We consider two-sided Jacobi matrices whose coefficients are obtained by continuous sampling along the orbits of a homeomorphim of a compact metric space. Given an ergodic probability measure, we study the topological structure of the associated almost sure spectrum. We establish a gap labelling theorem in the spirit of Johnson and Schwartzman. That is, we show that the constant value the integrated density of states takes in a gap of the spectrum must belong to the countable Schwartzman group of the base dynamics. This result is a natural companion to a recent result of Alkorn and Zhang, which established a Johnson-type theorem for the families of Jacobi matrices in question.
{"title":"Johnson–Schwartzman gap labelling for ergodic Jacobi matrices","authors":"D. Damanik, J. Fillman, Zhenghe Zhang","doi":"10.4171/jst/449","DOIUrl":"https://doi.org/10.4171/jst/449","url":null,"abstract":"We consider two-sided Jacobi matrices whose coefficients are obtained by continuous sampling along the orbits of a homeomorphim of a compact metric space. Given an ergodic probability measure, we study the topological structure of the associated almost sure spectrum. We establish a gap labelling theorem in the spirit of Johnson and Schwartzman. That is, we show that the constant value the integrated density of states takes in a gap of the spectrum must belong to the countable Schwartzman group of the base dynamics. This result is a natural companion to a recent result of Alkorn and Zhang, which established a Johnson-type theorem for the families of Jacobi matrices in question.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42091875","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}
In this paper, we study eigenvalue fluctuations of the finite volume Anderson model in the mesoscopic scale. We carry out this study in a regime of exponential localization and prove a central limit theorem for the eigenvalue counting function in a shrinking interval.
{"title":"Limit theorems on the mesoscopic scale for the Anderson model","authors":"Yoel Grinshpon","doi":"10.4171/jst/456","DOIUrl":"https://doi.org/10.4171/jst/456","url":null,"abstract":"In this paper, we study eigenvalue fluctuations of the finite volume Anderson model in the mesoscopic scale. We carry out this study in a regime of exponential localization and prove a central limit theorem for the eigenvalue counting function in a shrinking interval.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42316613","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}
We study ergodic Schr"odinger operators defined over product dynamical systems in which one factor is periodic and the other factor is either a subshift over a finite alphabet or an irrational rotation of the circle. In the case in which one factor is a Boshernitzan subshift, we prove that either the resulting operators are periodic or the resulting spectra must be Cantor sets. The main ingredient is a suitable stability result for Boshernitzan's criterion under taking products. We also discuss the stability of purely singular continuous spectrum, which, given the zero-measure spectrum result, amounts to stability results for eigenvalue exclusion. In particular, we examine situations in which the existing criteria for the exclusion of eigenvalues are stable under periodic perturbations. As a highlight of this, we show that any simple Toeplitz subshift over a binary alphabet exhibits uniform absence of eigenvalues on the hull for any periodic perturbation whose period is commensurate with the coding sequence. In the case of a full shift, we give an effective criterion to compute exactly the spectrum of a random Anderson model perturbed by a potential of period two, and we further show that the naive generalization of this criterion does not hold for period three. Next, we consider quasi-periodic potentials with potentials generated by trigonometric polynomials with periodic background. We show that the quasiperiodic cocycle induced by passing to blocks of period length is subcritical when the coupling constant is small and supercritical when the coupling constant is large. Thus, the spectral type is absolutely continuous for small coupling and pure point (for a.e. frequency and phase) when the coupling is large.
