Extended States For The Schrödinger Operator With Quasi Periodic Potential In Dimension Two

Author : Yulia Karpeshina,Roman Shterenberg
Publisher : American Mathematical Soc.
Page : 139 pages
File Size : 53,9 Mb
Release : 2019-04-10
Category : Schrödinger equation
ISBN : 9781470435431

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Extended States for the Schrödinger Operator with Quasi-Periodic Potential in Dimension Two by Yulia Karpeshina,Roman Shterenberg Pdf

The authors consider a Schrödinger operator H=−Δ+V(x⃗ ) in dimension two with a quasi-periodic potential V(x⃗ ). They prove that the absolutely continuous spectrum of H contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves ei⟨ϰ⃗ ,x⃗ ⟩ in the high energy region. Second, the isoenergetic curves in the space of momenta ϰ⃗ corresponding to these eigenfunctions have the form of slightly distorted circles with holes (Cantor type structure). A new method of multiscale analysis in the momentum space is developed to prove these results. The result is based on a previous paper on the quasiperiodic polyharmonic operator (−Δ)l+V(x⃗ ), l>1. Here the authors address technical complications arising in the case l=1. However, this text is self-contained and can be read without familiarity with the previous paper.

Author : Julija E. Karpešina,Roman Shterenberg
Publisher : Unknown
Page : 128 pages
File Size : 41,8 Mb
Release : 2019
Category : Electronic
ISBN : 1470450690

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Extended States for the Schrödinger Operator with Quasi-periodic Potential in Dimension Two by Julija E. Karpešina,Roman Shterenberg Pdf

Author : Raúl E. Curto,In Sung Hwang,Woo Young Lee
Publisher : American Mathematical Soc.
Page : 100 pages
File Size : 52,5 Mb
Release : 2019-09-05
Category : Functions of bounded variation
ISBN : 9781470436247

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Matrix Functions of Bounded Type: An Interplay Between Function Theory and Operator Theory by Raúl E. Curto,In Sung Hwang,Woo Young Lee Pdf

In this paper, the authors study matrix functions of bounded type from the viewpoint of describing an interplay between function theory and operator theory. They first establish a criterion on the coprime-ness of two singular inner functions and obtain several properties of the Douglas-Shapiro-Shields factorizations of matrix functions of bounded type. They propose a new notion of tensored-scalar singularity, and then answer questions on Hankel operators with matrix-valued bounded type symbols. They also examine an interpolation problem related to a certain functional equation on matrix functions of bounded type; this can be seen as an extension of the classical Hermite-Fejér Interpolation Problem for matrix rational functions. The authors then extend the H∞-functional calculus to an H∞¯¯¯¯¯¯¯¯¯+H∞-functional calculus for the compressions of the shift. Next, the authors consider the subnormality of Toeplitz operators with matrix-valued bounded type symbols and, in particular, the matrix-valued version of Halmos's Problem 5 and then establish a matrix-valued version of Abrahamse's Theorem. They also solve a subnormal Toeplitz completion problem of 2×2 partial block Toeplitz matrices. Further, they establish a characterization of hyponormal Toeplitz pairs with matrix-valued bounded type symbols and then derive rank formulae for the self-commutators of hyponormal Toeplitz pairs.

Author : Charles Collot,Pierre Raphaël,Jeremie Szeftel
Publisher : American Mathematical Soc.
Page : 93 pages
File Size : 51,7 Mb
Release : 2019-09-05
Category : Electronic
ISBN : 9781470436261

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On the Stability of Type I Blow Up for the Energy Super Critical Heat Equation by Charles Collot,Pierre Raphaël,Jeremie Szeftel Pdf

The authors consider the energy super critical semilinear heat equation The authors first revisit the construction of radially symmetric self similar solutions performed through an ode approach and propose a bifurcation type argument which allows for a sharp control of the spectrum of the corresponding linearized operator in suitable weighted spaces. They then show how the sole knowledge of this spectral gap in weighted spaces implies the finite codimensional nonradial stability of these solutions for smooth well localized initial data using energy bounds. The whole scheme draws a route map for the derivation of the existence and stability of self-similar blow up in nonradial energy super critical settings.

Author : J. I. Hall
Publisher : American Mathematical Soc.
Page : 186 pages
File Size : 53,8 Mb
Release : 2019-09-05
Category : Categories (Mathematics)
ISBN : 9781470436223

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Moufang Loops and Groups with Triality are Essentially the Same Thing by J. I. Hall Pdf

In 1925 Élie Cartan introduced the principal of triality specifically for the Lie groups of type D4, and in 1935 Ruth Moufang initiated the study of Moufang loops. The observation of the title in 1978 was made by Stephen Doro, who was in turn motivated by the work of George Glauberman from 1968. Here the author makes the statement precise in a categorical context. In fact the most obvious categories of Moufang loops and groups with triality are not equivalent, hence the need for the word “essentially.”

