Weyl–Titchmarsh theory for discrete symplectic systems with general linear dependence on spectral parameter

In this paper we develop the Weyl–Titchmarsh theory for discrete symplectic systems with general linear dependence on the spectral parameter. We generalize and complete several recent results concerning these systems, which have the spectral parameter only in the second equation. Our new theory includes characterizations of the Weyl discs and Weyl circles, their limiting behaviour, properties of square summable solutions including the analysis of the exact number of linearly independent square summable solutions and limit point/circle criteria. Some illustrative examples are also provided.     

Generalised Weyl–Heisenberg frame operators

We seek to characterise, in simple and unsophisticated terms, frame operators of Weyl–Heisenberg frames. We succeed only partially, using the newly introduced concepts of window operators and tile vertices. However, we are able to completely characterise the frame operator in each of two newly introduced classes: window Weyl–Heisenberg frames (a special class of Weyl–Heisenberg frames) and generalised Weyl–Heisenberg frames (a class more general than Weyl–Heisenberg frames).     

Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes

We give a universal approach to the deformation-obstruction theory of objects of the derived category of coherent sheaves over a smooth projective family. We recover and generalise the obstruction class of Lowen and Lieblich, and prove that it is a product of Atiyah and Kodaira–Spencer classes. This allows us to obtain deformation-invariant virtual cycles on moduli spaces of objects of the derived category on threefolds.     

Inverse scattering theory for one-dimensional Schrödinger operators with steplike finite-gap potentials

We develop direct and inverse scattering theory for one-dimensional Schrödinger operators with steplike potentials which are asymptotically close to different finite-gap potentials on different half-axes. We give a complete characterization of the scattering data, which allows unique solvability of the inverse scattering problem in the class of perturbations with finite second moment.     

Fischer Decomposition and Cauchy Kernel for Dunkl–Dirac operators

In this paper we present a Fischer decomposition for Dirac operator and an explicit construction of a Cauchy kernel for Dunkl-monogenic functions.     

Global wellposedness for a one-dimensional Chern–Simons–Dirac system in Lp

The global wellposedness in L^p(?) for the Chern–Simons–Dirac equation in the 1+1 space and time dimension is discussed. We consider two types of quadratic nonlinearity: the null case and the non-null case. We show the time global wellposedness for the Chern–Simon–Dirac equation in the framework of L^p(?), where 1≤p≤∞ for the null case. For the scaling critical case, p = 1, mass concentration phenomena of the solutions may occur in considering the time global solvability. We invoke the Delgado–Candy estimate which plays a crucial role in preventing concentration phenomena of the global solution. Our method is related to the original work of Candy (2011), who showed the time global wellposedness for the single Dirac equation with cubic nonlinearity in the critical space L²(?).     

On Titchmarsh–Weyl Functions and Eigenfunction Expansions of First-Order Symmetric Systems

We study general (not necessarily Hamiltonian) first-order symmetric system J y′(t)?B(t)y(t) = Δ(t) f(t) on an interval ${\mathcal{I}=[a,b) }$ with the regular endpoint a. It is assumed that the deficiency indices n _±(T _min) of the minimal relation T _min associated with this system in ${L^2_\Delta(\mathcal{I})}$ satisfy ${n_-(T_{\rm min})\leq n_+(T_{\rm min})}$ . We are interested in boundary conditions playing the role similar to that of separated self-adjoint boundary conditions for Hamiltonian systems. Instead we define λ-depending boundary conditions with Nevanlinna type spectral parameter τ = τ(λ) at the singular endpoint b. With this boundary value problem we associate the matrix m-function m(·) of the size ${N_\Sigma = {\rm dim} {\rm ker} (iJ+I)}$ . Its role is similar to that of the Titchmarsh–Weyl coefficient for the Hamiltonian system. In turn, it allows one to define the Fourier transform ${V: L^2_\Delta(\mathcal{I}) \to L^2(\Sigma)}$ where Σ (·) is a spectral matrix function of m(·). If V is an isometry, then the (exit space) self-adjoint extension ${\tilde{T}}$ of T _min induced by the boundary problem is unitarily equivalent to the multiplication operator in L ²(Σ). Hence the multiplicity of spectrum of ${\tilde{T}}$ does not exceed N _Σ. We also parameterize a set of spectral functions Σ(·) by means of the set of boundary parameters τ. Similar parameterizations for various classes of boundary value problems have earlier been obtained by Kac and Krein, Fulton, Hinton and Shaw, and others.     

