Pith. Sign in

RECORD

Spectral density and Sobolev inequalities for pure and mixed states

PaperPDF from arXiv

data files

Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
X Bluesky LinkedIn Reddit Hacker News
Michel Rumin

Functional Analysis · Submitted 16 Jun 2009

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

The authors' main claim We prove some general Sobolev-type and related inequalities for positive operators A of given ultracontractive spectral decay, without assuming e^{-tA} is submarkovian.

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗
Cite ▾
arXiv ID0906.2902 (version 1)
Pith IDQGNY6KQG, Pith's permanent record of this paper and its review
CitationMichel Rumin. "Spectral density and Sobolev inequalities for pure and mixed states." arXiv:0906.2902 (2009). https://pith.science/paper/QGNY6KQG
Claim paper

The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Open the compiled record

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,655 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.23 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.43 · compares with prior results 0.14 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We prove some general Sobolev-type and related inequalities for positive operators A of given ultracontractive spectral decay, without assuming e^{-tA} is submarkovian. These inequalities hold on functions, or pure states, as usual, but also on mixed states, or density operators in the quantum mechanical sense. This provides universal bounds of Faber-Krahn type on domains, that apply to their whole Dirichlet spectrum distribution, not only the first eigenvalue. Another application is given to relate the Novikov-Shubin numbers of coverings of finite simplicial complexes to the vanishing of the torsion of some l^{p,2}-cohomology.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We prove some general Sobolev-type and related inequalities for positive operators A of given ultracontractive spectral decay, without assuming e^{-tA} is submarkovian. contribution 0.71 0.95 0.75 asserted 1.00
s1 These inequalities hold on functions, or pure states, as usual, but also on mixed states, or density operators in the quantum mechanical sense. result 0.50 0.76 0.20 asserted 1.00
s2 This provides universal bounds of Faber-Krahn type on domains, that apply to their whole Dirichlet spectrum distribution, not only the first eigenvalue. result 0.71 0.82 0.20 asserted 1.00
s3 Another application is given to relate the Novikov-Shubin numbers of coverings of finite simplicial complexes to the vanishing of the torsion of some l^{p,2}-cohomology. result 0.65 0.80 0.07 asserted 0.98

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

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

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

We prove some general Sobolev-type and related inequalities for positive operators A of given ultracontractive spectral decay, without assuming e^{-tA} is submarkovian. These inequalities hold on functions, or pure states, as usual, but also on mixed states, or density operators in the quantum mechanical sense. This provides universal bounds of Faber-Krahn type on domains, that apply to their whole Dirichlet spectrum distribution, not only the first eigenvalue. Another application is given to relate the Novikov-Shubin numbers of coverings of finite simplicial complexes to the vanishing of the torsion of some l^{p,2}-cohomology.

Human review

How Pith checks papers → · Full audit of this paper →

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.