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Upper eigenvalue bounds for the Kirchhoff Laplacian on embedded metric graphs 嵌入度量图上Kirchhoff拉普拉斯算子的上特征值界
IF 1 3区 数学
Journal of Spectral Theory Pub Date : 2020-04-07 DOI: 10.4171/jst/388
Marvin Plumer
{"title":"Upper eigenvalue bounds for the Kirchhoff Laplacian on embedded metric graphs","authors":"Marvin Plumer","doi":"10.4171/jst/388","DOIUrl":"https://doi.org/10.4171/jst/388","url":null,"abstract":"We derive upper bounds for the eigenvalues of the Kirchhoff Laplacian on a compact metric graph depending on the graph's genus g. These bounds can be further improved if $g = 0$, i.e. if the metric graph is planar. Our results are based on a spectral correspondence between the Kirchhoff Laplacian and a particular a certain combinatorial weighted Laplacian. In order to take advantage of this correspondence, we also prove new estimates for the eigenvalues of the weighted combinatorial Laplacians that were previously known only in the weighted case.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43688693","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: 7
Spectral invariants of Dirichlet-to-Neumann operators on surfaces 曲面上Dirichlet到Neumann算子的谱不变量
IF 1 3区 数学
Journal of Spectral Theory Pub Date : 2020-03-04 DOI: 10.4171/jst/382
J. Lagac'e, Simon St-Amant
{"title":"Spectral invariants of Dirichlet-to-Neumann operators on surfaces","authors":"J. Lagac'e, Simon St-Amant","doi":"10.4171/jst/382","DOIUrl":"https://doi.org/10.4171/jst/382","url":null,"abstract":"We obtain a complete asymptotic expansion for the eigenvalues of the Dirichlet-to-Neumann maps associated with Schrodinger operators on Riemannian surfaces with boundary. For the zero potential, we recover the well-known spectral asymptotics for the Steklov problem. For nonzero porentials, we obtain new geometric invariants determined by the spectrum. In particular, for constant potentials, which give rise to the parameter-dependent Steklov problem, the total geodesic curvature on each connected component of the boundary is a spectral invariant. Under the constant curvature assumption, this allows us to obtain some interior information from the spectrum of these boundary operators.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42490012","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
Spectral theory of the thermal Hamiltonian: 1D case 热哈密顿量的谱理论:一维情况
IF 1 3区 数学
Journal of Spectral Theory Pub Date : 2020-02-24 DOI: 10.4171/jst/376
G. Nittis, Vicente Lenz
{"title":"Spectral theory of the thermal Hamiltonian: 1D case","authors":"G. Nittis, Vicente Lenz","doi":"10.4171/jst/376","DOIUrl":"https://doi.org/10.4171/jst/376","url":null,"abstract":"In 1964 J. M. Luttinger introduced a model for the quantum thermal transport. In this paper we study the spectral theory of the Hamiltonian operator associated to the Luttinger's model, with a special focus at the one-dimensional case. It is shown that the (so called) thermal Hamiltonian has a one-parameter family of self-adjoint extensions and the spectrum, the time-propagator group and the Green function are explicitly computed. Moreover, the scattering by convolution-type potentials is analyzed. Finally, also the associated classical problem is completely solved, thus providing a comparison between classical and quantum behavior. This article aims to be a first contribution in the construction of a complete theory for the thermal Hamiltonian.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42968507","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
Disjointness-preserving operators and isospectral Laplacians 不相容算子与等谱拉普拉斯算子
IF 1 3区 数学
Journal of Spectral Theory Pub Date : 2020-02-21 DOI: 10.4171/jst/379
W. Arendt, J. Kennedy
