Pub Date : 2023-11-09DOI: 10.1142/s0129055x23500368
Jussi Behrndt, Markus Holzmann, Christian Stelzer, Georg Stenzel
In this paper, we develop a systematic approach to treat Dirac operators [Formula: see text] with singular electrostatic, Lorentz scalar, and anomalous magnetic interactions of strengths [Formula: see text], respectively, supported on points in [Formula: see text], curves in [Formula: see text], and surfaces in [Formula: see text] that is based on boundary triples and their associated Weyl functions. First, we discuss the one-dimensional case which also serves as a motivation for the multidimensional setting. Afterwards, in the two- and three-dimensional situation we construct quasi, generalized, and ordinary boundary triples and their Weyl functions, and provide a detailed characterization of the associated Sobolev spaces, trace theorems, and the mapping properties of integral operators which play an important role in the analysis of [Formula: see text]. We make a substantial step towards more rough interaction supports [Formula: see text] and consider general compact Lipschitz hypersurfaces. We derive conditions for the interaction strengths such that the operators [Formula: see text] are self-adjoint, obtain a Krein-type resolvent formula, and characterize the essential and discrete spectrum. These conditions include purely Lorentz scalar and purely non-critical anomalous magnetic interactions as well as the confinement case, the latter having an important application in the mathematical description of graphene. Using a certain ordinary boundary triple, we also show the self-adjointness of [Formula: see text] for arbitrary critical combinations of the interaction strengths under the condition that [Formula: see text] is [Formula: see text]-smooth and derive its spectral properties. In particular, in the critical case, a loss of Sobolev regularity in the operator domain and a possible additional point of the essential spectrum are observed.
{"title":"Boundary triples and Weyl functions for Dirac operators with singular interactions","authors":"Jussi Behrndt, Markus Holzmann, Christian Stelzer, Georg Stenzel","doi":"10.1142/s0129055x23500368","DOIUrl":"https://doi.org/10.1142/s0129055x23500368","url":null,"abstract":"In this paper, we develop a systematic approach to treat Dirac operators [Formula: see text] with singular electrostatic, Lorentz scalar, and anomalous magnetic interactions of strengths [Formula: see text], respectively, supported on points in [Formula: see text], curves in [Formula: see text], and surfaces in [Formula: see text] that is based on boundary triples and their associated Weyl functions. First, we discuss the one-dimensional case which also serves as a motivation for the multidimensional setting. Afterwards, in the two- and three-dimensional situation we construct quasi, generalized, and ordinary boundary triples and their Weyl functions, and provide a detailed characterization of the associated Sobolev spaces, trace theorems, and the mapping properties of integral operators which play an important role in the analysis of [Formula: see text]. We make a substantial step towards more rough interaction supports [Formula: see text] and consider general compact Lipschitz hypersurfaces. We derive conditions for the interaction strengths such that the operators [Formula: see text] are self-adjoint, obtain a Krein-type resolvent formula, and characterize the essential and discrete spectrum. These conditions include purely Lorentz scalar and purely non-critical anomalous magnetic interactions as well as the confinement case, the latter having an important application in the mathematical description of graphene. Using a certain ordinary boundary triple, we also show the self-adjointness of [Formula: see text] for arbitrary critical combinations of the interaction strengths under the condition that [Formula: see text] is [Formula: see text]-smooth and derive its spectral properties. In particular, in the critical case, a loss of Sobolev regularity in the operator domain and a possible additional point of the essential spectrum are observed.","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":" 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135191528","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-03DOI: 10.1142/s0129055x24300048
A. Zuevsky
{"title":"On a Category of <i>V</i>-Structures for Foliations","authors":"A. Zuevsky","doi":"10.1142/s0129055x24300048","DOIUrl":"https://doi.org/10.1142/s0129055x24300048","url":null,"abstract":"","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":"99 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135869152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-01DOI: 10.1142/s0129055x24300012
Gregory Eskin
For a wave equation with time-independent Lorentzian metric consider an initial-boundary value problem in $mathbb{R}times Omega$, where $x_0in mathbb{R}$, is the time variable and $Omega$ is a bounded domain in $mathbb{R}^n$. Let $GammasubsetpartialOmega$ be a subdomain of $partialOmega$. We say that the boundary measurements are given on $mathbb{R}timesGamma$ if we know the Dirichlet and Neumann data on $mathbb{R}times Gamma$. The inverse boundary value problem consists of recovery of the metric from the boundary data. In author's previous works a localized variant of the boundary control method was developed that allows the recovery of the metric locally in a neighborhood of any point of $Omega$ where the spatial part of the wave operator is elliptic. This allow the recovery of the metric in the exterior of the ergoregion. Our goal is to recover the black hole. In some cases the ergoregion coincides with the black hole. In the case of two space dimensions we recover the black hole inside the ergoregion assuming that the ergosphere, i.e. the boundary of the ergoregion, is not characteristic at any point of the ergosphere.