{"title":"Spectral characteristics of Schrödinger operators generated by product systems","authors":"D. Damanik, J. Fillman, P. Gohlke","doi":"10.4171/jst/445","DOIUrl":"https://doi.org/10.4171/jst/445","url":null,"abstract":"We study ergodic Schr\"odinger operators defined over product dynamical systems in which one factor is periodic and the other factor is either a subshift over a finite alphabet or an irrational rotation of the circle. In the case in which one factor is a Boshernitzan subshift, we prove that either the resulting operators are periodic or the resulting spectra must be Cantor sets. The main ingredient is a suitable stability result for Boshernitzan's criterion under taking products. We also discuss the stability of purely singular continuous spectrum, which, given the zero-measure spectrum result, amounts to stability results for eigenvalue exclusion. In particular, we examine situations in which the existing criteria for the exclusion of eigenvalues are stable under periodic perturbations. As a highlight of this, we show that any simple Toeplitz subshift over a binary alphabet exhibits uniform absence of eigenvalues on the hull for any periodic perturbation whose period is commensurate with the coding sequence. In the case of a full shift, we give an effective criterion to compute exactly the spectrum of a random Anderson model perturbed by a potential of period two, and we further show that the naive generalization of this criterion does not hold for period three. Next, we consider quasi-periodic potentials with potentials generated by trigonometric polynomials with periodic background. We show that the quasiperiodic cocycle induced by passing to blocks of period length is subcritical when the coupling constant is small and supercritical when the coupling constant is large. Thus, the spectral type is absolutely continuous for small coupling and pure point (for a.e. frequency and phase) when the coupling is large.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49339054","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}
Let $omegasubsetmathbb{R}^n$ be a bounded domain with Lipschitz boundary. For $varepsilon>0$ and $ninmathbb{N}$ consider the infinite cone $Omega_{varepsilon}:=big{(x_1,x')in (0,infty)timesmathbb{R}^n: x'invarepsilon x_1omegabig}subsetmathbb{R}^{n+1}$ and the operator $Q_{varepsilon}^{alpha}$ acting as the Laplacian $umapsto-Delta u$ on $Omega_{varepsilon}$ with the Robin boundary condition $partial_nu u=alpha u$ at $partialOmega_varepsilon$, where $partial_nu$ is the outward normal derivative and $alpha>0$. We look at the dependence of the eigenvalues of $Q_varepsilon^alpha$ on the parameter $varepsilon$: this problem was previously addressed for $n=1$ only (in that case, the only admissible $omega$ are finite intervals). In the present work we consider arbitrary dimensions $nge2$ and arbitrarily shaped"cross-sections"$omega$ and look at the spectral asymptotics as $varepsilon$ becomes small, i.e. as the cone becomes"sharp"and collapses to a half-line. It turns out that the main term of the asymptotics of individual eigenvalues is determined by the single geometric quantity $N_omega:=dfrac{mathrm{Vol}_{n-1} partialomega }{mathrm{Vol}_n omega}$. More precisely, for any fixed $jin mathbb{N}$ and $alpha>0$ the $j$th eigenvalue $E_j(Q^alpha_varepsilon)$ of $Q^alpha_varepsilon$ exists for all sufficiently small $varepsilon>0$ and satisfies $E_j(Q^alpha_varepsilon)=-dfrac{N_omega^2,alpha^2}{(2j+n-2)^2,varepsilon^2}+Oleft(dfrac{1}{varepsilon}right)$ as $varepsilonto 0^+$. The paper also covers some aspects of Sobolev spaces on infinite cones, which can be of independent interest.
{"title":"Asymptotics of Robin eigenvalues on sharp infinite cones","authors":"Konstantin Pankrashkin, Marco Vogel","doi":"10.4171/JST/452","DOIUrl":"https://doi.org/10.4171/JST/452","url":null,"abstract":"Let $omegasubsetmathbb{R}^n$ be a bounded domain with Lipschitz boundary. For $varepsilon>0$ and $ninmathbb{N}$ consider the infinite cone $Omega_{varepsilon}:=big{(x_1,x')in (0,infty)timesmathbb{R}^n: x'invarepsilon x_1omegabig}subsetmathbb{R}^{n+1}$ and the operator $Q_{varepsilon}^{alpha}$ acting as the Laplacian $umapsto-Delta u$ on $Omega_{varepsilon}$ with the Robin boundary condition $partial_nu u=alpha u$ at $partialOmega_varepsilon$, where $partial_nu$ is the outward normal derivative and $alpha>0$. We look at the dependence of the eigenvalues of $Q_varepsilon^alpha$ on the parameter $varepsilon$: this problem was previously addressed for $n=1$ only (in that case, the only admissible $omega$ are finite intervals). In the present work we consider arbitrary dimensions $nge2$ and arbitrarily shaped\"cross-sections\"$omega$ and