Author : Ines Kath,Martin Olbrich
Publisher : American Mathematical Soc.
Page : 84 pages
File Size : 46,6 Mb
Release : 2020-02-13
Category : Education
ISBN : 9781470441036

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Compact Quotients of Cahen-Wallach Spaces by Ines Kath,Martin Olbrich Pdf

Indecomposable symmetric Lorentzian manifolds of non-constant curvature are called Cahen-Wallach spaces. Their isometry classes are described by continuous families of real parameters. The authors derive necessary and sufficient conditions for the existence of compact quotients of Cahen-Wallach spaces in terms of these parameters.

Author : Oskari Ajanki,László Erdős,Torben Krüger
Publisher : American Mathematical Soc.
Page : 133 pages
File Size : 49,6 Mb
Release : 2019-12-02
Category : Education
ISBN : 9781470436834

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Quadratic Vector Equations on Complex Upper Half-Plane by Oskari Ajanki,László Erdős,Torben Krüger Pdf

The authors consider the nonlinear equation −1m=z+Sm with a parameter z in the complex upper half plane H, where S is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in H is unique and its z-dependence is conveniently described as the Stieltjes transforms of a family of measures v on R. In a previous paper the authors qualitatively identified the possible singular behaviors of v: under suitable conditions on S we showed that in the density of v only algebraic singularities of degree two or three may occur. In this paper the authors give a comprehensive analysis of these singularities with uniform quantitative controls. They also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the authors' companion paper they present a complete stability analysis of the equation for any z∈H, including the vicinity of the singularities.

Author : Dominic Joyce
Publisher : American Mathematical Soc.
Page : 139 pages
File Size : 45,5 Mb
Release : 2019-09-05
Category : Electronic
ISBN : 9781470436452

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Algebraic Geometry over C∞-Rings by Dominic Joyce Pdf

If X is a manifold then the R-algebra C∞(X) of smooth functions c:X→R is a C∞-ring. That is, for each smooth function f:Rn→R there is an n-fold operation Φf:C∞(X)n→C∞(X) acting by Φf:(c1,…,cn)↦f(c1,…,cn), and these operations Φf satisfy many natural identities. Thus, C∞(X) actually has a far richer structure than the obvious R-algebra structure. The author explains the foundations of a version of algebraic geometry in which rings or algebras are replaced by C∞-rings. As schemes are the basic objects in algebraic geometry, the new basic objects are C∞-schemes, a category of geometric objects which generalize manifolds and whose morphisms generalize smooth maps. The author also studies quasicoherent sheaves on C∞-schemes, and C∞-stacks, in particular Deligne-Mumford C∞-stacks, a 2-category of geometric objects generalizing orbifolds. Many of these ideas are not new: C∞-rings and C∞ -schemes have long been part of synthetic differential geometry. But the author develops them in new directions. In earlier publications, the author used these tools to define d-manifolds and d-orbifolds, “derived” versions of manifolds and orbifolds related to Spivak's “derived manifolds”.

Author : Elizabeth Milićević,Petra Schwer,Anne Thomas
Publisher : American Mathematical Soc.
Page : 101 pages
File Size : 47,5 Mb
Release : 2019-12-02
Category : Education
ISBN : 9781470436766

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Dimensions of Affine Deligne–Lusztig Varieties: A New Approach Via Labeled Folded Alcove Walks and Root Operators by Elizabeth Milićević,Petra Schwer,Anne Thomas Pdf

Let G be a reductive group over the field F=k((t)), where k is an algebraic closure of a finite field, and let W be the (extended) affine Weyl group of G. The associated affine Deligne–Lusztig varieties Xx(b), which are indexed by elements b∈G(F) and x∈W, were introduced by Rapoport. Basic questions about the varieties Xx(b) which have remained largely open include when they are nonempty, and if nonempty, their dimension. The authors use techniques inspired by geometric group theory and combinatorial representation theory to address these questions in the case that b is a pure translation, and so prove much of a sharpened version of a conjecture of Görtz, Haines, Kottwitz, and Reuman. The authors' approach is constructive and type-free, sheds new light on the reasons for existing results in the case that b is basic, and reveals new patterns. Since they work only in the standard apartment of the building for G(F), their results also hold in the p-adic context, where they formulate a definition of the dimension of a p-adic Deligne–Lusztig set. The authors present two immediate applications of their main results, to class polynomials of affine Hecke algebras and to affine reflection length.

