This paper is devoted to a comparison between the normalized first (non-trivial) Neumann eigenvalue $|Omega| mu_1(Omega)$ for a Lipschitz open set $Omega$ in the plane, and the normalized first (non-trivial) Steklov eigenvalue $P(Omega) sigma_1(Omega)$. More precisely, we study the ratio $F(Omega):=|Omega| mu_1(Omega)/P(Omega) sigma_1(Omega)$. We prove that this ratio can take arbitrarily small or large values if we do not put any restriction on the class of sets $Omega$. Then we restrict ourselves to the class of plane convex domains for which we get explicit bounds. We also study the case of thin convex domains for which we give more precise bounds. The paper finishes with the plot of the corresponding Blaschke-Santal'o diagrams $(x,y)=left(|Omega| mu_1(Omega), P(Omega) sigma_1(Omega) right)$.
{"title":"A comparison between Neumann and Steklov eigenvalues","authors":"A. Henrot, Marco Michetti","doi":"10.4171/jst/429","DOIUrl":"https://doi.org/10.4171/jst/429","url":null,"abstract":"This paper is devoted to a comparison between the normalized first (non-trivial) Neumann eigenvalue $|Omega| mu_1(Omega)$ for a Lipschitz open set $Omega$ in the plane, and the normalized first (non-trivial) Steklov eigenvalue $P(Omega) sigma_1(Omega)$. More precisely, we study the ratio $F(Omega):=|Omega| mu_1(Omega)/P(Omega) sigma_1(Omega)$. We prove that this ratio can take arbitrarily small or large values if we do not put any restriction on the class of sets $Omega$. Then we restrict ourselves to the class of plane convex domains for which we get explicit bounds. We also study the case of thin convex domains for which we give more precise bounds. The paper finishes with the plot of the corresponding Blaschke-Santal'o diagrams $(x,y)=left(|Omega| mu_1(Omega), P(Omega) sigma_1(Omega) right)$.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44726175","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":"Coexistence of absolutely continuous and pure point spectrum for kicked quasiperiodic potentials","authors":"Kristian Bjerklöv, R. Krikorian","doi":"10.4171/JST/370","DOIUrl":"https://doi.org/10.4171/JST/370","url":null,"abstract":"","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45884013","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":"Invertibility issues for a class of Wiener–Hopf plus Hankel operators","authors":"V. Didenko, B. Silbermann","doi":"10.4171/JST/359","DOIUrl":"https://doi.org/10.4171/JST/359","url":null,"abstract":"","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46102758","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 abstract scattering structure of the non homogeneous linearized Vlasov-Poisson equations from the viewpoint of trace class properties which are emblematic of the abstract scattering theory [13, 14, 15, 19]. In dimension 1+1, we derive an original reformulation which is trace class. It yields the existence of the Moller wave operators. The non homogeneous background electric field is periodic with 4 + e bounded derivatives. Mathematics Subject Classification (2010). Primary: 47A40; Secondary: 35P25.
{"title":"Trace class properties of the non homogeneous linear Vlasov–Poisson equation in dimension 1+1","authors":"B. Després","doi":"10.4171/JST/354","DOIUrl":"https://doi.org/10.4171/JST/354","url":null,"abstract":"We consider the abstract scattering structure of the non homogeneous linearized Vlasov-Poisson equations from the viewpoint of trace class properties which are emblematic of the abstract scattering theory [13, 14, 15, 19]. In dimension 1+1, we derive an original reformulation which is trace class. It yields the existence of the Moller wave operators. The non homogeneous background electric field is periodic with 4 + e bounded derivatives. Mathematics Subject Classification (2010). Primary: 47A40; Secondary: 35P25.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47141570","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 prove that the spectral structure on the $N$-dimensional standard sphere of radius $(N-1)^{1/2}$ compatible with a projection onto the first $n$-coordinates converges to the spectral structure on the $n$-dimensional Gaussian space with variance $1$ as $Nto infty$. We also show the analogue for the first Dirichlet eigenvalue and its eigenfunction on a ball in the sphere and on a half-space in the Gaussian space.