{"title":"Determination of black holes by boundary measurements","authors":"Gregory Eskin","doi":"10.1142/s0129055x24300012","DOIUrl":"https://doi.org/10.1142/s0129055x24300012","url":null,"abstract":"For a wave equation with time-independent Lorentzian metric consider an initial-boundary value problem in $mathbb{R}times Omega$, where $x_0in mathbb{R}$, is the time variable and $Omega$ is a bounded domain in $mathbb{R}^n$. Let $GammasubsetpartialOmega$ be a subdomain of $partialOmega$. We say that the boundary measurements are given on $mathbb{R}timesGamma$ if we know the Dirichlet and Neumann data on $mathbb{R}times Gamma$. The inverse boundary value problem consists of recovery of the metric from the boundary data. In author's previous works a localized variant of the boundary control method was developed that allows the recovery of the metric locally in a neighborhood of any point of $Omega$ where the spatial part of the wave operator is elliptic. This allow the recovery of the metric in the exterior of the ergoregion. Our goal is to recover the black hole. In some cases the ergoregion coincides with the black hole. In the case of two space dimensions we recover the black hole inside the ergoregion assuming that the ergosphere, i.e. the boundary of the ergoregion, is not characteristic at any point of the ergosphere.","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":"46 12","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136103381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-01DOI: 10.1142/s0129055x2450003x
Benjamin Schulz
{"title":"On Topology Changes in Quantum Field Theory and Quantum Gravity","authors":"Benjamin Schulz","doi":"10.1142/s0129055x2450003x","DOIUrl":"https://doi.org/10.1142/s0129055x2450003x","url":null,"abstract":"","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135272929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-01DOI: 10.1142/s0129055x24500028
Indranil Biswas, Saikat Chatterjee, Praphulla Koushik, Frank Neumann
Let $mathbb{X}=[X_1rightrightarrows X_0]$ be a Lie groupoid equipped with a connection, given by a smooth distribution $mathcal{H} subset T X_1$ transversal to the fibers of the source map. Under the assumption that the distribution $mathcal{H}$is integrable, we define a version of de Rham cohomology for the pair $(mathbb{X}, mathcal{H})$, and we study connections on principal $G$-bundles over $(mathbb{X}, mathcal{H})$ in terms of the associated Atiyah sequence of vector bundles. We also discuss associated constructions for differentiable stacks. Finally, we develop the corresponding Chern-Weil theory and describe characteristic classes of principal G-bundles over a pair $(mathbb{X}, mathcal{H})$.