Weyl’s theorem for totally hereditarily normaloid operators

A Banach space operatorT ∈B(χ) is said to behereditarily normaloid, denotedT ∈ ℋN, if every part ofT is normaloid;T ∈ ℋN istotally hereditarily normaloid, denotedT ∈ ℑHN, if every invertible part ofT is also normaloid. Class ℑHN is large; it contains a number of the commonly considered classes of operators. The operatorT isalgebraically totally hereditarily normaloid, denotedT ∈a — ℑHN, both non-constant polynomialp such thatp(T) ∈ ℑHN. For operatorsT ∈a − ℑHN, bothT andT* satisfy Weyl’s theorem; if also either ind(T−μ)≥0 or ind(T−μ)≤0 for all complexμ such thatT−μ is Fredholm, thenf(T) andf(T*) satisfy Weyl’s theorem for all analytic functionsf ∈ ℋ(σ(T)). For operatorsT ∈a — ℑHN such thatT has SVEP,T* satisfiesa-Weyl’s theorem.     

Titchmarsh–Weyl Formula for the Spectral Density of a Class of Jacobi Matrices in the Critical Case

Functional Analysis and Its Applications - We consider a class of Jacobi matrices with unbounded entries in the so-called critical (double root, Jordan block) case. We prove a formula which relates...     

Spectral theory of Sturm–Liouville difference operators

We present several classes of explicit self-adjoint Sturm–Liouville difference operators with either a non-Hermitian leading coefficient function, or a non-Hermitian potential function, or a non-definite weight function, or a non-self-adjoint boundary condition. These examples are obtained using a general procedure for constructing difference operators realizing discrete Sturm–Liouville problems, and the minimum conditions for such difference operators to be self-adjoint with respect to a natural quadratic form. It is shown that a discrete Sturm–Liouville problem admits a difference operator realization if and only if it does not have all complex numbers as eigenvalues. Spectral properties of self-adjoint Sturm–Liouville difference operators are studied. In particular, several eigenvalue comparison results are proved.     

Some inequalities for Kodaira–Iitaka dimension on subvarieties

In this paper an investigation is presented of the relationship between the Kodaira–Iitaka dimension of a line bundle on an ambient normal variety and the Kodaira–Iitaka dimension of related line bundles on normal subvarieties. The results that are found extend in some directions those that were presented by Peternell et al. (Int J Math 10:1065–1079, 1999) Those results in turn generalized the well known easy addition theorem for Kodaira–Iitaka dimension on an algebraic fiber space.     

Numerical computation and analysis of the Titchmarsh–Weyl mα(λ) function for some simple potentials

This article is concerned with the Titchmarsh–Weyl m_α(λ) function for the differential equation d²y/dx²+[λ−q(x)]y=0. The test potential q(x)=x², for which the relevant m_α(λ) functions are meromorphic, having simple poles at the points λ=4k+1 and λ=4k+3, is studied in detail. We are able to calculate the m_α(λ) function both far from and near to these poles. The calculation is then extended to several other potentials, some of which do not have analytical solutions. Numerical data are given for the Titchmarsh–Weyl m_α(λ) function for these potentials to illustrate the computational effectiveness of the method used.     

Regularity and index theory for Dirac–Schrödinger systems with Lipschitz coefficients

Dirac–Schrödinger systems play a central role when modeling Dirac bundles and Dirac–Schrödinger operators near the boundary, along ends or near other singularities of Riemannian manifolds. In this article we develop the Fredholm theory of Dirac–Schrödinger systems with Lipschitz coefficients.     

Weyl formula for multi-quasi-elliptic operators. of Schrödinger type

In this work we consider a general class of Schr?dinger type operators, associated with multi-quasi-elliptic symbols. We give a precise estimate of the remainder of the so-called Weyl asymptotic formula for the eigenvalues of these operators. In order to reach our aim, we use the Weyl–H?rmander calculus, with locally temperate metrics and weights, and interpolation techniques. Received: February 14, 2000; in final form: October 29, 2000?Published online: July 13, 2001     

Schur–Weyl duality for higher levels

We extend Schur–Weyl duality to an arbitrary level l ≥ 1, level one recovering the classical duality between the symmetric and general linear groups. In general, the symmetric group is replaced by the degenerate cyclotomic Hecke algebra over parametrized by a dominant weight of level l for the root system of type A_∞. As an application, we prove that the degenerate analogue of the quasi-hereditary cover of the cyclotomic Hecke algebra constructed by Dipper, James and Mathas is Morita equivalent to certain blocks of parabolic category for the general linear Lie algebra.      

Singular parameters for the Birman–Murakami–Wenzl algebra

In this paper, we classify the singular parameters for the Birman–Murakami–Wenzl algebra over an arbitrary field. Equivalently, we give a criterion for the Birman–Murakami–Wenzl algebra being Morita equivalent to the direct sum of the Hecke algebras associated to certain symmetric groups.     

Muckenhoupt–Wheeden conjectures for sparse operators

We provide an explicit example of a pair of weights and a dyadic sparse operator for which the Hardy–Littlewood maximal function is bounded from \(L^p(v)\) to \(L^p(u)\) and from \(L^{p'}(u^{1-p'})\) to \(L^{p'}(v^{1-p'})\) while the sparse operator is not bounded on the same spaces. Our construction also provides an example of a single weight for which the weak-type endpoint does not hold for sparse operators.