{"title":"Spectral convergence of high-dimensional spheres to Gaussian spaces","authors":"Asuka Takatsu","doi":"10.4171/jst/424","DOIUrl":"https://doi.org/10.4171/jst/424","url":null,"abstract":"We prove that the spectral structure on the $N$-dimensional standard sphere of radius $(N-1)^{1/2}$ compatible with a projection onto the first $n$-coordinates converges to the spectral structure on the $n$-dimensional Gaussian space with variance $1$ as $Nto infty$. We also show the analogue for the first Dirichlet eigenvalue and its eigenfunction on a ball in the sphere and on a half-space in the Gaussian space.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43917842","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 [GUW] we introduced a class of “semi-classical functions of isotropic type”, starting with a model case and applying Fourier integral operators associated with canonical transformations. These functions are a substantial generalization of the “oscillatory functions of Lagrangian type” that have played major role in semi-classical and micro-local analysis. In this paper we exhibit more clearly the nature of these isotropic functions by obtaining oscillatory integral expressions for them. Then we use these to prove that the classes of isotropic functions are equivariant with respect to the action of general FIOs (under the usual clean-intersection hypothesis). The simplest examples of isotropic states are the “coherent states”, a class of oscillatory functions that has played a pivotal role in mathematics and theoretical physics beginning with their introduction by of Schrödinger in the 1920’s. We prove that every oscillatory function of isotropic type can be expressed as a superposition of coherent states, and examine some implications of that fact. We also show that certain functions of elliptic operators have isotropic functions for Schwartz kernels. This lead us to a result on an eigenvalue counting function that appears to be new (Corollary 4.5). In memory of Mikhail Shubin.
{"title":"Integral representations of isotropic semiclassical functions and applications","authors":"V. Guillemin, A. Uribe, Zuoqin Wang","doi":"10.4171/jst/400","DOIUrl":"https://doi.org/10.4171/jst/400","url":null,"abstract":"In [GUW] we introduced a class of “semi-classical functions of isotropic type”, starting with a model case and applying Fourier integral operators associated with canonical transformations. These functions are a substantial generalization of the “oscillatory functions of Lagrangian type” that have played major role in semi-classical and micro-local analysis. In this paper we exhibit more clearly the nature of these isotropic functions by obtaining oscillatory integral expressions for them. Then we use these to prove that the classes of isotropic functions are equivariant with respect to the action of general FIOs (under the usual clean-intersection hypothesis). The simplest examples of isotropic states are the “coherent states”, a class of oscillatory functions that has played a pivotal role in mathematics and theoretical physics beginning with their introduction by of Schrödinger in the 1920’s. We prove that every oscillatory function of isotropic type can be expressed as a superposition of coherent states, and examine some implications of that fact. We also show that certain functions of elliptic operators have isotropic functions for Schwartz kernels. This lead us to a result on an eigenvalue counting function that appears to be new (Corollary 4.5). In memory of Mikhail Shubin.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42034891","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 note, we establish $L^p$-bounds for the semigroup $e^{-tq^w(x,D)}$, $t ge 0$, generated by a quadratic differential operator $q^w(x,D)$ on $mathbb{R}^n$ that is the Weyl quantization of a complex-valued quadratic form $q$ defined on the phase space $mathbb{R}^{2n}$ with non-negative real part $textrm{Re} , q ge 0$ and trivial singular space. Specifically, we show that $e^{-tq^w(x,D)}$ is bounded $L^p(mathbb{R}^n) rightarrow L^q(mathbb{R}^n)$ for all $t > 0$ whenever $1 le p le q le infty$, and we prove bounds on $||e^{-tq^w(x,D)}||_{L^p rightarrow L^q}$ in both the large $t gg 1$ and small $0 < t ll 1$ time regimes. Regarding $L^p rightarrow L^q$ bounds for the evolution semigroup at large times, we show that $||e^{-tq^w(x,D)}||_{L^p rightarrow L^q}$ is exponentially decaying as $t rightarrow infty$, and we determine the precise rate of exponential decay, which is independent of $(p,q)$. At small times $0 < t ll 1$, we establish bounds on $||e^{-tq^w(x,D)}||_{L^p rightarrow L^q}$ for $(p,q)$ with $1 le p le q le infty$ that are polynomial in $t^{-1}$.
