Resolvent estimates and spectrum of the Dirac operator with periodic potential

Some estimates are given of the norm of the resolvent of the Dirac operator on ann-dimensional torus (n 2) for complex values of the quasimomentum. It is shown that the spectrum of the periodic Dirac operator with potential 3$$ " align="middle" border="0"> , >3, is absolutely continuous.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 1, pp. 3–22, April, 1995.     

High‐energy and smoothness asymptotic expansion of the scattering amplitude for the Dirac equation and application

We obtain an explicit formula for the diagonal singularities of the scattering amplitude for the Dirac equation with short‐range electromagnetic potentials. Using this expansion we uniquely reconstruct an electric potential and magnetic field from the high‐energy limit of the scattering amplitude. Moreover, supposing that the electric potential and magnetic field are asymptotic sums of homogeneous terms we give the unique reconstruction procedure for these asymptotics from the scattering amplitude, known for some energy E. Furthermore, we prove that the set of the averaged scattering solutions to the Dirac equation is dense in the set of all solutions to the Dirac equation that are in L²(Ω), where Ω is any connected bounded open set in with smooth boundary, and we show that if we know an electric potential and a magnetic field for , then the scattering amplitude, given for some energy E, uniquely determines these electric potential and magnetic field everywhere in . Combining this uniqueness result with the reconstruction procedure for the asymptotics of the electric potential and the magnetic field we show that the scattering amplitude, known for some E, uniquely determines a electric potential and a magnetic field, that are asymptotic sums of homogeneous terms, which converges to the electric potential and the magnetic field respectively. Moreover, we discuss the symmetries of the kernel of the scattering matrix, which follow from the parity, charge‐conjugation and time‐reversal transformations for the Dirac operator. Copyright © 2014 John Wiley & Sons, Ltd.     

Exponential Decay of Solution Energy for Equations Associated with Some Operator Models of Mechanics

We consider the equation in a Hilbert space , where A is a uniformly positive self-adjoint operator and B is a dissipative operator. The main result is the proof of a theorem stating the exponential energy decay for solutions of this equation (or the exponential stability of the semigroup associated with the equation) under the additional assumption that B is sectorial and is subordinate to A in the sense of quadratic forms.     

Optimal Domains and Integral Representations of Convolution Operators in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(<Emphasis Type="Italic">G</Emphasis>)

Given , a compact abelian group G and a function , we identify the maximal (i.e. optimal) domain of the convolution operator (as an operator from L^p(G) to itself). This is the largest Banach function space (with order continuous norm) into which L^p(G) is embedded and to which has a continuous extension, still with values in L^p(G). Of course, the optimal domain depends on p and g. Whereas is compact, this is not always so for the extension of to its optimal domain. Several characterizations of precisely when this is the case are presented.     

Exponential Stability of Semigroups Related to Operator Models in Mechanics

In this paper, we consider equations of the form , where is a function with values in the Hilbert space , the operator B is symmetric, and the operator A is uniformly positive and self-adjoint in . The linear operator generating the C ₀-semigroup in the energy space is associated with this equation. We prove that this semigroup is exponentially stable if the operator B is uniformly positive and the operator A dominates B in the sense of quadratic forms.     

Quantum Defect for the Dirac Operator with a Nonanalytic Potential

We evaluate the quantum defects for the continuous and discrete spectra of the radial Dirac operator with the potential V(r) = –A/r + q(r), where A > 0 and .     

On Kirchberg's Inequality for Compact Kähler Manifolds of Even Complex Dimension

In 1986 Kirchberg showed that each eigenvalue of the Dirac operator on a compact Kähler manifold of even complex dimension satisfies the inequality , where by S we denote the scalar curvature. It is conjectured that the manifolds for the limiting case of this inequality are products T ²×N, where T ² is a flat torus and N is the twistor space of a quaternionic Kähler manifold of positive scalar curvature. In 1990 Lichnerowicz announced an affirmative answer for this conjecture (cf. [11]), but his proof seems to work only when assuming that the Ricci tensor is parallel. The aim of this note is to prove several results about manifolds satisfying the limiting case of Kirchberg's inequality and to prove the above conjecture in some particular cases.     

The Gleason theorem for the field of rational numbers and residue fields

Charges taking values in a fieldF and defined on orthomodular partially ordered sets (logics) of all projectors in some finite-dimensional linear space overF are considered. In the cases whereF is the field of rational numbers or a residue field, the Gleason representation , where is a linear operator, is proved.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 584–591, October, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01265.     

