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Complete asymptotic expansions of the spectral function for symbolic perturbations of almost periodic Schrödinger operators in dimension one 一维概周期Schrödinger算子符号扰动谱函数的完全渐近展开
IF 1 3区 数学
Journal of Spectral Theory Pub Date : 2020-11-18 DOI: 10.4171/jst/396
J. Galkowski
{"title":"Complete asymptotic expansions of the spectral function for symbolic perturbations of almost periodic Schrödinger operators in dimension one","authors":"J. Galkowski","doi":"10.4171/jst/396","DOIUrl":"https://doi.org/10.4171/jst/396","url":null,"abstract":"In this article we consider asymptotics for the spectral function of Schrodinger operators on the real line. Let $P:L^2(mathbb{R})to L^2(mathbb{R})$ have the form $$ P:=-tfrac{d^2}{dx^2}+W, $$ where $W$ is a self-adjoint first order differential operator with certain modified almost periodic structure. We show that the kernel of the spectral projector, $mathbb{1}_{(-infty,lambda^2]}(P)$ has a full asymptotic expansion in powers of $lambda$. In particular, our class of potentials $W$ is stable under perturbation by formally self-adjoint first order differential operators with smooth, compactly supported coefficients. Moreover, it includes certain potentials with dense pure point spectrum. The proof combines the gauge transform methods of Parnovski-Shterenberg and Sobolev with Melrose's scattering calculus.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42173205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Construction of quasimodes for non-selfadjoint operators via propagation of Hagedorn wave-packets 通过Hagedorn波包的传播构造非自伴算子的拟模
IF 1 3区 数学
Journal of Spectral Theory Pub Date : 2020-10-28 DOI: 10.4171/jst/418
V'ictor Arnaiz
{"title":"Construction of quasimodes for non-selfadjoint operators via propagation of Hagedorn wave-packets","authors":"V'ictor Arnaiz","doi":"10.4171/jst/418","DOIUrl":"https://doi.org/10.4171/jst/418","url":null,"abstract":"We construct quasimodes for non-selfadjoint semiclassical operators using propagation of Hagedorn wave-packets. Assuming that the imaginary part of the principal symbol of the operator is non-negative and vanishes on certain points of the phase-space satisfying a finite-type dynamical condition, we construct quasimodes that concentrate on these non-damped points. More generally, we apply this technique to construct quasimodes for non-selfadjoint semiclassical perturbations of the harmonic oscillator that concentrate on periodic orbits or invariant tori.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44589594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
A sharp isoperimetric inequality for the second eigenvalue of the Robin plate 罗宾平板第二特征值的尖锐等周不等式
IF 1 3区 数学
Journal of Spectral Theory Pub Date : 2020-10-20 DOI: 10.4171/jst/413
L. Chasman, J. Langford
{"title":"A sharp isoperimetric inequality for the second eigenvalue of the Robin plate","authors":"L. Chasman, J. Langford","doi":"10.4171/jst/413","DOIUrl":"https://doi.org/10.4171/jst/413","url":null,"abstract":"Among all $C^{infty}$ bounded domains with equal volume, we show that the second eigenvalue of the Robin plate is uniquely maximized by an open ball, so long as the Robin parameter lies within a particular range of negative values. Our methodology combines recent techniques introduced by Freitas and Laugesen to study the second eigenvalue of the Robin membrane problem and techniques employed by Chasman to study the free plate problem. In particular, we choose eigenfunctions of the ball as trial functions in the Rayleigh quotient for a general domain; such eigenfunctions are comprised of ultraspherical Bessel and modified Bessel functions. Much of our work hinges on developing an understanding of delicate properties of these special functions, which may be of independent interest.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42176224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Spectral fluctuations for the multi-dimensional Anderson model 多维Anderson模型的谱涨落
IF 1 3区 数学
Journal of Spectral Theory Pub Date : 2020-10-18 DOI: 10.4171/jst/412
Yoel Grinshpon, Moshe J. White
{"title":"Spectral fluctuations for the multi-dimensional Anderson model","authors":"Yoel Grinshpon, Moshe J. White","doi":"10.4171/jst/412","DOIUrl":"https://doi.org/10.4171/jst/412","url":null,"abstract":"In this paper, we examine fluctuations of polynomial linear statistics for the Anderson model on $mathbb{Z}^d$ for any potential with finite moments. We prove that if normalized by the square root of the size of the truncated operator, these fluctuations converge to a Gaussian limit. For a vast majority of potentials and polynomials, we show that the variance of the limiting distribution is strictly positive, and we classify in full the rare cases in which this does not happen.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44019959","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the spectrum of the periodic focusing Zakharov–Shabat operator 关于周期性聚焦Zakharov-Shabat算子的频谱
IF 1 3区 数学
Journal of Spectral Theory Pub Date : 2020-10-08 DOI: 10.4171/JST/432
G. Biondini, Jeffrey Oregero, A. Tovbis
{"title":"On the spectrum of the periodic focusing Zakharov–Shabat operator","authors":"G. Biondini, Jeffrey Oregero, A. Tovbis","doi":"10.4171/JST/432","DOIUrl":"https://doi.org/10.4171/JST/432","url":null,"abstract":"The spectrum of the focusing Zakharov-Shabat operator on the circle is studied, and its explicit dependence on the presence of a semiclassical parameter is also considered. Several new results are obtained. In particular: (i) it is proved that the resolvent set is comprised of two connected components, (ii) new bounds on the location of the Floquet and Dirichlet spectra are obtained, some of which depend explicitly on the value of the semiclassical parameter, (iii) it is proved that the spectrum localizes to a\"cross\"in the spectral plane in the semiclassical limit. The results are illustrated by discussing several examples in which the spectrum is computed analytically or numerically.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48255612","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Zero measure spectrum for multi-frequency Schrödinger operators 零测量频谱多频率Schrödinger运营商
