The spectrum of the normalized complex Laplacian for electrical networks is analyzed. We show that eigenvalues lie in a larger region compared to the case of the real Laplacian. We show the existence of eigenvalues with negative real part and absolute value greater than $2$. An estimate from below for the first non-vanishing eigenvalue in modulus is provided. We supplement the estimates with examples, showing sharpness.
{"title":"Eigenvalues of the normalized complex Laplacian for finite electrical networks","authors":"A. Muranova, R. Schippa","doi":"10.5445/IR/1000130241","DOIUrl":"https://doi.org/10.5445/IR/1000130241","url":null,"abstract":"The spectrum of the normalized complex Laplacian for electrical networks is analyzed. We show that eigenvalues lie in a larger region compared to the case of the real Laplacian. We show the existence of eigenvalues with negative real part and absolute value greater than $2$. An estimate from below for the first non-vanishing eigenvalue in modulus is provided. We supplement the estimates with examples, showing sharpness.","PeriodicalId":236439,"journal":{"name":"arXiv: Spectral Theory","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116778798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-11DOI: 10.13140/RG.2.2.27579.03364/1
Xinye Chen
The graph neural network has developed by leaps and bounds in recent years. This note summarizes the spectral graph neural network and related fundamentals of spectral graph theory and discusses the technical details of the main graph neural networks defined on the spectral domain.
{"title":"A Note on Spectral Graph Neural Network.","authors":"Xinye Chen","doi":"10.13140/RG.2.2.27579.03364/1","DOIUrl":"https://doi.org/10.13140/RG.2.2.27579.03364/1","url":null,"abstract":"The graph neural network has developed by leaps and bounds in recent years. This note summarizes the spectral graph neural network and related fundamentals of spectral graph theory and discusses the technical details of the main graph neural networks defined on the spectral domain.","PeriodicalId":236439,"journal":{"name":"arXiv: Spectral Theory","volume":"45 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120906982","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The second nonzero eigenvalue of the Laplacian on $S^{2}$ becomes maximal as the surface degenerates to two disjoint spheres, by a result of Nadirashvili. On spheres in all dimensions, an upper bound on the eigenvalue was derived by Petrides (the odd-dimensional case was proved earlier by Girouard, Nadirashvili, and Polterovich). Druet showed that the inequality is not sharp on higher dimensional sphere. In this paper, we will provide a simpler proof of these inequalities in all dimensions by adapting the trial function construction of Freitas and Laugesen from hyperbolic space.
{"title":"Maximization of the second Laplacian eigenvalue on the sphere","authors":"Hanna N. Kim","doi":"10.1090/proc/15908","DOIUrl":"https://doi.org/10.1090/proc/15908","url":null,"abstract":"The second nonzero eigenvalue of the Laplacian on $S^{2}$ becomes maximal as the surface degenerates to two disjoint spheres, by a result of Nadirashvili. On spheres in all dimensions, an upper bound on the eigenvalue was derived by Petrides (the odd-dimensional case was proved earlier by Girouard, Nadirashvili, and Polterovich). Druet showed that the inequality is not sharp on higher dimensional sphere. In this paper, we will provide a simpler proof of these inequalities in all dimensions by adapting the trial function construction of Freitas and Laugesen from hyperbolic space.","PeriodicalId":236439,"journal":{"name":"arXiv: Spectral Theory","volume":"490 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134056152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a systematic construction of signed harmonic functions for discrete Laplacian operators with Dirichlet conditions in the quarter plane. In particular, we prove that the set of harmonic functions is an algebra generated by a single element, which conjecturally corresponds to the unique positive harmonic function.
{"title":"Constructing discrete harmonic functions in wedges","authors":"V. Hoang, K. Raschel, Pierre Tarrago","doi":"10.1090/tran/8615","DOIUrl":"https://doi.org/10.1090/tran/8615","url":null,"abstract":"We propose a systematic construction of signed harmonic functions for discrete Laplacian operators with Dirichlet conditions in the quarter plane. In particular, we prove that the set of harmonic functions is an algebra generated by a single element, which conjecturally corresponds to the unique positive harmonic function.","PeriodicalId":236439,"journal":{"name":"arXiv: Spectral Theory","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134383324","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $Omega subset mathbb{R}^d$ be bounded with $C^1$ boundary. In this paper we consider Schrodinger operators $-Delta+ W$ on $Omega$ with $W(x)approxmathrm{dist}(x, partialOmega)^{-2}$ as $mathrm{dist}(x, partialOmega)to 0$. Under weak assumptions on $W$ we derive a two-term asymptotic formula for the sum of the eigenvalues of such operators.
