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The very unusual properties of the resolvent, heat kernel, and zeta function for the operator $-d^2/dr^2 - 1/(4r^2)$

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Klaus Kirsten, Paul Loya, Jinsung Park

Mathematical Physics · Submitted 1 Nov 2005

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Only for the Friedrichs realization standard properties are reproduced, for all other realizations highly nonstandard properties are observed.

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arXiv IDmath-ph/0511002 (version 2)
Pith IDVPSNQTVH, Pith's permanent record of this paper and its review
CitationJinsung Park, Klaus Kirsten, Paul Loya. "The very unusual properties of the resolvent, heat kernel, and zeta function for the operator $-d^2/dr^2 - 1/(4r^2)$." arXiv:math-ph/0511002 (2005). https://pith.science/paper/VPSNQTVH
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,311 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.39 · human participants 0.01 · controlled comparison 0.17 · randomized assignment 0.01 · states quantitative result 0.31 · compares with prior results 0.47 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

In this article we analyze the resolvent, the heat kernel and the spectral zeta function of the operator $-d^2/dr^2 - 1/(4r^2)$ over the finite interval. The structural properties of these spectral functions depend strongly on the chosen self-adjoint realization of the operator, a choice being made necessary because of the singular potential present. Only for the Friedrichs realization standard properties are reproduced, for all other realizations highly nonstandard properties are observed. In particular, for $k\in \N$ we find terms like $(\log t)^{-k}$ in the small-$t$ asymptotic expansion of the heat kernel. Furthermore, the zeta function has $s=0$ as a logarithmic branch point.

background contribution method result interpretation main claim

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 In this article we analyze the resolvent, the heat kernel and the spectral zeta function of the operator $-d^2/dr^2 - 1/(4r^2)$ over the finite interval. contribution 1.00 0.93 0.06 asserted 0.96
s1 The structural properties of these spectral functions depend strongly on the chosen self-adjoint realization of the operator, a choice being made necessary because of the singular potential present. background 0.78 0.25 0.10 asserted 1.00
s2 Only for the Friedrichs realization standard properties are reproduced, for all other realizations highly nonstandard properties are observed. result 0.98 0.63 0.61 asserted 1.00
s3 In particular, for $k\in \N$ we find terms like $(\log t)^{-k}$ in the small-$t$ asymptotic expansion of the heat kernel. result 1.00 0.95 0.13 asserted 1.00
s4 Furthermore, the zeta function has $s=0$ as a logarithmic branch point. result 1.00 0.84 0.16 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

In this article we analyze the resolvent, the heat kernel and the spectral zeta function of the operator $-d^2/dr^2 - 1/(4r^2)$ over the finite interval. The structural properties of these spectral functions depend strongly on the chosen self-adjoint realization of the operator, a choice being made necessary because of the singular potential present. Only for the Friedrichs realization standard properties are reproduced, for all other realizations highly nonstandard properties are observed. In particular, for $k\in \N$ we find terms like $(\log t)^{-k}$ in the small-$t$ asymptotic expansion of the heat kernel. Furthermore, the zeta function has $s=0$ as a logarithmic branch point.

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2 arXiv papers cite this in 3 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Compares with it 1 paper
  • In this way, we provide a model for which [equation] As in the model presented in [ref] , whose $ζ$ function satisfies [ref], the $ζ$ function for the Coulomb-type background is not analytic at $s=0$ either, so definition [ref] does not provide a finite regularization of the effective action.

    A New Example of the Effects of a Singular Background on the Zeta Function · Introduction · 0.95

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