A simple derivation of Borland's locality result

A new proof of Borland's localization theorem is given, which is simpler, less rigorous, but gives more quantitative results.     

On a Generalization of the Trèves Criterion for the First Integrals of the KdV Hierarchy to Higher GD Hierarchies

First, a new proof of the necessity of the Trèves criterion for a differential polynomialto be a first integral of the KdV hierarchy is given. This proof is much shorter than the original one and admits generalizations. Second, a similar criterion is suggested and its necessity proven for the next of the GD hierarchies which is generated by the third-order linear operator.     

The spectral property of time delay

This paper discusses the spectral property of time delay. If one considers the scattering by a potential v, then the spectral property is the statement that the trace of the time-delay operator is proportional to the change in state density produced by the interaction v. We give a new and elementary proof of this result. As a consequence it is shown that Levinson's theorem is implied by the spectral property.     

The Shannon-McMillan Theorem for AF C*-systems

We give a new proof of quantum Shannon-McMillan theorem, extending it to AF C*-systems. Our proof is based on the variational principle, instead of the classical Shannon-McMillan theorem.     

On the Structure of Verma Modules over Virasoro and Neveu-Schwarz Algebras

In the paper we present a different proof of the theorem of B. L. Feigin and D. B. Fuchs about the structure of Verma modules over Virasoro algebra. We state some new results about the structure of Verma modules over Neveu-Schwarz. The proof has two advantages: First, it is simpler in the most interesting cases (for example in the so called minimal models), second, it can be generalized for Neveu-Schwarz algebra for some class of Verma modules. Received: 15 December 1995 / Accepted: 4 March 1996     

A Poisson–Lie Framework for Rational Reductions of the KP Hierarchy

We give a simple proof of I. Krichever's theorem on rational reductions of the Kadomtsev–Petviashvili hierarchy by using the Poisson–Lie structure on the group of pseudo-differential symbols.     

Generalized no-broadcasting theorem

We prove a generalized version of the no-broadcasting theorem, applicable to essentially any nonclassical finite-dimensional probabilistic model satisfying a no-signaling criterion, including ones with "superquantum" correlations. A strengthened version of the quantum no-broadcasting theorem follows, and its proof is significantly simpler than existing proofs of the no-broadcasting theorem.     

Tanaka Theorem for Inelastic Maxwell Models

We show that the Euclidean Wasserstein distance is contractive for inelastic homogeneous Boltzmann kinetic equations in the Maxwellian approximation and its associated Kac-like caricature. This property is as a generalization of the Tanaka theorem to inelastic interactions. Even in the elastic classical Boltzmann equation, we give a simpler proof of the Tanaka theorem than the ones in [29, 31]. Consequences are drawn on the asymptotic behavior of solutions in terms only of the Euclidean Wasserstein distance.     

Anderson localization for one- and quasi-one-dimensional systems

We prove almost-sure exponential localization of all the eigenfunctions and nondegeneracy of the spectrum for random discrete Schrödinger operators on one- and quasi-one-dimensional lattices. This paper provides a much simpler proof of these results than previous approaches and extends to a much wider class of systems; we remark in particular that the singular continuous spectrum observed in some quasiperiodic systems disappears under arbitrarily small local perturbations of the potential. Our results allow us to prove that, e.g., for strong disorder, the smallest positive Lyapunov exponent of some products of random matrices does not vanish as the size of the matrices increases to infinity.     

A complex minkowski space approach to twistors

This paper is basically a review of known results in twistor theory. Its value is intended to lie in the connections presented between twistor concepts and structures in complex Minkowski space. The relationship of twistor theory to complex null infinity and a new proof of the Kerr theorem are presented; these results are to some extent original.     

Reflectingness and closed timelike curves in space-time

It is shown by Clarke and Felice (J Phys A Math Gen 15:2415–2417, 1982) that under certain conditions a stationary space-time will produce a closed timelike curve through each point of the space-time. Here a simpler proof and a different version of their theorem are provided.     