{"title":"Disjointness-preserving operators and isospectral Laplacians","authors":"W. Arendt, J. Kennedy","doi":"10.4171/jst/379","DOIUrl":"https://doi.org/10.4171/jst/379","url":null,"abstract":"All the known counterexamples to Kac' famous question \"can one hear the shape of a drum\", i.e., does isospectrality of two Laplacians on domains imply that the domains are congruent, consist of pairs of domains composed of copies of isometric building blocks arranged in different ways, such that the unitary operator intertwining the Laplacians acts as a sum of overlapping \"local\" isometries mapping the copies to each other. \u0000We prove and explore a complementary positive statement: if an operator intertwining two appropriate realisations of the Laplacian on a pair of domains preserves disjoint supports, then under additional assumptions on it generally far weaker than unitarity, the domains are congruent. We show this in particular for the Dirichlet, Neumann and Robin Laplacians on spaces of continuous functions and on $L^2$-spaces.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46220657","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
Carleman estimate for complex second order elliptic operators with discontinuous Lipschitz coefficients 具有不连续Lipschitz系数的复二阶椭圆算子的Carleman估计
IF 1 3区 数学
Journal of Spectral Theory Pub Date : 2020-01-13 DOI: 10.4171/jst/410
E. Francini, S. Vessella, J.-N. Wang
{"title":"Carleman estimate for complex second order elliptic operators with discontinuous Lipschitz coefficients","authors":"E. Francini, S. Vessella, J.-N. Wang","doi":"10.4171/jst/410","DOIUrl":"https://doi.org/10.4171/jst/410","url":null,"abstract":"In this paper, we derive a local Carleman estimate for the complex second order elliptic operator with Lipschitz coefficients having jump discontinuities. Combing the result in [BL] and the arguments in [DcFLVW], we present an elementary method to derive the Carleman estimate under the optimal regularity assumption on the coefficients.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47544291","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
Improved sharp spectral inequalities for Schrödinger operators on the semi-axis 改进了半轴上Schrödinger算子的尖锐谱不等式
IF 1 3区 数学
Journal of Spectral Theory Pub Date : 2019-12-31 DOI: 10.4171/jst/434
L. Schimmer
{"title":"Improved sharp spectral inequalities for Schrödinger operators on the semi-axis","authors":"L. Schimmer","doi":"10.4171/jst/434","DOIUrl":"https://doi.org/10.4171/jst/434","url":null,"abstract":"We prove a Lieb--Thirring inequality for Schr\"odinger operators on the semi-axis with Robin boundary condition at the origin. The result improves on a bound obtained by P. Exner, A. Laptev and M. Usman [Commun. Math. Phys. 362(2), 531--541 (2014)]. The main difference in our proof is that we use the double commutation method in place of the single commutation method. We also establish an improved inequality in the case of a Dirichlet boundary condition.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45277999","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
The $L^p$ boundedness of the wave operators for matrix Schrödinger equations 矩阵Schrödinger方程波动算子的L^p有界性
IF 1 3区 数学
Journal of Spectral Theory Pub Date : 2019-12-30 DOI: 10.4171/jst/417
R. Weder
{"title":"The $L^p$ boundedness of the wave operators for matrix Schrödinger equations","authors":"R. Weder","doi":"10.4171/jst/417","DOIUrl":"https://doi.org/10.4171/jst/417","url":null,"abstract":"We prove that the wave operators for $n times n$ matrix Schrodinger equations on the half line, with general selfadjoint boundary condition, are bounded in the spaces $L^p(mathbb R^+, mathbb C^n), 1 frac{5}{2},$ and that the scattering matrix is the identity at zero and infinite energy, we prove that the wave operators are bounded in $L^1(mathbb R^+, mathbb C^n),$ and in $L^infty(mathbb R^+, mathbb C^n).$ We also prove that the wave operators for $ntimes n$ matrix Schrodinger equations on the line are bounded in the spaces $L^p(mathbb R, mathbb C^n), 1 frac{5}{2},$ and that the scattering matrix is the identity at zero and infinite energy, we prove that the wave operators are bounded in $L^1(mathbb R, mathbb C^n),$ and in $L^infty(mathbb R, mathbb C^n).$ We obtain our results for $ntimes n$ matrix Schrodinger equations on the line from the results for $2ntimes 2n$ matrix Schrodinger equations on the half line.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2019-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45646548","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
The BFK-gluing formula and the curvature tensors on a 2-dimensional compact hypersurface 二维紧致超曲面上的bfk -胶合公式与曲率张量
IF 1 3区 数学
Journal of Spectral Theory Pub Date : 2019-12-24 DOI: 10.4171/jst/320
K. Kirsten, Yoonweon Lee