Author : Oktay Veliev
Publisher : Springer Nature
Page : 420 pages
File Size : 42,6 Mb
Release : 2024-06-30
Category : Electronic
ISBN : 9783031490354

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Multidimensional Periodic Schrödinger Operator by Oktay Veliev Pdf

Author : Chen Wan
Publisher : American Mathematical Soc.
Page : 90 pages
File Size : 54,5 Mb
Release : 2019-12-02
Category : Education
ISBN : 9781470436865

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A Local Relative Trace Formula for the Ginzburg-Rallis Model: The Geometric Side by Chen Wan Pdf

Following the method developed by Waldspurger and Beuzart-Plessis in their proofs of the local Gan-Gross-Prasad conjecture, the author is able to prove the geometric side of a local relative trace formula for the Ginzburg-Rallis model. Then by applying such formula, the author proves a multiplicity formula of the Ginzburg-Rallis model for the supercuspidal representations. Using that multiplicity formula, the author proves the multiplicity one theorem for the Ginzburg-Rallis model over Vogan packets in the supercuspidal case.

Author : Tvrtko Tadić
Publisher : American Mathematical Soc.
Page : 170 pages
File Size : 46,9 Mb
Release : 2019-12-02
Category : Education
ISBN : 9781470436858

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Time-Like Graphical Models by Tvrtko Tadić Pdf

The author studies continuous processes indexed by a special family of graphs. Processes indexed by vertices of graphs are known as probabilistic graphical models. In 2011, Burdzy and Pal proposed a continuous version of graphical models indexed by graphs with an embedded time structure— so-called time-like graphs. The author extends the notion of time-like graphs and finds properties of processes indexed by them. In particular, the author solves the conjecture of uniqueness of the distribution for the process indexed by graphs with infinite number of vertices. The author provides a new result showing the stochastic heat equation as a limit of the sequence of natural Brownian motions on time-like graphs. In addition, the author's treatment of time-like graphical models reveals connections to Markov random fields, martingales indexed by directed sets and branching Markov processes.

Author : Sergio Albeverio,Raphael Höegh-Krohn
Publisher : Cambridge University Press
Page : 568 pages
File Size : 46,7 Mb
Release : 1992-06-26
Category : Mathematics
ISBN : 0521419301

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Ideas and Methods in Quantum and Statistical Physics: Volume 2 by Sergio Albeverio,Raphael Höegh-Krohn Pdf

A collection of essays by many of the closest co-workers of Raphael Høegh-Krohn.

Author : Fritz Gesztesy
Publisher : American Mathematical Soc.
Page : 472 pages
File Size : 52,6 Mb
Release : 2007
Category : Mathematics
ISBN : 0821842498

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Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday by Fritz Gesztesy Pdf

This Festschrift had its origins in a conference called SimonFest held at Caltech, March 27-31, 2006, to honor Barry Simon's 60th birthday. It is not a proceedings volume in the usual sense since the emphasis of the majority of the contributions is on reviews of the state of the art of certain fields, with particular focus on recent developments and open problems. The bulk of the articles in this Festschrift are of this survey form, and a few review Simon's contributions to aparticular area. Part 1 contains surveys in the areas of Quantum Field Theory, Statistical Mechanics, Nonrelativistic Two-Body and $N$-Body Quantum Systems, Resonances, Quantum Mechanics with Electric and Magnetic Fields, and the Semiclassical Limit. Part 2 contains surveys in the areas of Random andErgodic Schrodinger Operators, Singular Continuous Spectrum, Orthogonal Polynomials, and Inverse Spectral Theory. In several cases, this collection of surveys portrays both the history of a subject and its current state of the art. A substantial part of the contributions to this Festschrift are survey articles on the state of the art of certain areas with special emphasis on open problems. This will benefit graduate students as well as researchers who want to get a quick, yet comprehensiveintroduction into an area covered in this volume.

Author : Jean-Dominique Deuschel
Publisher : Springer Science & Business Media
Page : 470 pages
File Size : 54,6 Mb
Release : 2005-01-12
Category : Computers
ISBN : 3540230335

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Interacting Stochastic Systems by Jean-Dominique Deuschel Pdf

The Research Network on "Interacting stochastic systems of high complexity" set up by the German Research Foundation aimed at exploring and developing connections between research in infinite-dimensional stochastic analysis, statistical physics, spatial population models from mathematical biology, complex models of financial markets or of stochastic models interacting with other sciences. This book presents a structured collection of papers on the core topics, written at the close of the 6-year programme by the research groups who took part in it. The structure chosen highlights the interweaving of certain themes and certain interconnections discovered through the joint work. This yields a reference work on results and methods that will be useful to all who work between applied probability and the physical, economic, and life sciences.