{"title":"Eigenfunctions growth of R-limits on graphs","authors":"Siegfried Beckus, Latif Eliaz","doi":"10.4171/jst/389","DOIUrl":"https://doi.org/10.4171/jst/389","url":null,"abstract":"","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41711288","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 ground state of the semi-relativistic Pauli–Fierz Hamiltonian $$ H = |textbf{p} - textbf{A(x)}| + H_f + Vtextbf{(x)}. $$ Here $textbf{A(x)}$ denotes the quantized radiation field with an ultraviolet cutoff function and $H_f$ the free field Hamiltonian with dispersion relation $|textbf{k}|$. The Hamiltonian $H$ describes the dynamics of a massless and semi-relativistic charged particle interacting with the quantized radiation field with an ultraviolet cutoff function. In 2016, the first two authors proved the existence of the ground state $Phi_m$ of the massive Hamiltonian $H_m$ is proven. Here, the massive Hamiltonian $H_m$ is defined by $H$ with dispersion relation $sqrt{textbf{k}^2+m^2}$ $(m>0)$. In this paper, the existence of the ground state of $H$ is proven. To this aim, we estimate a singular and non-local pull-through formula and show the equicontinuity of ${a(k)Phi_m}_{0
{"title":"Spectrum of the semi-relativistic Pauli–Fierz model II","authors":"Takeru Hidaka, Fumio Hiroshima, Itaru Sasaki","doi":"10.4171/jst/386","DOIUrl":"https://doi.org/10.4171/jst/386","url":null,"abstract":"We consider the ground state of the semi-relativistic Pauli–Fierz Hamiltonian $$ H = |textbf{p} - textbf{A(x)}| + H_f + Vtextbf{(x)}. $$ Here $textbf{A(x)}$ denotes the quantized radiation field with an ultraviolet cutoff function and $H_f$ the free field Hamiltonian with dispersion relation $|textbf{k}|$. The Hamiltonian $H$ describes the dynamics of a <i>massless</i> and semi-relativistic charged particle interacting with the quantized radiation field with an ultraviolet cutoff function. In 2016, the first two authors proved the existence of the ground state $Phi_m$ of the massive Hamiltonian $H_m$ is proven. Here, the massive Hamiltonian $H_m$ is defined by $H$ with dispersion relation $sqrt{textbf{k}^2+m^2}$ $(m>0)$. In this paper, the existence of the ground state of $H$ is proven. To this aim, we estimate a singular and non-local pull-through formula and show the equicontinuity of ${a(k)Phi_m}_{0<m<m_0}$ ${phi_m}_{0<m<m_0}$,=\"\" $a(k)$=\"\" $h$=\"\" $m_0$,=\"\" annihilation=\"\" compactness=\"\" denotes=\"\" existence=\"\" formal=\"\" ground=\"\" is=\"\" kernel=\"\" of=\"\" operator.=\"\" set=\"\" showing=\"\" shown.<=\"\" some=\"\" span=\"\" state=\"\" the=\"\" where=\"\" with=\"\">\u0000</m<m_0}$>","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532060","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":"The anisotropic Calderón problem on 3-dimensional conformally Stäckel manifolds","authors":"Thierry Daudé, N. Kamran, F. Nicoleau","doi":"10.4171/jst/384","DOIUrl":"https://doi.org/10.4171/jst/384","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":"41819919","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":"Erratum to “Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains”","authors":"S. Larson","doi":"10.4171/jst/383","DOIUrl":"https://doi.org/10.4171/jst/383","url":null,"abstract":"","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47574052","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}
Summary: In this work, we study the multiplicity of the singular spectrum for operators of the form A ω = A + ∑ n ω n C n on a separable Hilbert space H , where A is a self-adjoint operator and { C n } n is a countable collection of non-negative finite-rank operators. When { ω n } n are independent real random variables with absolutely continuous distributions, we show that the multiplicity of the singular spectrum is almost surely bounded above by the maximum algebraic multiplicity of the eigenvalues of the operator √ C n ( A ω − z ) − 1 √ C n for all n and almost all ( z, ω ) . The result is optimal in the sense that there are operators for which the bound is achieved. We also provide an effective bound on the multiplicity of the singular spectrum for some special cases.
