Reviews in Mathematical Physics最新文献

{"title":"Quantizing graphs, one way or two?","authors":"Jon Harrison","doi":"10.1142/s0129055x24600018","DOIUrl":"https://doi.org/10.1142/s0129055x24600018","url":null,"abstract":"<p>Quantum graphs were introduced to model free electrons in organic molecules using a self-adjoint Hamiltonian on a network of intervals. A second graph quantization describes wave propagation on a graph by specifying scattering matrices at the vertices. A question that is frequently raised is the extent to which these models are the same or complementary. In particular, are all energy-independent unitary vertex scattering matrices associated with a self-adjoint Hamiltonian? Here we review results related to this issue. In addition, we observe that a self-adjoint Dirac operator with four component spinors produces a secular equation for the graph spectrum that matches the secular equation associated with wave propagation on the graph when the Dirac operator describes particles with zero mass and the vertex conditions do not allow spin rotation at the vertices.</p>","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":"159 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140165681","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}

{"title":"On uniform decay of the Maxwell fields on black hole space-times","authors":"Sari Ghanem","doi":"10.1142/s0129055x24500120","DOIUrl":"https://doi.org/10.1142/s0129055x24500120","url":null,"abstract":"<p>In this paper, we study the Maxwell equations in the domain of outer-communication of the Schwarzschild black hole. We prove that if the middle components of the non-stationary solutions of the Maxwell equations verify a Morawetz-type estimate supported on a compact region in space around the trapped surface, then the components of the Maxwell fields decay uniformly in the entire exterior of the Schwarzschild black hole, including the event horizon. This is shown by making only use of Sobolev inequalities combined with energy estimates using the Maxwell equations directly. The proof does not pass through the scalar wave equation on the Schwarzschild black hole, does not need to decouple the middle components for the Maxwell fields, and would be in particular useful for the non-abelian case of the Yang–Mills equations where the decoupling of the middle components cannot occur. In fact, the estimates for the hereby argument are still valid for the Yang–Mills fields except for the Lie derivatives of the fields that are involved in the proof.</p>","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":"46 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140166086","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}

{"title":"Analytical solutions to the 1D compressible isothermal Navier–Stokes equations with Maxwell’s law","authors":"Jianwei Dong, Lijuan Bo","doi":"10.1142/s0129055x24500168","DOIUrl":"https://doi.org/10.1142/s0129055x24500168","url":null,"abstract":"<p>In this paper, we present some analytical solutions to the one-dimensional compressible isothermal Navier–Stokes equations with Maxwell’s law in the real line. First, we construct two analytical solutions by using a self-similar ansatz, one blows up in finite time and the other exists globally-in-time. Second, we construct two global analytical solutions with different large initial data by using a non-self-similar ansatz.</p>","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":"33 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140165682","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}

{"title":"Schottky cohomology for rank two bosonic vertex operator algebra","authors":"A. Zuevsky","doi":"10.1142/s0129055x24500144","DOIUrl":"https://doi.org/10.1142/s0129055x24500144","url":null,"abstract":"<p>Using the Schottky procedure of forming a genus <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>g</mi></math></span><span></span> Riemann surface by attaching handles to the Riemann sphere, we construct coboundary operators and corresponding cohomology for the double complexes of rational functions associated to a particular example of the rank two bosonic vertex operator algebra <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>V</mi></math></span><span></span>.</p>","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":"106 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140166259","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}

{"title":"Gibbs states and their classical limit","authors":"Christiaan J. F. van de Ven","doi":"10.1142/s0129055x24500090","DOIUrl":"https://doi.org/10.1142/s0129055x24500090","url":null,"abstract":"<p>A continuous bundle of <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>C</mi></mrow><mrow><mo stretchy=\"false\">∗</mo></mrow></msup></math></span><span></span>-algebras provides a rigorous framework to study the thermodynamic limit of quantum theories. If the bundle admits the additional structure of a strict deformation quantization (in the sense of Rieffel) one is allowed to study the <i>classical limit</i> of the quantum system, i.e. a mathematical formalism that examines the convergence of algebraic quantum states to probability measures on phase space (typically a Poisson or symplectic manifold). In this manner, we first prove the existence of the classical limit of Gibbs states illustrated with a class of Schrödinger operators in the regime where Planck’s constant <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>ℏ</mi></math></span><span></span> appearing in front of the Laplacian approaches zero. We additionally show that the ensuing limit corresponds to the unique probability measure satisfying the so-called classical or static KMS-condition. Subsequently, we conduct a similar study on the free energy of mean-field quantum spin systems in the regime of large particles, and discuss the existence of the classical limit of the relevant Gibbs states. Finally, a short section is devoted to single site quantum spin systems in the large spin limit.</p>","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":"41 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140165650","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}

