Integral Equations and Operator Theory最新文献_第2页

Conditions for Semi-Boundedness and Discreteness of the Spectrum to Schrödinger Operator and Some Nonlinear PDEs 薛定谔算子和一些非线性多线性方程谱的半边界性和不确定性条件

IF 0.8 3区数学

Integral Equations and Operator Theory Pub Date : 2024-08-08 DOI: 10.1007/s00020-024-02773-8

Leonid Zelenko

{"title":"Conditions for Semi-Boundedness and Discreteness of the Spectrum to Schrödinger Operator and Some Nonlinear PDEs","authors":"Leonid Zelenko","doi":"10.1007/s00020-024-02773-8","DOIUrl":"https://doi.org/10.1007/s00020-024-02773-8","url":null,"abstract":"For the Schrödinger operator (H=-Delta + V({{textbf{x}}})cdot ), acting in the space (L_2({{textbf{R}}}^d),(dge 3)), necessary and sufficient conditions for semi-boundedness and discreteness of its spectrum are obtained without the assumption that the potential (V({{textbf{x}}})) is bounded below. By reducing the problem to study the existence of regular solutions of the Riccati PDE, the necessary conditions for the discreteness of the spectrum of operator H are obtained under the assumption that it is bounded below. These results are similar to the ones obtained by the author in [26] for the one-dimensional case. Furthermore, sufficient conditions for the semi-boundedness and discreteness of the spectrum of H are obtained in terms of a non-increasing rearrangement, mathematical expectation, and standard deviation from the latter for the positive part (V_+({{textbf{x}}})) of the potential (V({{textbf{x}}})) on compact domains that go to infinity, under certain restrictions for its negative part (V_-({{textbf{x}}})). Choosing optimally the vector field associated with the difference between the potential (V({{textbf{x}}})) and its mathematical expectation on the balls that go to infinity, we obtain a condition for semi-boundedness and discreteness of the spectrum for H in terms of solutions of the Neumann problem for the nonhomogeneous (d/(d-1))-Laplace equation. This type of optimization refers to a divergence constrained transportation problem.","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"161 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934721","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

Horizontal Fourier Transform of the Polyanalytic Fock Kernel 多解析 Fock 内核的水平傅里叶变换

IF 0.8 3区数学

Integral Equations and Operator Theory Pub Date : 2024-07-04 DOI: 10.1007/s00020-024-02772-9

Erick Lee-Guzmán, Egor A. Maximenko, Gerardo Ramos-Vazquez, Armando Sánchez-Nungaray

{"title":"Horizontal Fourier Transform of the Polyanalytic Fock Kernel","authors":"Erick Lee-Guzmán, Egor A. Maximenko, Gerardo Ramos-Vazquez, Armando Sánchez-Nungaray","doi":"10.1007/s00020-024-02772-9","DOIUrl":"https://doi.org/10.1007/s00020-024-02772-9","url":null,"abstract":"Let (n,mge 1) and (alpha >0). We denote by (mathcal {F}_{alpha ,m}) the m-analytic Bargmann–Segal–Fock space, i.e., the Hilbert space of all m-analytic functions defined on (mathbb {C}^n) and square integrables with respect to the Gaussian weight (exp (-alpha |z|^2)). We study the von Neumann algebra (mathcal {A}) of bounded linear operators acting in (mathcal {F}_{alpha ,m}) and commuting with all “horizontal” Weyl translations, i.e., Weyl unitary operators associated to the elements of (mathbb {R}^n). The reproducing kernel of (mathcal {F}_{1,m}) was computed by Youssfi [Polyanalytic reproducing kernels in (mathbb {C}^n), Complex Anal. Synerg., 2021, 7, 28]. Multiplying the elements of (mathcal {F}_{alpha ,m}) by an appropriate weight, we transform this space into another reproducing kernel Hilbert space whose kernel K is invariant under horizontal translations. Using the well-known Fourier connection between Laguerre and Hermite functions, we compute the Fourier transform of K in the “horizontal direction” and decompose it into the sum of d products of Hermite functions, with (d=left( {begin{array}{c}n+m-1 nend{array}}right) ). Finally, applying the scheme proposed by Herrera-Yañez, Maximenko, Ramos-Vazquez [Translation-invariant operators in reproducing kernel Hilbert spaces, Integr. Equ. Oper. Theory, 2022, 94, 31], we show that (mathcal {F}_{alpha ,m}) is isometrically isomorphic to the space of vector-functions (L^2(mathbb {R}^n)^d), and (mathcal {A}) is isometrically isomorphic to the algebra of matrix-functions (L^infty (mathbb {R}^n)^{dtimes d}).","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"5 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141547404","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

Essential positivity for Toeplitz operators on the Fock space 福克空间上的托普利兹算子的基本实在性

IF 0.8 3区数学

Integral Equations and Operator Theory Pub Date : 2024-06-26 DOI: 10.1007/s00020-024-02770-x