{"title":"Connections on Lie Groupoids and Chern-Weil Theory","authors":"Indranil Biswas, Saikat Chatterjee, Praphulla Koushik, Frank Neumann","doi":"10.1142/s0129055x24500028","DOIUrl":"https://doi.org/10.1142/s0129055x24500028","url":null,"abstract":"Let $mathbb{X}=[X_1rightrightarrows X_0]$ be a Lie groupoid equipped with a connection, given by a smooth distribution $mathcal{H} subset T X_1$ transversal to the fibers of the source map. Under the assumption that the distribution $mathcal{H}$is integrable, we define a version of de Rham cohomology for the pair $(mathbb{X}, mathcal{H})$, and we study connections on principal $G$-bundles over $(mathbb{X}, mathcal{H})$ in terms of the associated Atiyah sequence of vector bundles. We also discuss associated constructions for differentiable stacks. Finally, we develop the corresponding Chern-Weil theory and describe characteristic classes of principal G-bundles over a pair $(mathbb{X}, mathcal{H})$.","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":"57 3-4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135271788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-01DOI: 10.1142/s0129055x24500041
Sami Baraket, Rima Chetouane, Rached Jaidane, Wafa Mtaouaa
{"title":"Sign-Changing Solutions for a Weighted Schrodinger-Kirchhoff Equation with Double Exponential Nonlinearities Growth","authors":"Sami Baraket, Rima Chetouane, Rached Jaidane, Wafa Mtaouaa","doi":"10.1142/s0129055x24500041","DOIUrl":"https://doi.org/10.1142/s0129055x24500041","url":null,"abstract":"","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":"72 5-6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135272439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-01DOI: 10.1142/s0129055x24300036
Pascal Millet
We present in detail the geometric framework necessary to understand the Teukolsky equation and we develop in particular the case of Kerr spacetime.
我们详细介绍了理解Teukolsky方程所必需的几何框架，并特别发展了Kerr时空的情况。
{"title":"Geometric background for the Teukolsky equation revisited","authors":"Pascal Millet","doi":"10.1142/s0129055x24300036","DOIUrl":"https://doi.org/10.1142/s0129055x24300036","url":null,"abstract":"We present in detail the geometric framework necessary to understand the Teukolsky equation and we develop in particular the case of Kerr spacetime.","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":"30 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136103245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-01DOI: 10.1142/s0129055x24500016
Young Jin Suh, SUDHAKAR K. Chaubey, Mohammad Nazrul Islam Khan
{"title":"Lorentzian Manifolds : A Characterization with a Type of Semi-Symmetric Non-Metric Connection","authors":"Young Jin Suh, SUDHAKAR K. Chaubey, Mohammad Nazrul Islam Khan","doi":"10.1142/s0129055x24500016","DOIUrl":"https://doi.org/10.1142/s0129055x24500016","url":null,"abstract":"","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":"73 1-2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135272436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-01DOI: 10.1142/s0129055x24300024
Jonathan Sorce
{"title":"Renormalization and the type classification of von Neumann algebras","authors":"Jonathan Sorce","doi":"10.1142/s0129055x24300024","DOIUrl":"https://doi.org/10.1142/s0129055x24300024","url":null,"abstract":"","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":"57 7-8","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135271786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-27DOI: 10.1142/s0129055x2330008x
Hajime Moriya
The Kubo–Martin–Schwinger (KMS) condition is a well-founded general definition of equilibrium states on quantum systems. The time invariance property of equilibrium states is one of its basic consequences. From the time invariance of any equilibrium state it follows that the spontaneous breakdown of time-translation symmetry is impossible. Moreover, triviality of the temporal long-range order is derived from the KMS condition. Therefore, the manifestation of genuine quantum time crystals is impossible as long as the standard notion of spontaneous symmetry breakdown is considered.
{"title":"Nonexistence of spontaneous symmetry breakdown of time-translation symmetry on general quantum systems: Any macroscopic order parameter moves not!","authors":"Hajime Moriya","doi":"10.1142/s0129055x2330008x","DOIUrl":"https://doi.org/10.1142/s0129055x2330008x","url":null,"abstract":"The Kubo–Martin–Schwinger (KMS) condition is a well-founded general definition of equilibrium states on quantum systems. The time invariance property of equilibrium states is one of its basic consequences. From the time invariance of any equilibrium state it follows that the spontaneous breakdown of time-translation symmetry is impossible. Moreover, triviality of the temporal long-range order is derived from the KMS condition. Therefore, the manifestation of genuine quantum time crystals is impossible as long as the standard notion of spontaneous symmetry breakdown is considered.","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":"167 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136233033","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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