Stability of the magnetic Couette-Taylor flow

In this paper we consider the magnetic Couette-Taylor problem, that is, a conducting fluid between two infinite rotating cylinders, subject to a magnetic field parallel to the rotation axis. This configuration admits an equilibrium solution of the form It is shown that this equilibrium is Ljapounov stable under small perturbations in where provided that the parameters a, b, , are small. The methods of proof are a combination of an energy method, based on Bloch space analysis and small data techniques.Received: February 5, 2003; revised: September 29, 2003Dedicated to Prof. H. Amann on the occasion of his 65. birthday     

Disjointness Preserving Operators on Complex Riesz Spaces

It is proven that ifE and F are complex Riesz spaces and ifT is an order bounded disjointness preserving operator fromE intoF , then This fundamental result of M. Meyer is obtained by elementary means using as the main tool the functional calculus derived from the Freudenthal spectral theorem. It is also shown that ifT is an order bounded disjointness preserving operator, a formula of the form holds. It implies a polar decomposition of an order bounded disjointness preserving operator as the product of a Riesz homomorphism and an orthomorphism. Results of P. Meyer-Nieberg in this regard are generalized.     

Dirac equation and spin 1 representations,a connection with symmetries of the Maxwell equations

A unitary operator on the space of spinors that makes it possible to associate each transformation in this space with a transformation in the space of electromagnetic field strengths is found. A connection is established by means of this operator between representations in the space of spinors and the space of field strengths for the Lorentsz, Poincaré, and conformal groups. Unusual symmetries of the Dirac equation are found on this basis. It is noted that the Pauli—Gürsey symmetry operators (without the ⁵ operator) of the Dirac equation withm=0 form the same representation D(1/2, 0)D(0, 1/2) of the O(1, 3) algebra of the Lorentz group as the spin matrices of the standard spinor representation. It is shown that besides the standard (spinor) representation of the Poincaré group, the massless Dirac equation is invariant with respect to two other representations of this group, namely, the vector and tensor representations specified by the generators of the representations D(1/2, 1/2) and D(1, 0) D(0, 0) of the Lorentz group, respectively. Unusual families of representations of the conformal algebra associated with these representations of the group O(1, 3) are investigated. Analogous O(1, 2) and P(1, 2) invariance algebras are established for the Dirac equation withm>0.Institute of Nuclear Research, Ukrainian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 3, pp. 388–406, March, 1992.     

Absolute Continuity of a Two-Dimensional Magnetic Periodic Schrödinger Operator ith Potentials of the Type of Measure Derivative

A two-dimensional magnetic periodic Schrödinger operator with a variable metric is considered. An electric potential is assumed to be a distribution formally given by an expression , where d is a periodic signed measure with a locally finite variation. We also assume that the perturbation generated by the electric potential is strongly subject (in the sense of forms) to the free operator. Under this natural assumption, we prove that the spectrum of the Schrödinger operator is absolutely continuous. Bibliography: 15 titles.     

On the Summing Property of the Sobolev Embedding Operators

We prove that the Sobolev embedding operator S _d,k,p : , where 1/s=1/p-k/d , is (v,1) -absolutely summing for appropriate v > 1 . The result is optimal for s 2 .     

Stability of the magnetic Couette-Taylor flow

In this paper we consider the magnetic Couette-Taylor problem, that is, a conducting fluid between two infinite rotating cylinders, subject to a magnetic field parallel to the rotation axis. This configuration admits an equilibrium solution of the form $ (0,ar + br^{{ - 1}} ,0,0,0,\alpha + \beta \log r). $ (0,ar + br^{{ - 1}} ,0,0,0,\alpha + \beta \log r). It is shown that this equilibrium is Ljapounov stable under small perturbations in $ \mathcal{L}^{2} (\Gamma ), $ \mathcal{L}^{2} (\Gamma ), where $ \Gamma = \{ (r,\varphi ,z)/r_{1} < r < r_{2} ,\varphi \in [0,2\pi ],z \in \mathbb{R}\} , $ \Gamma = \{ (r,\varphi ,z)/r_{1} < r < r_{2} ,\varphi \in [0,2\pi ],z \in \mathbb{R}\} , provided that the parameters a, b, , are small. The methods of proof are a combination of an energy method, based on Bloch space analysis and small data techniques.     

