Non-self-adjoint graphs

Amru Hussein; David Krejcirik; Petr Siegl

RECORD

Non-self-adjoint graphs

Amru Hussein, David Krejcirik, Petr Siegl

Spectral Theory · Submitted 20 Aug 2013

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

The authors' main claim Among other things, we describe a simple way how to relate the similarity transforms between Laplacians on certain graphs with elementary similarity transforms between matrices defining the boundary conditions.

how it is stated · asserted 0.99 read the record →

Comment (0) Read on arXiv ↗

Cite ▾

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, 2,910 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.19 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.07 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

On finite metric graphs we consider Laplace operators, subject to various classes of non-self-adjoint boundary conditions imposed at graph vertices. We investigate spectral properties, existence of a Riesz basis of projectors and similarity transforms to self-adjoint Laplacians. Among other things, we describe a simple way how to relate the similarity transforms between Laplacians on certain graphs with elementary similarity transforms between matrices defining the boundary conditions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	On finite metric graphs we consider Laplace operators, subject to various classes of non-self-adjoint boundary conditions imposed at graph vertices.	background 0.54	0.64	0.04	asserted 0.79
s1	We investigate spectral properties, existence of a Riesz basis of projectors and similarity transforms to self-adjoint Laplacians.	contribution 0.99	0.90	0.16	asserted 0.86
s2	Among other things, we describe a simple way how to relate the similarity transforms between Laplacians on certain graphs with elementary similarity transforms between matrices defining the boundary conditions.	contribution 0.69	0.92	0.31	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.87 · 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

On finite metric graphs we consider Laplace operators, subject to various classes of non-self-adjoint boundary conditions imposed at graph vertices. We investigate spectral properties, existence of a Riesz basis of projectors and similarity transforms to self-adjoint Laplacians. Among other things, we describe a simple way how to relate the similarity transforms between Laplacians on certain graphs with elementary similarity transforms between matrices defining the boundary conditions.

Human review

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

How the literature uses it

1 arXiv paper cite this in 1 sentence

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

What it cites · 8 sentences without a confident label

Builds on 1 paper

Considering the same boundary conditions at the central vertex on the compact star graph with two edges one obtains that the operators is similar to a self-adjoint Laplacian for any self-adjoint boundary condition imposed at both endpoints simultaneously, in particular for Dirichlet boundary conditions as considered in [ref] .

Surprising spectra of PT-symmetric point interactions · Regular case · 0.83

Uses the method or tool of 1 paper

An alternative method how to derive other closed formulae for the metric operators $Θ$ was developed in [ref] and further employed in [ref] , where one can additionally find an explicit formula for the self-adjoint operator to which $-Δ(A,B)$ is similar.

On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators · Riesz basis on compact graphs · 0.85

Compares with 1 paper

This reproduces some of the result form [ref] .

Surprising spectra of PT-symmetric point interactions · Irregular compact case · 0.93

Cites as support 1 paper

Furthermore, if $(A,B)$ does not admit such a parametrisation, then the numerical range of $-Δ(A,B)$ is entire $$, see [ref] .

Maximal-quasi-accretive Laplacians on finite metric graphs · m-sectorial boundary conditions · 0.98

Cites as background 1 paper

This intuitive picture is formalised here by recalling from [ref] some notation and basic definitions.

The inverse scattering problem for metric graphs and the traveling salesman problem · The Laplacian on finite metric graphs · 0.95

Discussion

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