Robert Fulsche

引用次数: 0

A Quantum Harmonic Analysis Approach to Segal Algebras 西格尔代数的量子谐波分析方法

IF 0.8 3区数学

Integral Equations and Operator Theory Pub Date : 2024-06-22 DOI: 10.1007/s00020-024-02771-w

Eirik Berge, Stine Marie Berge, Robert Fulsche

引用次数: 0

Tropical Reproducing Kernels and Optimization 热带复制内核与优化

IF 0.8 3区数学

Integral Equations and Operator Theory Pub Date : 2024-05-29 DOI: 10.1007/s00020-024-02769-4

Pierre-Cyril Aubin-Frankowski, Stéphane Gaubert

{"title":"Tropical Reproducing Kernels and Optimization","authors":"Pierre-Cyril Aubin-Frankowski, Stéphane Gaubert","doi":"10.1007/s00020-024-02769-4","DOIUrl":"https://doi.org/10.1007/s00020-024-02769-4","url":null,"abstract":"Hilbertian kernel methods and their positive semidefinite kernels have been extensively used in various fields of applied mathematics and machine learning, owing to their several equivalent characterizations. We here unveil an analogy with concepts from tropical geometry, proving that tropical positive semidefinite kernels are also endowed with equivalent viewpoints, stemming from Fenchel–Moreau conjugations. This tropical analogue of Aronszajn’s theorem shows that these kernels correspond to a feature map, define monotonous operators, and generate max-plus function spaces endowed with a reproducing property. They furthermore include all the Hilbertian kernels classically studied as well as Monge arrays. However, two relevant notions of tropical reproducing kernels must be distinguished, based either on linear or sesquilinear interpretations. The sesquilinear interpretation is the most expressive one, since reproducing spaces then encompass classical max-plus spaces, such as those of (semi)convex functions. In contrast, in the linear interpretation, the reproducing kernels are characterized by a restrictive condition, von Neumann regularity. Finally, we provide a tropical analogue of the “representer theorems”, showing that a class of infinite dimensional regression and interpolation problems admit solutions lying in finite dimensional spaces. We illustrate this theorem by an application to optimal control, in which tropical kernels allow one to represent the value function.","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"12 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141167439","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

Local Spectral Multiplicity of Selfadjoint Couplings with General Interface Conditions 具有一般界面条件的自相关耦合的局部谱多重性

IF 0.8 3区数学

Integral Equations and Operator Theory Pub Date : 2024-05-22 DOI: 10.1007/s00020-024-02767-6

Sergey Simonov, Harald Woracek

引用次数: 0

On Commutators of Compact Operators: Generalizations and Limitations of Anderson’s Approach 论紧凑算子的换元：安德森方法的概括与局限

IF 0.8 3区数学

Integral Equations and Operator Theory Pub Date : 2024-05-17 DOI: 10.1007/s00020-024-02764-9

Jireh Loreaux, Sasmita Patnaik, Srdjan Petrovic, Gary Weiss

{"title":"On Commutators of Compact Operators: Generalizations and Limitations of Anderson’s Approach","authors":"Jireh Loreaux, Sasmita Patnaik, Srdjan Petrovic, Gary Weiss","doi":"10.1007/s00020-024-02764-9","DOIUrl":"https://doi.org/10.1007/s00020-024-02764-9","url":null,"abstract":"We offer a new perspective and some advances on the 1971 Pearcy-Topping problem: is every compact operator a commutator of compact operators? Our goal is to analyze and generalize the 1970’s work in this area of Joel Anderson. We reduce the general problem to a simpler sequence of finite matrix equations with norm constraints, while at the same time developing strategies for counterexamples. Our approach is to ask which compact operators are commutators AB − BA of compact operators A,B; and to analyze the implications of Joel Anderson’s contributions to this problem. By extending the techniques of Anderson, we obtain new classes of operators that are commutators of compact operators beyond those obtained by the second and the fourth author. We also found obstructions to extending Anderson’s techniques to obtain any positive compact operator as a commutator of compact operators. Some of these constraints involve general block-tridiagonal matrix forms for operators and some involve B(H)-ideal constraints. Finally, we provide some necessary conditions for the Pearcy-Topping problem involving singular numbers and B(H)-ideal constraints.","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"48 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062504","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

Subhomogeneous Operator Systems and Classification of Operator Systems Generated by $$Lambda $$ -Commuting Unitaries 亚均质算子系统和由 $$Lambda $$ - 通约单元生成的算子系统分类

IF 0.8 3区数学

Integral Equations and Operator Theory Pub Date : 2024-05-05 DOI: 10.1007/s00020-024-02765-8

Ran Kiri

引用次数: 0

Real Structure in Operator Spaces, Injective Envelopes and G-spaces 算子空间、注入包络和 G 空间中的实结构

IF 0.8 3区数学

Integral Equations and Operator Theory Pub Date : 2024-04-24 DOI: 10.1007/s00020-024-02766-7

David P. Blecher, Arianna Cecco, Mehrdad Kalantar

引用次数: 0

Sufficient Criteria for Stabilization Properties in Banach Spaces 巴拿赫空间稳定特性的充分标准