{"title":"$L^p$-bounds for semigroups generated by non-elliptic quadratic differential operators","authors":"F. White","doi":"10.4171/jst/426","DOIUrl":"https://doi.org/10.4171/jst/426","url":null,"abstract":"In this note, we establish $L^p$-bounds for the semigroup $e^{-tq^w(x,D)}$, $t ge 0$, generated by a quadratic differential operator $q^w(x,D)$ on $mathbb{R}^n$ that is the Weyl quantization of a complex-valued quadratic form $q$ defined on the phase space $mathbb{R}^{2n}$ with non-negative real part $textrm{Re} , q ge 0$ and trivial singular space. Specifically, we show that $e^{-tq^w(x,D)}$ is bounded $L^p(mathbb{R}^n) rightarrow L^q(mathbb{R}^n)$ for all $t > 0$ whenever $1 le p le q le infty$, and we prove bounds on $||e^{-tq^w(x,D)}||_{L^p rightarrow L^q}$ in both the large $t gg 1$ and small $0 < t ll 1$ time regimes. Regarding $L^p rightarrow L^q$ bounds for the evolution semigroup at large times, we show that $||e^{-tq^w(x,D)}||_{L^p rightarrow L^q}$ is exponentially decaying as $t rightarrow infty$, and we determine the precise rate of exponential decay, which is independent of $(p,q)$. At small times $0 < t ll 1$, we establish bounds on $||e^{-tq^w(x,D)}||_{L^p rightarrow L^q}$ for $(p,q)$ with $1 le p le q le infty$ that are polynomial in $t^{-1}$.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49294073","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 introduce a parameterized family of Poisson transforms on trees of bounded degree, construct explicit inverses for generic pa-rameters, and characterize moderate growth of Laplace eigenfunctions by H¨older regularity of their boundary values.
{"title":"Poisson transforms for trees of bounded degree","authors":"Kai-Uwe Bux, J. Hilgert, T. Weich","doi":"10.4171/jst/414","DOIUrl":"https://doi.org/10.4171/jst/414","url":null,"abstract":". We introduce a parameterized family of Poisson transforms on trees of bounded degree, construct explicit inverses for generic pa-rameters, and characterize moderate growth of Laplace eigenfunctions by H¨older regularity of their boundary values.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43132309","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}
Consider an elliptic self-adjoint pseudodifferential operator A acting on m-columns of half-densities on a closed manifold M , whose principal symbol is assumed to have simple eigenvalues. We show that the spectrum of A decomposes, up to an error with superpolynomial decay, into m distinct series, each associated with one of the eigenvalues of the principal symbol of A. These spectral results are then applied to the study of propagation of singularities in hyperbolic systems. The key technical ingredient is the use of the carefully devised pseudodifferential projections introduced in the first part of this work, which decompose L2(M) into almost-orthogonal almost-invariant subspaces under the action of both A and the hyperbolic evolution.
{"title":"Invariant subspaces of elliptic systems II: Spectral theory","authors":"Matteo Capoferri, D. Vassiliev","doi":"10.4171/JST/402","DOIUrl":"https://doi.org/10.4171/JST/402","url":null,"abstract":"Consider an elliptic self-adjoint pseudodifferential operator A acting on m-columns of half-densities on a closed manifold M , whose principal symbol is assumed to have simple eigenvalues. We show that the spectrum of A decomposes, up to an error with superpolynomial decay, into m distinct series, each associated with one of the eigenvalues of the principal symbol of A. These spectral results are then applied to the study of propagation of singularities in hyperbolic systems. The key technical ingredient is the use of the carefully devised pseudodifferential projections introduced in the first part of this work, which decompose L2(M) into almost-orthogonal almost-invariant subspaces under the action of both A and the hyperbolic evolution.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46104489","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 survey our recent results on the Benjamin-Ono equation on the torus. As an application of the methods developed we construct large families of periodic or quasiperiodic solutions, which are not C∞-smooth.
{"title":"On the Benjamin–Ono equation on $mathbb{T}$ and its periodic and quasiperiodic solutions","authors":"P. G'erard, T. Kappeler, P. Topalov","doi":"10.4171/jst/398","DOIUrl":"https://doi.org/10.4171/jst/398","url":null,"abstract":"In this paper, we survey our recent results on the Benjamin-Ono equation on the torus. As an application of the methods developed we construct large families of periodic or quasiperiodic solutions, which are not C∞-smooth.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46867002","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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