A proof of the Nielsen-Ninomiya theorem

The Nielsen-Ninomiya theorem asserts the impossibility of constructing lattice models of non-selfinteracting chiral fermions. A new proof is given here. This proof fills a technical gap in the two proofs presented by the authors of the theorem. It also serves as prelude to an investigation of the chiral properties of the general lattice model.     

Multiple-conjugation expansion in distortion elimination by phase conjugation

The multiple-conjugation expansion results from a method of phase-conjugate scattering analysis that is convenient in connection with strong aberrators and weak phase conjugators. Based on this expansion, a new proof is obtained of a recently established theorem according to which a complete cancellation of distortion is possible in the absence of gains or losses. The proof makes use of the reciprocity property and of the generalized optical theorem of potential scattering.     

Brillouin's theorem and the Hellmann-Feynman theorem for Hartree-Fock wave functions

It is shown that there is an intimate relation between the Hellmann-Feynman theorem and Brillouin's theorem. A more general form of Brillouin's theorem is provided, which applies to excited states of arbitrary symmetry and multiplicity. This new form leads to a simple proof of the Hellmann-Feynman theorem. This theorem is valid when all the orbitals that occur in the wave function are determined by a complete, and not a partial, variational procedure. Arguing in the opposite direction it is shown that the complete satisfaction of the generalized Brillouin's theorem provides an alternative scheme for obtaining the Hartree-Fock orbitals.     

The Hamiltonian structure associated to evolution-type Lagrangians

It is demonstrated that, for a certain class of Lagrangians, which includes those for the Korteweg-de Vries (KdV) hierarchy, the Hamiltonian structure provided by the Hamilton-Cartan formalism is precisely the one discovered by Gardner for the KdV equation. A simple geometric relation between the Cartan 2-forms for this class of Lagrangians and the Cartan 1-forms for the associated stationary problems is given. This relation provides a new proof of the theorem of Bogoyavlenski-Novikov and Gel'fand-Dikii on the integrability of the stationary Korteweg-de Vries equations.Research supported by the Natural Sciences and Engineering Research Council of Canada.     

K-causal structure of space-time in general relativity

Using K-causal relation introduced by Sorkin and Woolgar [1], we generalize results of Garcia-Parrado and Senovilla [2,3] on causal maps. We also introduce causality conditions with respect to K-causality which are analogous to those in classical causality theory and prove their inter-relationships. We introduce a new causality condition following the work of Bombelli and Noldus [4] and show that this condition lies in between global hyperbolicity and causal simplicity. This approach is simpler and more general as compared to traditional causal approach [5,6] and it has been used by Penrose et al [7] in giving a new proof of positivity of mass theorem. C ⁰-space-time structures arise in many mathematical and physical situations like conical singularities, discontinuous matter distributions, phenomena of topology-change in quantum field theory etc.      

The Poincaré-Lyapounov-Nekhoroshev Theorem

We give a detailed and mainly geometric proof of a theorem by N. N. Nekhoroshev for hamiltonian systems in n degrees of freedom with k constants of motion in involution, where 1≤k≤n. This state's persistence of k-dimensional invariant tori, and local existence of partial action-angle coordinates, under suitable nondegeneracy conditions. Thus it admits as special cases the Poincaré-Lyapounov theorem (corresponding to k=1) and the Liouville-Arnold one (corresponding to k=n) and interpolates between them. The crucial tool for the proof is a generalization of the Poincaré map, also introduced by Nekhoroshev.     

A new proof of the propagation theorem forN-body quantum systems

A new proof of I. Sigal's and A. Soffer's propagation theorem is given. This theorem describes a large class of operators which are Kato-smooth with respect to anN-body Schrödinger operator.On leave from Division of Mathematical Methods in Physics, Warsaw University, Hoza 74, PL-00682 Warsaw, Poland     

The Density of Surface States as the Total Time Delay

For a scattering problem of tight-binding Bloch electrons by a weak random surface potential, a generalized Levinson theorem is put forward showing the equality of the total density of surface states and the density of the total time delay. The proof uses explicit formulas for the wave operators in the new rescaled energy and interaction representation, as well as an index theorem for adequate associated operator algebras.