Roberto A. Prado, César R. de Oliveira, Edmundo C. de Oliveira
{"title":"Density of States and Lifshitz Tails for Discrete 1D Random Dirac Operators","authors":"Roberto A. Prado, César R. de Oliveira, Edmundo C. de Oliveira","doi":"10.1007/s11040-021-09403-4","DOIUrl":"10.1007/s11040-021-09403-4","url":null,"abstract":"<div><p>We study the density of states and Lifshitz tails for a family of random Dirac operators on the one-dimensional lattice <span>(mathbb {Z})</span>. These operators consist of the sum of a discrete free Dirac operator with a random potential. The potential is a diagonal matrix formed by two different scalar potentials, which are sequences of independent and identically distributed random variables according to a Borel probability measure of compact support in <span>(mathbb {R})</span>. The existence of the density of state measure for these Dirac operators is obtained through two approaches by finite-volume quantities. By using one of these approaches, we show that the distribution function of the density of states decays exponentially for energies near the spectral band edges, i.e., we establish Lifshitz tails for these operators. Lifshitz tails are established first for Dirac operators restricted to appropriate subspaces of energies and, using this, extended to the full operators, including the occurrence of internal tails in the case of spectral gap.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4590762","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":"Solution of the System of Two Coupled First-Order ODEs with Second-Degree Polynomial Right-Hand Sides","authors":"Francesco Calogero, Farrin Payandeh","doi":"10.1007/s11040-021-09400-7","DOIUrl":"10.1007/s11040-021-09400-7","url":null,"abstract":"<div><p>The <i>explicit</i> solution <span>(x_{n}left (tright ) ,)</span> <i>n</i> = 1,2, of the <i>initial-values</i> problem is exhibited of a <i>subclass</i> of the <i>autonomous</i> system of 2 coupled <i>first-order</i> ODEs with <i>second-degree</i> polynomial right-hand sides, hence featuring 12 <i>a priori arbitrary</i> (time-independent) coefficients: \u0000</p><div><div><span>$$ dot{x}_{n}=c_{n1}left( x_{1}right)^{2}+c_{n2}x_{1}x_{2}+c_{n3}left( x_{2}right)^{2}+c_{n4}x_{1}+c_{n5}x_{2}+c_{n6}~,~~~n=1,2~. $$</span></div></div><p> The solution is <i>explicitly</i> provided if the 12 coefficients <i>c</i><sub><i>n</i><i>j</i></sub> (<i>n</i> = 1,2; <i>j</i> = 1,2,3,4,5,6) are expressed by <i>explicitly</i> provided formulas in terms of 10 <i>a priori arbitrary</i> parameters; the <i>inverse</i> problem to express these 10 parameters in terms of the 12 coefficients <i>c</i><sub><i>n</i><i>j</i></sub> is also <i>explicitly</i> solved, but it is found to imply—as it were, <i>a posteriori</i>—that the 12 coefficients <i>c</i><sub><i>n</i><i>j</i></sub> must then satisfy 4 <i>algebraic constraints</i>, which are <i>explicitly</i> exhibited. Special subcases are also identified the <i>general</i> solutions of which are <i>completely periodic</i> with a period independent of the initial data (“isochrony”), or are characterized by additional restrictions on the coefficients <i>c</i><sub><i>n</i><i>j</i></sub> which identify particularly interesting models.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11040-021-09400-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4668589","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":"Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes","authors":"Claudio Dappiaggi, Alessio Marta","doi":"10.1007/s11040-021-09402-5","DOIUrl":"10.1007/s11040-021-09402-5","url":null,"abstract":"<div><p>We consider the Klein-Gordon operator on an <i>n</i>-dimensional asymptotically anti-de Sitter spacetime (<i>M</i>,<i>g</i>) together with arbitrary boundary conditions encoded by a self-adjoint pseudodifferential operator on <i>∂</i><i>M</i> of order up to 2. Using techniques from <i>b</i>-calculus and a propagation of singularities theorem, we prove that there exist advanced and retarded fundamental solutions, characterizing in addition their structural and microlocal properties. We apply this result to the problem of constructing Hadamard two-point distributions. These are bi-distributions which are weak bi-solutions of the underlying equations of motion with a prescribed form of their wavefront set and whose anti-symmetric part is proportional to the difference between the advanced and the retarded fundamental solutions. In particular, under a suitable restriction of the class of admissible boundary conditions and setting to zero the mass, we prove their existence extending to the case under scrutiny a deformation argument which is typically used on globally hyperbolic spacetimes with empty boundary.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11040-021-09402-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4484318","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":"Bilinear Equation and Additional Symmetries for an Extension of the Kadomtsev–Petviashvili Hierarchy","authors":"Jiaping Lu, Chao-Zhong Wu","doi":"10.1007/s11040-021-09401-6","DOIUrl":"10.1007/s11040-021-09401-6","url":null,"abstract":"<div><p>An extension of the Kadomtsev–Petviashvili (KP) hierarchy defined via scalar pseudo-differential operators was studied in Szablikowski and Blaszak (J. Math. Phys. <b>49</b>(8), 082701, 20, 2008) and Wu and Zhou (J. Geom. Phys. <b>106</b>, 327–341, 2016). In this paper, we represent the extended KP hierarchy into the form of bilinear equation of (adjoint) Baker–Akhiezer functions, and construct its additional symmetries. As a byproduct, we derive the Virasoro symmetries for the constrained KP hierarchies.