Calogero Operator and Lie Superalgebras

We construct a supersymmetric analogue of the Calogero operator , which depends on the parameter k. This analogue is related to the root system of the Lie superalgebra . It becomes the standard Calogero operator for m = 0 and becomes the operator constructed by Veselov, Chalykh, and Feigin up to changing the variables and the parameter k for m = 1. For k = 1 and 1/2, the operator is the radial part of the second-order Laplace operator for the symmetric superspaces corresponding to the respective pairs . We show that for any m and n, the supersymmetric analogues of the Jack polynomials constructed by Kerov, Okounkov, and Olshanskii are eigenfunctions of the operator . For k = 1 and 1/2, the supersymmetric analogues of the Jack polynomials coincide with the spherical functions on the above superspaces. We also study the algebraic analogue of the Berezin integral.     

Maximum Norm Resolvent Estimates for Elliptic Finite Element Operators

We study a finite element approximation A _h, based on simplicial Lagrange elements, of a second order elliptic operator A under homogeneous Dirichlet boundary conditions in two and three dimensions, where h is thought of as a meshsize. The main result of the paper is a new resolvent estimate for the operator A _h in the L -norm. This estimate is uniform with respect to h for the case with at least quadratic elements. In the case with linear elements, the estimate contains on the right a factor proportional to (log log ), where = 1 or = in two or three dimensions, respectively.This revised version was published online in October 2005 with corrections to the Cover Date.     

Schrödinger Operators on Lattices. The Efimov Effect and Discrete Spectrum Asymptotics

The Hamiltonian of a system of three quantum mechanical particles moving on the three-dimensional lattice and interacting via zero-range attractive potentials is considered. For the two-particle energy operator h(k), with the two-particle quasi-momentum, the existence of a unique positive eigenvalue below the bottom of the continuous spectrum of h(k) for k 0 is proven, provided that h(0) has a zero energy resonance. The location of the essential and discrete spectra of the three-particle discrete Schrödinger operator H(K), being the three-particle quasi-momentum, is studied. The existence of infinitely many eigenvalues of H(0) is proven. It is found that for the number N(0, z) of eigenvalues of H(0) lying below the following limit exists with Moreover, for all sufficiently small nonzero values of the three-particle quasi-momentum K the finiteness of the number of eigenvalues of H(K) below the essential spectrum is established and the asymptotics for the number N(K, 0) of eigenvalues lying below zero is given. Communicated by Gian Michele GrafSubmitted 19/11/03, accepted 08/03/04     

Parameter choice by discrepancy principles for ill-posed problems leading to optimal convergence rates

Schock (Ref. 1) considered a general a posteriori parameter choice strategy for the Tikhonov regularization of the ill-posed operator equationTx=y which provides nearly the optimal rate of convergence if the minimal-norm least-squares solution belongs to the range of the operator (T ^* T) ^v , o<v1. Recently, Nair (Ref. 2) improved the result of Schock and also provided the optimal rate ifv=1. In this note, we further improve the result and show in particular that the optimal rate can be achieved for 1/2v1.The final version of this work was written while M. T. Nair was a Visiting Fellow at the Centre for Mathematics and Its Applications, Australian National University, Canberra, Australia. The work of S. George was supported by a Senior Research Fellowship from CSIR, India.     

Killing spinors on K?hler manifolds

In the paper Kählerian Killing spinors are defined and their basic properties are investigated. Each Kähler manifold that admits a Kählerian Killing spinor is Einstein of odd complex dimension. Kählerian Killing spinors are a special kind of Kählerian twistor spinors. Real Kählerian Killing spinors appear for example, on closed Kähler manifolds with the smallest possible first eigenvalue of the Dirac operator. For the complex projective spaces P ^2l–1 and the complex hyperbolic spaces H ^2l–1 withl>1 the dimension of the space of Kählerian Killing spinors is equal to ( ). It is shown that in complex dimension 3 the complex hyperbolic space H ³ is the only simple connected complete spin Kähler manifold admitting an imaginary Kählerian Killing spinor.     

Explicit finite volume criteria for localization in continuous random media and applications

We give finite volume criteria for localization of quantum or classical waves in continuous random media. We provide explicit conditions, depending on the parameters of the model, for starting the bootstrap multiscale analysis. A simple application to Anderson Hamiltonians on the continuum yields localization at the bottom of the spectrum in an interval of size C for large , where stands for the disorder parameter. A more sophisticated application proves localization for two-dimensional random Schrödinger operators in a constant magnetic field (random Landau Hamiltonians) up to a distance from the Landau levels for large B, where B is the strength of the magnetic field.