260611677,
  author       = {Pith},
  title        = {Pith review of: Feynman--Kac formula for the heat equation with a one-center point interaction in $d=3$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RLEFPNCE}},
  note         = {Machine review of arXiv:2606.11677}
}
abstract

We study Schr\"odinger operators with a one-center point interaction, formally defined by \begin{align*} -\Delta_\alpha=-\Delta+\alpha\,\delta_0(\cdot), \end{align*} for $\alpha\in\mathbb{R}$, and the associated heat equation \begin{align} \partial_t u=\tfrac{1}{2}\Delta_{\alpha} u,\quad u(0,x)=u_0(x)\in C_c^{\infty}(\mathbb{R}^3\setminus\{0\}).\label{eq:HEapp} \end{align} Here $\Delta$ denotes the Laplacian (self-adjoint on $L^2(\mathbb{R}^3)$) and $\delta_x$ the Dirac measure at $x$. The operator $-\Delta_\alpha$ can be realized either as a self-adjoint extension of $-\Delta|_{C_0^{\infty}(\mathbb{R}^3\setminus\{0\})}$ in $L^2(\mathbb{R}^3)$, or as the norm-resolvent limit of $-\Delta+\lambda_\varepsilon V(\cdot/\varepsilon)$ for suitable $\lambda_\varepsilon$ and $V:\mathbb{R}^3\to\mathbb{R}$. In this paper we construct, for each $t>0$ and $x\in\mathbb{R}^3\setminus\{0\}$, a probability law on path space and a normalizing function $G_t^\alpha(x)$ giving the following probabilistic representation of the solution to the associated equation: \begin{align*} u(t,x)=G_t^\alpha(x)\,\mathbb{E}\bigl[u_0\bigl(W^{t,x}(t)\bigr)\bigr], \end{align*} where $\{W^{t,x}(s):0\le s\le t\}$ is a continuous process depending on $(t,x,\alpha)$. The result provides a Feynman--Kac type formula for the heat equation with a one-point interaction in three dimensions.

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  1. A submartingale for the probability of avoiding the origin in one-point interaction ground-state diffusion: $d \in \{2,3\}$

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