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11040-021-09401-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4238213","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":"Kahan Discretizations of Skew-Symmetric Lotka-Volterra Systems and Poisson Maps","authors":"C. A. Evripidou, P. Kassotakis, P. Vanhaecke","doi":"10.1007/s11040-021-09399-x","DOIUrl":"10.1007/s11040-021-09399-x","url":null,"abstract":"<div><p>The Kahan discretization of the Lotka-Volterra system, associated with any skew-symmetric graph Γ, leads to a family of rational maps, parametrized by the step size. When these maps are Poisson maps with respect to the quadratic Poisson structure of the Lotka-Volterra system, we say that the graph Γ has the Kahan-Poisson property. We show that if Γ is connected, it has the Kahan-Poisson property if and only if it is a cloning of a graph with vertices <span>(1,2,dots ,n)</span>, with an arc <i>i</i> → <i>j</i> precisely when <i>i</i> < <i>j</i>, and with all arcs having the same value. We also prove a similar result for augmented graphs, which correspond with deformed Lotka-Volterra systems and show that the obtained Lotka-Volterra systems and their Kahan discretizations are superintegrable as well as Liouville integrable.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11040-021-09399-x","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5147330","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":"Discrete mKdV Equation via Darboux Transformation","authors":"Joseph Cho, Wayne Rossman, Tomoya Seno","doi":"10.1007/s11040-021-09398-y","DOIUrl":"10.1007/s11040-021-09398-y","url":null,"abstract":"<div><p>We introduce an efficient route to obtaining the discrete potential mKdV equation emerging from a particular discrete motion of discrete planar curves.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11040-021-09398-y","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4000475","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":"Asymptotic Behavior of Density in the Boundary-Driven Exclusion Process on the Sierpinski Gasket","authors":"Joe P. Chen, Patrícia Gonçalves","doi":"10.1007/s11040-021-09392-4","DOIUrl":"10.1007/s11040-021-09392-4","url":null,"abstract":"<div><p>We derive the macroscopic laws that govern the evolution of the density of particles in the exclusion process on the Sierpinski gasket in the presence of a variable speed boundary. We obtain, at the hydrodynamics level, the heat equation evolving on the Sierpinski gasket with either Dirichlet or Neumann boundary conditions, depending on whether the reservoirs are fast or slow. For a particular strength of the boundary dynamics we obtain linear Robin boundary conditions. As for the fluctuations, we prove that, when starting from the stationary measure, namely the product Bernoulli measure in the equilibrium setting, they are governed by Ornstein-Uhlenbeck processes with the respective boundary conditions.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11040-021-09392-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4339805","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":"Convergence of Discrete Period Matrices and Discrete Holomorphic Integrals for Ramified Coverings of the Riemann Sphere","authors":"Alexander I. Bobenko, Ulrike Bücking","doi":"10.1007/s11040-021-09394-2","DOIUrl":"10.1007/s11040-021-09394-2","url":null,"abstract":"<div><p>We consider the class of compact Riemann surfaces which are ramified coverings of the Riemann sphere <span>(hat {mathbb {C}})</span>. Based on a triangulation of this covering we define discrete (multivalued) harmonic and holomorphic functions. We prove that the corresponding discrete period matrices converge to their continuous counterparts. In order to achieve an error estimate, which is linear in the maximal edge length of the triangles, we suitably adapt the triangulations in a neighborhood of every branch point. Finally, we also prove a convergence result for discrete holomorphic integrals for our adapted triangulations of the ramified covering.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11040-021-09394-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4087019","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":"Wegner Estimate for Random Divergence-Type Operators Monotone in the Randomness","authors":"Alexander Dicke","doi":"10.1007/s11040-021-09396-0","DOIUrl":"10.1007/s11040-021-09396-0","url":null,"abstract":"<div><p>In this note, a Wegner estimate for random divergence-type operators that are monotone in the randomness is proven. The proof is based on a recently shown unique continuation estimate for the gradient and the ensuing eigenvalue liftings. The random model which is studied here contains quite general random perturbations, among others, some that have a non-linear dependence on the random parameters.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11040-021-09396-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4756699","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":"Boundary from Bulk Integrability in Three Dimensions: 3D Reflection Maps from Tetrahedron Maps","authors":"Akihito Yoneyama","doi":"10.1007/s11040-021-09393-3","DOIUrl":"10.1007/s11040-021-09393-3","url":null,"abstract":"<div><p>We establish a general method for obtaining set-theoretical solutions to the 3D reflection equation by using known ones to the Zamolodchikov tetrahedron equation, where the former equation was proposed by Isaev and Kulish as a boundary analog of the latter. By applying our method to Sergeev’s electrical solution and a two-component solution associated with the discrete modified KP equation, we obtain new solutions to the 3D reflection equation. Our approach is closely related to a relation between the transition maps of Lusztig’s parametrizations of the totally positive part of <i>S</i><i>L</i><sub>3</sub> and <i>S</i><i>O</i><sub>5</sub>, which is obtained via folding the Dynkin diagram of <i>A</i><sub>3</sub> into one of <i>B</i><sub>2</sub>.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11040-021-09393-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4723758","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}