{"title":"The BFK-gluing formula and the curvature tensors on a 2-dimensional compact hypersurface","authors":"K. Kirsten, Yoonweon Lee","doi":"10.4171/jst/320","DOIUrl":"https://doi.org/10.4171/jst/320","url":null,"abstract":"In the proof of the BFK-gluing formula for zeta-determinants of Laplacians there appears a real polynomial whose constant term is an important ingredient in the gluing formula. This polynomial is determined by geometric data on an arbitrarily small collar neighborhood of a cutting hypersurface. In this paper we express the coefficients of this polynomial in terms of the scalar and principal curvatures of the cutting hypersurface embedded in the manifold when this hypersurface is 2-dimensional. Similarly, we express some coefficients of the heat trace asymptotics of the Dirichlet-to-Neumann operator in terms of the scalar and principal curvatures of the cutting hypersurface.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2019-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42265496","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
Quantum evolution and sub-Laplacian operators on groups of Heisenberg type 海森堡型群上的量子演化和次拉普拉斯算子
IF 1 3区 数学
Journal of Spectral Theory Pub Date : 2019-10-31 DOI: 10.4171/jst/375
C. Fermanian-Kammerer, Veronique Fischer
{"title":"Quantum evolution and sub-Laplacian operators on groups of Heisenberg type","authors":"C. Fermanian-Kammerer, Veronique Fischer","doi":"10.4171/jst/375","DOIUrl":"https://doi.org/10.4171/jst/375","url":null,"abstract":"In this paper we analyze the evolution of the time averaged energy densities associated with a family of solutions to a Schrodinger equation on a Lie group of Heisenberg type. We use a semi-classical approach adapted to the stratified structure of the group and describe the semi-classical measures (also called quantum limits) that are associated with this family. This allows us to prove an Egorov's type Theorem describing the quantum evolution of a pseudodifferential semi-classical operator through the semi-group generated by a sub-Laplacian.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2019-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44955325","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: 11
The wave trace and Birkhoff billiards 波迹和伯克霍夫台球
IF 1 3区 数学
Journal of Spectral Theory Pub Date : 2019-10-14 DOI: 10.4171/jst/440
Amir Vig
{"title":"The wave trace and Birkhoff billiards","authors":"Amir Vig","doi":"10.4171/jst/440","DOIUrl":"https://doi.org/10.4171/jst/440","url":null,"abstract":"The purpose of this article is to develop a Hadamard-Riesz type parametrix for the wave propagator in bounded planar domains with smooth, strictly convex boundary. This parametrix then allows us to rederive an oscillatory integral representation for the wave trace appearing in cite{MaMe82} and compute its principal symbol explicitly in terms of geometric data associated to the billiard map. This results in new formulas for the wave invariants. The order of the principal symbol, which appears to be inconsistent in the works of cite{MaMe82} and cite{Popov1994}, is also corrected. In those papers, the principal symbol was never explicitly computed and to our knowledge, this paper contains the first precise formulas for the principal symbol of the wave trace. The wave trace formulas we provide are localized near both simple lengths corresponding to nondegenerate periodic orbits and degenerate lengths associated to one parameter families of periodic orbits tangent to a single rational caustic. Existence of a Hadamard-Riesz type parametrix for the wave propagator appears to be new in the literature, with the exception of the author's prior work cite{Vig18} in the special case of elliptical domains. It allows us to circumvent the symbol calculus in cite{DuGu75} and cite{HeZe12} when computing trace formulas, which are instead derived from our explicit parametrix and a rescaling argument via Hadamard's variational formula for the wave trace. These techniques also appear to be new in the literature.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2019-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48820104","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
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