{"title":"Semiclassical asymptotics for a class of singular Schrödinger operators","authors":"R. Frank, S. Larson","doi":"10.4171/ecr/18-1/9","DOIUrl":"https://doi.org/10.4171/ecr/18-1/9","url":null,"abstract":"Let $Omega subset mathbb{R}^d$ be bounded with $C^1$ boundary. In this paper we consider Schrodinger operators $-Delta+ W$ on $Omega$ with $W(x)approxmathrm{dist}(x, partialOmega)^{-2}$ as $mathrm{dist}(x, partialOmega)to 0$. Under weak assumptions on $W$ we derive a two-term asymptotic formula for the sum of the eigenvalues of such operators.","PeriodicalId":236439,"journal":{"name":"arXiv: Spectral Theory","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131047578","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-02DOI: 10.1142/S0129055X21500264
D. R. Dolai
In this work we obtain the integrated density of states for the Schrodinger operators with decaying random potentials acting on $ell^2(mathbb{Z}^d)$. We also study the asymptotic of the largest and smallest eigenvalues of its finite volume approximation
{"title":"The IDS and asymptotic of the largest eigenvalue of random Schrödinger operators with decaying random potential","authors":"D. R. Dolai","doi":"10.1142/S0129055X21500264","DOIUrl":"https://doi.org/10.1142/S0129055X21500264","url":null,"abstract":"In this work we obtain the integrated density of states for the Schrodinger operators with decaying random potentials acting on $ell^2(mathbb{Z}^d)$. We also study the asymptotic of the largest and smallest eigenvalues of its finite volume approximation","PeriodicalId":236439,"journal":{"name":"arXiv: Spectral Theory","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132685345","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the so-called first trace formula holds for all Schrodinger operators on the line with real-valued integrable potentials.
我们证明了所谓的第一迹公式对实值可积势直线上的所有薛定谔算子都成立。
{"title":"On the first trace formula for Schrödinger operators","authors":"R. Hryniv, Y. Mykytyuk","doi":"10.4171/JST/348","DOIUrl":"https://doi.org/10.4171/JST/348","url":null,"abstract":"We prove that the so-called first trace formula holds for all Schrodinger operators on the line with real-valued integrable potentials.","PeriodicalId":236439,"journal":{"name":"arXiv: Spectral Theory","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115688983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Spectral inclusion and spectral pollution results are proved for sequences of linear operators of the form $T_0 + i gamma s_n$ on a Hilbert space, where $s_n$ is strongly convergent to the identity operator and $gamma > 0$. We work in both an abstract setting and a more concrete Sturm-Liouville framework. The results provide rigorous justification for a method of computing eigenvalues in spectral gaps.
证明了Hilbert空间上$T_0 + i gamma s_n$形式的线性算子序列的谱包含和谱污染结果，其中$s_n$强收敛于单位算子和$gamma > 0$。我们在一个抽象的环境和一个更具体的Sturm-Liouville框架中工作。结果为谱隙中特征值的计算方法提供了严格的依据。
{"title":"Spectral inclusion and pollution for a class of dissipative perturbations","authors":"A. Stepanenko","doi":"10.1063/5.0028440","DOIUrl":"https://doi.org/10.1063/5.0028440","url":null,"abstract":"Spectral inclusion and spectral pollution results are proved for sequences of linear operators of the form $T_0 + i gamma s_n$ on a Hilbert space, where $s_n$ is strongly convergent to the identity operator and $gamma > 0$. We work in both an abstract setting and a more concrete Sturm-Liouville framework. The results provide rigorous justification for a method of computing eigenvalues in spectral gaps.","PeriodicalId":236439,"journal":{"name":"arXiv: Spectral Theory","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129156215","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the complex absorbing potential (CAP) method for computing scattering resonances applies to the case of exponentially decaying potentials. That means that the eigenvalues of $-Delta + V - iepsilon x^2$, $|V(x)|leq C e^{-2gamma |x|}$ converge, as $ epsilonto 0+ $, to the poles of the meromorphic continuation of $ ( -Delta + V -lambda^2 )^{-1} $ uniformly on compact subsets of $textrm{Re},lambda>0$, $textrm{Im},lambda>-gamma$, $arglambda > -pi/8$.
我们证明了计算散射共振的复吸收势(CAP)方法适用于指数衰减势的情况。这意味着$-Delta + V - iepsilon x^2$, $|V(x)|leq C e^{-2gamma |x|}$的特征值与$ epsilonto 0+ $一样，在$textrm{Re},lambda>0$, $textrm{Im},lambda>-gamma$, $arglambda > -pi/8$的紧子集上一致收敛于$ ( -Delta + V -lambda^2 )^{-1} $的亚纯延拓的极点。
{"title":"Resonances as viscosity limits for exponentially decaying potentials","authors":"Haoren Xiong","doi":"10.1063/5.0016405","DOIUrl":"https://doi.org/10.1063/5.0016405","url":null,"abstract":"We show that the complex absorbing potential (CAP) method for computing scattering resonances applies to the case of exponentially decaying potentials. That means that the eigenvalues of $-Delta + V - iepsilon x^2$, $|V(x)|leq C e^{-2gamma |x|}$ converge, as $ epsilonto 0+ $, to the poles of the meromorphic continuation of $ ( -Delta + V -lambda^2 )^{-1} $ uniformly on compact subsets of $textrm{Re},lambda>0$, $textrm{Im},lambda>-gamma$, $arglambda > -pi/8$.","PeriodicalId":236439,"journal":{"name":"arXiv: Spectral Theory","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126101494","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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