look at the spectral asymptotics as $varepsilon$ becomes small, i.e. as the cone becomes\"sharp\"and collapses to a half-line. It turns out that the main term of the asymptotics of individual eigenvalues is determined by the single geometric quantity $N_omega:=dfrac{mathrm{Vol}_{n-1} partialomega }{mathrm{Vol}_n omega}$. More precisely, for any fixed $jin mathbb{N}$ and $alpha>0$ the $j$th eigenvalue $E_j(Q^alpha_varepsilon)$ of $Q^alpha_varepsilon$ exists for all sufficiently small $varepsilon>0$ and satisfies $E_j(Q^alpha_varepsilon)=-dfrac{N_omega^2,alpha^2}{(2j+n-2)^2,varepsilon^2}+Oleft(dfrac{1}{varepsilon}right)$ as $varepsilonto 0^+$. The paper also covers some aspects of Sobolev spaces on infinite cones, which can be of independent interest.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46653152","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}
We generalize many recent uniqueness results on the fractional Calder'on problem to cover the cases of all domains with nonempty exterior. The highlight of our work is the characterization of uniqueness and nonuniqueness of partial data inverse problems for the fractional conductivity equation on domains that are bounded in one direction for conductivities supported in the whole Euclidean space and decaying to a constant background conductivity at infinity. We generalize the uniqueness proof for the fractional Calder'on problem by Ghosh, Salo and Uhlmann to a general abstract setting in order to use the full strength of their argument. This allows us to observe that there are also uniqueness results for many inverse problems for higher order local perturbations of a lower order fractional Laplacian. We give concrete example models to illustrate these curious situations and prove Poincar'e inequalities for the fractional Laplacians of any order on domains that are bounded in one direction. We establish Runge approximation results in these general settings, improve regularity assumptions also in the cases of bounded sets and prove general exterior determination results. Counterexamples to uniqueness in the inverse fractional conductivity problem with partial data are constructed in another companion work.
{"title":"Fractional Calderón problems and Poincaré inequalities on unbounded domains","authors":"J. Railo, Philipp Zimmermann","doi":"10.4171/jst/444","DOIUrl":"https://doi.org/10.4171/jst/444","url":null,"abstract":"We generalize many recent uniqueness results on the fractional Calder'on problem to cover the cases of all domains with nonempty exterior. The highlight of our work is the characterization of uniqueness and nonuniqueness of partial data inverse problems for the fractional conductivity equation on domains that are bounded in one direction for conductivities supported in the whole Euclidean space and decaying to a constant background conductivity at infinity. We generalize the uniqueness proof for the fractional Calder'on problem by Ghosh, Salo and Uhlmann to a general abstract setting in order to use the full strength of their argument. This allows us to observe that there are also uniqueness results for many inverse problems for higher order local perturbations of a lower order fractional Laplacian. We give concrete example models to illustrate these curious situations and prove Poincar'e inequalities for the fractional Laplacians of any order on domains that are bounded in one direction. We establish Runge approximation results in these general settings, improve regularity assumptions also in the cases of bounded sets and prove general exterior determination results. Counterexamples to uniqueness in the inverse fractional conductivity problem with partial data are constructed in another companion work.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45629369","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}
We investigate the spectrum of 2-dimensional canonical systems in the limit circle case. It is discrete and, by the Krein-de Branges formula, cannot be more dense than the integers. But in many cases it will be more sparse. The spectrum of a particular selfadjoint realisation coincides with the zeroes of one entry of the monodromy matrix of the system. Classical function theory thus establishes an immediate connection between the growth of the monodromy matrix and the distribution of the spectrum. We prove a generic and flexibel upper estimate for the monodromy matrix, use it to prove a bound for the case of a continuous Hamiltonian, and construct examples which show that this bound is sharp. The first two results run along the lines of earlier work of R.Romanov, but significantly improve upon these results. This is seen even on the rough scale of exponential order.