IF 1 3区 数学
Journal of Spectral Theory Pub Date : 2020-09-24 DOI: 10.4171/jst/411
J. Chaika, D. Damanik, J. Fillman, P. Gohlke
{"title":"Zero measure spectrum for multi-frequency Schrödinger operators","authors":"J. Chaika, D. Damanik, J. Fillman, P. Gohlke","doi":"10.4171/jst/411","DOIUrl":"https://doi.org/10.4171/jst/411","url":null,"abstract":"Building on works of Berthe--Steiner--Thuswaldner and Fogg--Nous we show that on the two-dimensional torus, Lebesgue almost every translation admits a natural coding such that the associated subshift satisfies the Boshernitzan criterion. As a consequence we show that for these torus translations, every quasi-periodic potential can be approximated uniformly by one for which the associated Schrodinger operator has Cantor spectrum of zero Lebesgue measure. We also describe a framework that can allow this to be extended to higher-dimensional tori.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48050638","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Semiclassical Gevrey operators and magnetic translations 半经典格夫里算子和磁平移
IF 1 3区 数学
Journal of Spectral Theory Pub Date : 2020-09-19 DOI: 10.4171/jst/394
M. Hitrik, R. Lascar, J. Sjoestrand, Maher Zerzeri
{"title":"Semiclassical Gevrey operators and magnetic translations","authors":"M. Hitrik, R. Lascar, J. Sjoestrand, Maher Zerzeri","doi":"10.4171/jst/394","DOIUrl":"https://doi.org/10.4171/jst/394","url":null,"abstract":"We study semiclassical Gevrey pseudodifferential operators acting on the Bargmann space of entire functions with quadratic exponential weights. Using some ideas of the time frequency analysis, we show that such operators are uniformly bounded on a natural scale of exponentially weighted spaces of holomorphic functions, provided that the Gevrey index is $geq 2$.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42936045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
A proof of the triangular Ashbaugh–Benguria–Payne–Pólya–Weinberger inequality 一个三角形Ashbaugh-Benguria-Payne-Pólya-Weinberger不等式的证明
IF 1 3区 数学
Journal of Spectral Theory Pub Date : 2020-09-02 DOI: 10.4171/JST/409
R. Arbon, Mohammed Mannan, Michael Psenka, Seyoon Ragavan
{"title":"A proof of the triangular Ashbaugh–Benguria–Payne–Pólya–Weinberger inequality","authors":"R. Arbon, Mohammed Mannan, Michael Psenka, Seyoon Ragavan","doi":"10.4171/JST/409","DOIUrl":"https://doi.org/10.4171/JST/409","url":null,"abstract":"In this paper, we show that for all triangles in the plane, the equilateral triangle maximizes the ratio of the first two Dirichlet-Laplacian eigenvalues. This is an extension of work by Siudeja, who proved the inequality in the case of acute triangles. The proof utilizes inequalities due to Siudeja and Freitas, together with improved variational bounds.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47049413","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
The density of states and local eigenvalue statistics for random band matrices of fixed width 固定宽度随机带矩阵的态密度和局部特征值统计
IF 1 3区 数学
Journal of Spectral Theory Pub Date : 2020-08-30 DOI: 10.4171/jst/405
Benjamin C. Brodie, P. Hislop
{"title":"The density of states and local eigenvalue statistics for random band matrices of fixed width","authors":"Benjamin C. Brodie, P. Hislop","doi":"10.4171/jst/405","DOIUrl":"https://doi.org/10.4171/jst/405","url":null,"abstract":"We prove that the local eigenvalue statistics for $d=1$ random band matrices with fixed bandwidth and, for example, Gaussian entries, is given by a Poisson point process and we identify the intensity of the process. The proof relies on an extension of the localization bounds of Schenker cite{schenker} and the Wegner and Minami estimates. These two estimates are proved using averaging over the diagonal disorder. The new component is a proof of the uniform convergence and the smoothness of the density of states function. The limit function, known to be the semicircle law with a band-width dependent error cite{bmp,dps,dl,mpk}, is identified as the intensity of the limiting Poisson point process. The proof of these results for the density of states relies on a new result that simplifies and extends some of the ideas used by Dolai, Krishna, and Mallick cite{dkm}. These authors proved regularity properties of the density of states for random Schrodinger operators (lattice and continuum) in the localization regime. The proof presented here applies to the random Schrodinger operators on a class of infinite graphs treated by in cite{dkm} and extends the results of cite{dkm} to probability measures with unbounded support. The method also applies to fixed bandwidth RBM for $d=2,3$ provided certain localization bounds are known.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":"15 17","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41279439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Eigenvalue estimates for the one-particle density matrix 单粒子密度矩阵的特征值估计
IF 1 3区 数学
Journal of Spectral Theory Pub Date : 2020-08-25 DOI: 10.4171/jst/407
A. Sobolev
{"title":"Eigenvalue estimates for the one-particle density matrix","authors":"A. Sobolev","doi":"10.4171/jst/407","DOIUrl":"https://doi.org/10.4171/jst/407","url":null,"abstract":"It is shown that the eigenvalues $lambda_k, k=1, 2, dots,$ of the one-particle density matrix satisfy the bound $lambda_kle C k^{-8/3}$ with a positive constant $C$.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41327297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
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