{"title":"Kolmogorov extension theorem for non-probability measures on Cayley trees","authors":"F. H. Haydarov","doi":"10.1142/s0129055x24500107","DOIUrl":"https://doi.org/10.1142/s0129055x24500107","url":null,"abstract":"<p>In this paper, we shall discuss the extendability of probability and non-probability measures on Cayley trees to a <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>σ</mi></math></span><span></span>-additive measure on Borel fields which has a fundamental role in the theory of Gibbs measures.</p>","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":"7 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140172866","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}

{"title":"Spectral asymptotics of elliptic operators on manifolds","authors":"Ivan G. Avramidi","doi":"10.1142/s0129055x24500077","DOIUrl":"https://doi.org/10.1142/s0129055x24500077","url":null,"abstract":"&lt;p&gt;The study of spectral properties of natural geometric elliptic partial differential operators acting on smooth sections of vector bundles over Riemannian manifolds is a central theme in global analysis, differential geometry and mathematical physics. Instead of studying the spectrum of a differential operator &lt;span&gt;&lt;math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; directly one usually studies its spectral functions, that is, spectral traces of some functions of the operator, such as the spectral zeta function &lt;span&gt;&lt;math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;ζ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mstyle&gt;&lt;mtext mathvariant=\"normal\"&gt;Tr&lt;/mtext&gt;&lt;/mstyle&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo stretchy=\"false\"&gt;−&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; and the heat trace &lt;span&gt;&lt;math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi mathvariant=\"normal\"&gt;Θ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mstyle&gt;&lt;mtext mathvariant=\"normal\"&gt;Tr exp&lt;/mtext&gt;&lt;/mstyle&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mo stretchy=\"false\"&gt;−&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;. The kernel &lt;span&gt;&lt;math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;′&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; of the heat semigroup &lt;span&gt;&lt;math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mstyle&gt;&lt;mtext mathvariant=\"normal\"&gt;exp&lt;/mtext&gt;&lt;/mstyle&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mo stretchy=\"false\"&gt;−&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;, called the heat kernel, plays a major role in quantum field theory and quantum gravity, index theorems, non-commutative geometry, integrable systems and financial mathematics. We review some recent progress in the study of spectral asymptotics. We study more general spectral functions, such as &lt;span&gt;&lt;math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mstyle&gt;&lt;mtext mathvariant=\"normal\"&gt;Tr&lt;/mtext&gt;&lt;/mstyle&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;, that we call quantum heat traces. Also, we define new invariants of differential operators that depend not only on the eigenvalues but also on the eigenfunctions, and, therefore, contain much more information about the geometry of the manifold. Furthermore, we study some new invariants, such as &lt;span&gt;&lt;math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mstyle&gt;&lt;mtext mathvariant=\"normal\"&gt;Tr exp&lt;/mtext&gt;&lt;/mstyle&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mo stretchy=\"false\"&gt;−&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo stretchy=\"false\"&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;mstyle&gt;&lt;mtext mathvariant=\"normal\"&gt;exp&lt;/mtext&gt;&lt;/mstyle&gt;","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":"19 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140165679","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}

{"title":"On spectral and scattering theory for one-body quantum systems in crossed constant electric and magnetic fields","authors":"Tadayoshi Adachi, Yuta Tsujii","doi":"10.1142/s0129055x24500119","DOIUrl":"https://doi.org/10.1142/s0129055x24500119","url":null,"abstract":"","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":"227 ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139011355","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}

{"title":"Topological Aspects of Matters and Langlands Program","authors":"Kazuki Ikeda","doi":"10.1142/s0129055x24500053","DOIUrl":"https://doi.org/10.1142/s0129055x24500053","url":null,"abstract":"In the framework of Langlands program, we offer a unified description of the integer and fractional quantum Hall effect as well as the fractal nature of energy spectra of 2d Bloch electrons. We categorify topological invariants on the Brillouin Zone and address the several dualities in a coherent manner where analogs of the classical Fourier transform provide an essential crux of the matter. Based on the Langlands philosophy, we elucidate the duality of topological computation and that of Ising models in the same context.","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":" June","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135186669","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}

{"title":"Conformal super Virasoro algebra: matrix model and quantum deformed algebra","authors":"Fridolin Melong","doi":"10.1142/s0129055x24500065","DOIUrl":"https://doi.org/10.1142/s0129055x24500065","url":null,"abstract":"In this paper, we construct the super Virasoro algebra with an arbitrary conformal dimension $Delta$ from the generalized $mathcal{R}(p,q)$-deformed quantum algebra and investigate the $mathcal{R}(p,q)$-deformed super Virasoro algebra with the particular conformal dimension $Delta=1$. Furthermore, we perform the R(p,q)-conformal Virasoro n-algebra, the $mathcal{R}(p,q)$-conformal super Virasoro n-algebra ($n$ even) and discuss a toy model for the $mathcal{R}(p,q)$-conformal Virasoro constraints and R(p,q)-conformal super Virasoro constraints. Besides, we generalized the notion of the $mathcal{R}(p,q)$-elliptic hermitian matrix model with an arbitrary conformal dimension $Delta$. Finally, we deduce relevant particular cases generated by quantum algebras known in the literature.","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":" August","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135186667","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}