{"title":"A growth estimate for the monodromy matrix of a canonical system","authors":"R. Pruckner, H. Woracek","doi":"10.4171/jst/437","DOIUrl":"https://doi.org/10.4171/jst/437","url":null,"abstract":"We investigate the spectrum of 2-dimensional canonical systems in the limit circle case. It is discrete and, by the Krein-de Branges formula, cannot be more dense than the integers. But in many cases it will be more sparse. The spectrum of a particular selfadjoint realisation coincides with the zeroes of one entry of the monodromy matrix of the system. Classical function theory thus establishes an immediate connection between the growth of the monodromy matrix and the distribution of the spectrum. We prove a generic and flexibel upper estimate for the monodromy matrix, use it to prove a bound for the case of a continuous Hamiltonian, and construct examples which show that this bound is sharp. The first two results run along the lines of earlier work of R.Romanov, but significantly improve upon these results. This is seen even on the rough scale of exponential order.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44789327","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}
We find Weyl upper bounds for the quantum open baker's map in the semiclassical limit. For the number of eigenvalues in an annulus, we derive the asymptotic upper bound $mathcal O(N^delta)$ where $delta$ is the dimension of the trapped set of the baker's map and $(2 pi N)^{-1}$ is the semiclassical parameter, which improves upon the previous result of $mathcal O(N^{delta + epsilon})$. Furthermore, we derive a Weyl upper bound with explicit dependence on the inner radius of the annulus for quantum open baker's maps with Gevrey cutoffs.
我们找到了半经典极限下量子开贝克映射的Weyl上界。对于环空中特征值的数目，我们推导出了渐近上界$mathcal O(N^delta)$，其中$delta$为贝克映射的捕获集的维数，$(2 pi N)^{-1}$为半经典参数，改进了之前$mathcal O(N^{delta + epsilon})$的结果。进一步，我们导出了具有gevery截止点的量子开放baker映射的Weyl上界，该上界与环的内半径有显式的依赖关系。
{"title":"Weyl laws for open quantum maps","authors":"Zhen-Hu Li","doi":"10.4171/jst/441","DOIUrl":"https://doi.org/10.4171/jst/441","url":null,"abstract":"We find Weyl upper bounds for the quantum open baker's map in the semiclassical limit. For the number of eigenvalues in an annulus, we derive the asymptotic upper bound $mathcal O(N^delta)$ where $delta$ is the dimension of the trapped set of the baker's map and $(2 pi N)^{-1}$ is the semiclassical parameter, which improves upon the previous result of $mathcal O(N^{delta + epsilon})$. Furthermore, we derive a Weyl upper bound with explicit dependence on the inner radius of the annulus for quantum open baker's maps with Gevrey cutoffs.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48720145","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}
We consider the linear BCS equation, determining the BCS critical temperature, in the presence of a boundary, where Dirichlet boundary conditions are imposed. In the one-dimensional case with point interactions, we prove that the critical temperature is strictly larger than the bulk value, at least at weak coupling. In particular, the Cooper-pair wave function localizes near the boundary, an effect that cannot be modeled by effective Neumann boundary conditions on the order parameter as often imposed in Ginzburg-Landau theory. We also show that the relative shift in critical temperature vanishes if the coupling constant either goes to zero or to infinity.
{"title":"Boundary superconductivity in the BCS Model","authors":"C. Hainzl, B. Roos, R. Seiringer","doi":"10.4171/jst/439","DOIUrl":"https://doi.org/10.4171/jst/439","url":null,"abstract":"We consider the linear BCS equation, determining the BCS critical temperature, in the presence of a boundary, where Dirichlet boundary conditions are imposed. In the one-dimensional case with point interactions, we prove that the critical temperature is strictly larger than the bulk value, at least at weak coupling. In particular, the Cooper-pair wave function localizes near the boundary, an effect that cannot be modeled by effective Neumann boundary conditions on the order parameter as often imposed in Ginzburg-Landau theory. We also show that the relative shift in critical temperature vanishes if the coupling constant either goes to zero or to infinity.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44362843","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}
{"title":"A Faber–Krahn inequality for the Riesz potential operator for triangles and quadrilaterals","authors":"R. Mahadevan, Franco Olivares-Contador","doi":"10.4171/jst/390","DOIUrl":"https://doi.org/10.4171/jst/390","url":null,"abstract":"","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44758662","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}
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