摘要:本文研究了可分Hilbert空间H上形式为A ω = A +∑n ω n C n的算子的奇异谱的多重性，其中A是自伴随算子，{cn} n是非负有限秩算子的可数集合。当{ω n} n是具有绝对连续分布的独立实随机变量时，我们证明了奇异谱的多重性几乎肯定是由算子√C n (A ω−z)−1√C n对所有n和几乎所有(z， ω)的特征值的最大代数多重性所限定的。这个结果是最优的，因为有一些运算符的界是达到的。对于一些特殊情况，给出了奇异谱多重性的有效界。
{"title":"A theorem on the multiplicity of the singular spectrum of a general Anderson-type Hamiltonian","authors":"D. R. Dolai, Anish Mallick","doi":"10.4171/jst/374","DOIUrl":"https://doi.org/10.4171/jst/374","url":null,"abstract":"Summary: In this work, we study the multiplicity of the singular spectrum for operators of the form A ω = A + ∑ n ω n C n on a separable Hilbert space H , where A is a self-adjoint operator and { C n } n is a countable collection of non-negative finite-rank operators. When { ω n } n are independent real random variables with absolutely continuous distributions, we show that the multiplicity of the singular spectrum is almost surely bounded above by the maximum algebraic multiplicity of the eigenvalues of the operator √ C n ( A ω − z ) − 1 √ C n for all n and almost all ( z, ω ) . The result is optimal in the sense that there are operators for which the bound is achieved. We also provide an effective bound on the multiplicity of the singular spectrum for some special cases.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43871355","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":"Anderson localization for a generalized Maryland model with potentials given by skew shifts","authors":"Jia Shi, Xiaoping Yuan","doi":"10.4171/jst/373","DOIUrl":"https://doi.org/10.4171/jst/373","url":null,"abstract":"","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45961053","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}
J. Behrndt, M. Holzmann, V. Lotoreichik, G. Raikov
We consider the self-adjoint Landau Hamiltonian $H_0$ in $L^2(mathbb{R}^2)$ whose spectrum consists of infinitely degenerate eigenvalues $Lambda_q$, $q in mathbb{Z}_+$, and the perturbed operator $H_upsilon = H_0 + upsilondelta_Gamma$, where $Gamma subset mathbb{R}^2$ is a regular Jordan $C^{1,1}$-curve, and $upsilon in L^p(Gamma;mathbb{R})$, $p>1$, has a constant sign. We investigate ${rm Ker}(H_upsilon -Lambda_q)$, $q in mathbb{Z}_+$, and show that generically $$0 leq {rm dim , Ker}(H_upsilon -Lambda_q) - {rm dim , Ker}(T_q(upsilon delta_Gamma))
{"title":"The fate of Landau levels under $delta$-interactions","authors":"J. Behrndt, M. Holzmann, V. Lotoreichik, G. Raikov","doi":"10.4171/jst/422","DOIUrl":"https://doi.org/10.4171/jst/422","url":null,"abstract":"We consider the self-adjoint Landau Hamiltonian $H_0$ in $L^2(mathbb{R}^2)$ whose spectrum consists of infinitely degenerate eigenvalues $Lambda_q$, $q in mathbb{Z}_+$, and the perturbed operator $H_upsilon = H_0 + upsilondelta_Gamma$, where $Gamma subset mathbb{R}^2$ is a regular Jordan $C^{1,1}$-curve, and $upsilon in L^p(Gamma;mathbb{R})$, $p>1$, has a constant sign. We investigate ${rm Ker}(H_upsilon -Lambda_q)$, $q in mathbb{Z}_+$, and show that generically $$0 leq {rm dim , Ker}(H_upsilon -Lambda_q) - {rm dim , Ker}(T_q(upsilon delta_Gamma))<infty,$$ where $T_q(upsilon delta_Gamma) = p_q (upsilon delta_Gamma)p_q$, is an operator of Berezin-Toeplitz type, acting in $p_q L^2(mathbb{R}^2)$, and $p_q$ is the orthogonal projection on ${rm Ker},(H_0 -Lambda_q)$. If $upsilon neq 0$ and $q = 0$, we prove that ${rm Ker},(T_0(upsilon delta_Gamma)) = {0}$. If $q geq 1$, and $Gamma = mathcal{C}_r$ is a circle of radius $r$, we show that ${rm dim , Ker} (T_q(delta_{mathcal{C}_r})) leq q$, and the set of $r in (0,infty)$ for which ${rm dim , Ker}(T_q(delta_{mathcal{C}_r})) geq 1$, is infinite and discrete.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44674885","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 first formulate an inverse problem for a linear fractional Lam'e system. We determine the Lam'e parameters from exterior partial measurements of the Dirichlet-to-Neumann map. We further study an inverse obstacle problem as well as an inverse problem for a nonlinear fractional Lam'e system. Our arguments are based on the unique continuation property for the fractional operator as well as the associated Runge approximation property.
{"title":"On inverse problems arising in fractional elasticity","authors":"Li Li","doi":"10.4171/jst/428","DOIUrl":"https://doi.org/10.4171/jst/428","url":null,"abstract":"We first formulate an inverse problem for a linear fractional Lam'e system. We determine the Lam'e parameters from exterior partial measurements of the Dirichlet-to-Neumann map. We further study an inverse obstacle problem as well as an inverse problem for a nonlinear fractional Lam'e system. Our arguments are based on the unique continuation property for the fractional operator as well as the associated Runge approximation property.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42972862","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}
Operators with zero dimensional spectral measures appear naturally in the theory of ergodic Schr"odinger operators. We develop the concept of a complete family of Hausdorff measure functions in order to analyze and distinguish between these measures with any desired precision. We prove that the dimension of spectral measures of half-line operators with positive upper Lyapunov exponent are at most logarithmic for every possible boundary phase. We show that this is sharp by constructing an explicit operator whose spectral measure obtains this dimension. We also extend and improve some basic results from the theory of rank one perturbations and quantum dynamics to encompass generalized Hausdorff dimensions.
{"title":"Fine dimensional properties of spectral measures","authors":"M. Landrigan, M. Powell","doi":"10.4171/jst/436","DOIUrl":"https://doi.org/10.4171/jst/436","url":null,"abstract":"Operators with zero dimensional spectral measures appear naturally in the theory of ergodic Schr\"odinger operators. We develop the concept of a complete family of Hausdorff measure functions in order to analyze and distinguish between these measures with any desired precision. We prove that the dimension of spectral measures of half-line operators with positive upper Lyapunov exponent are at most logarithmic for every possible boundary phase. We show that this is sharp by constructing an explicit operator whose spectral measure obtains this dimension. We also extend and improve some basic results from the theory of rank one perturbations and quantum dynamics to encompass generalized Hausdorff dimensions.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47026487","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：