George Weiss

Bio: George Weiss is an academic researcher. The author has contributed to research in topics: Eigenfunction & Spectral theory of ordinary differential equations. The author has an hindex of 1, co-authored 1 publications receiving 1294 citations.

Analytic Number Theory

Abstract: Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large sieve Exponential sums The Dirichlet polynomials Zero-density estimates Sums over finite fields Character sums Sums over primes Holomorphic modular forms Spectral theory of automorphic forms Sums of Kloosterman sums Primes in arithmetic progressions The least prime in an arithmetic progression The Goldbach problem The circle method Equidistribution Imaginary quadratic fields Effective bounds for the class number The critical zeros of the Riemann zeta function The spacing of zeros of the Riemann zeta-function Central values of $L$-functions Bibliography Index.

Trace ideals and their applications

Abstract: Preliminaries Calkin's theory of operator ideals and symmetrically normed ideals convergence theorems for $\mathcal J_P$ Trace, determinant, and Lidskii's theorem $f(x)g(-i abla)$ Fredholm theory Scattering with a trace condition Bound state problems Lots of inequalities Regularized determinants and renormalization in quantum field theory An introduction to the theory on a Banach space Borel transforms, the Krein spectral shift, and all that Spectral theory of rank one perturbations Localization in the Anderson model following Aizenman-Molchanov The Xi function Addenda Bibliography Index.

Theory of dipole-exchange spin wave spectrum for ferromagnetic films with mixed exchange boundary conditions

Abstract: A theory is developed for dispersion characteristics of spin waves in ferromagnetic films taking into account both the dipole-dipole and the exchange interactions. An arbitrary orientation of the internal bias magnetic field is assumed. The general case of mixing exchange boundary conditions (surface spin pinning conditions) is considered. The simple analytical dispersion equations are obtained using the classical perturbation theory. The modification of the spin wave spectrum due to surface anisotropy (or pinning conditions) is discussed.

Floquet Theory for Partial Differential Equations

Abstract: 1. Holomorphic Fredholm Operator Functions.- 1.1. Lifting and open mapping theorems.- 1.2. Some classes of linear operators.- 1.3. Banach vector bundles.- 1.4. Fredholm operators that depend continuously on a parameter.- 1.5. Some information from complex analysis.- A. Interpolation of entire functions of finite order.- B. Some information from the complex analysis in several variables.- C. Some problems of infinite-dimensional complex analysis.- 1.6. Fredholm operators that depend holomorphically on a parameter.- 1.7. Image and cokernel of a Fredholm morphism in spaces of holomorphic sections.- 1.8. Image and cokernel of a Fredholm morphism in spaces of holomorphic sections with bounds.- 1.9. Comments and references.- 2. Spaces, Operators and Transforms.- 2.1. Basic spaces and operators.- 2.2. Fourier transform on the group of periods.- 2.3. Comments and references.- 3. Floquet Theory for Hypoelliptic Equations and Systems in the Whole Space.- 3.1. Floquet - Bloch solutions. Quasimomentums and Floquet exponents.- 3.2. Floquet expansion of solutions of exponential growth.- 3.3. Completeness of Floquet solutions in a class of solutions of faster growth.- 3.4. Other classes of equations.- A. Elliptic systems.- B. Hypoelliptic equations and systems.- C. Pseudodifferential equations.- D. Smoothness of coefficients.- 3.5. Comments and references.- 4. Properties of Solutions of Periodic Equations.- 4.1. Distribution of quasimomentums and decreasing solutions.- 4.2. Solvability of non-homogeneous equations.- 4.3. Bloch property.- 4.4. Quasimomentum dispersion relation. Bloch variety.- 4.5. Some problems of spectral theory.- 4.6. Positive solutions.- 4.7. Comments and references.- 5. Evolution Equations.- 5.1. Abstract hypoelliptic evolution equations on the whole axis.- 5.2. Some degenerate cases.- 5.3. Cauchy problem for abstract parabolic equations.- 5.4. Elliptic and parabolic boundary value problems in a cylinder.- A. Elliptic problems.- B. Parabolic problems.- 5.5. Comments and references.- 6. Other Classes of Problems.- 6.1. Equations with deviating arguments.- 6.2. Equations with coefficients that do not depend on some arguments.- 6.3. Invariant differential equations on Riemannian symmetric spaces of non-compact type.- 6.4. Comments and references.- Index of symbols.

Non-linear equations of korteweg-de vries type, finite-zone linear operators, and

Abstract: The basic content of this survey is an exposition of a recently developed method of constructing a broad class of periodic and almost-periodic solutions of non-linear equations of mathematical physics to which (in the rapidly decreasing case) the method of the inverse scattering problem is applicable. These solutions are such that the spectrum of their associated linear differential operators has a finite-zone structure. The set of linear operators with a given finite-zone spectrum is the Jacobian variety of a Riemann surface, which is determined by the structure of the spectrum. We give an explicit solution of the corresponding non-linear equations in the language of the theory of Abelian functions.