We prove the Morse relations for all geodesics connecting two non-conjugate points on a class of globally hyperbolic Lorentzian manifolds. We overcome the difficulties coming from the fact that the Morse index of every geodesic is infinite, and from the lack of the Palais-Smale condition, by using the Morse complex approach. Introduction Let M be a smooth connected manifold without boundary of ...

We introduce the concept of group state transfer on graphs, summarize its relationship to other concepts in theory quantum walks, set up a basic theory, and discuss examples. Let X be graph with adjacency matrix A consider walks vertex V(X) governed by continuous time-dependent unitary transition operator U(t)=exp(itA). For S,T⊆V(X), we say admits “group transfer” from S T at time τ if submatri...

We consider the long standing open question on whether one can actually compute spectra and pseudospectra of arbitrary (possibly non-self-adjoint) Schrödinger operators.We conclude that the answer is affirmative for “almost all” such operators, meaning that the operators must satisfy rather weak conditions such as the spectrum cannot be empty nor the whole plane. We include algorithms for the g...

The following lemma follows the standard paradigm of recurrence results in ergodic theory: given a topological space X which satisfies a suitable ’smallness’ condition (e.g. compactness, μ(X) < ∞), and a transformation T : X → X, there exist points of X which satisfy a certain ’almost-periodicity’ condition under the action of T . Lemma 1. Poincare Recurrence Lemma If (X,β, μ, T ) is an m.p.s. ...

We discuss conditions such that strong stability and strong asymptotic compactness of a (discrete or continuous) semiflow defined on a subset in the positive cone of an ordered Banach space is preserved under asymptotic domination. This is used to show that on a Banach lattice with order continuous norm strong stability and almost periodicity of a (discrete or strongly continuous) semigroup of ...

We consider the space of weakly almost periodic functions on a transformation semigroup (S, X , ?) and show that if X is a locally compact noncompact uniform space, and ? is a separately continuous, separately proper, and equicontinuous action of S on X, then every continuous function on X, vanishing at infinity is weakly almost periodic. We also use a number of diverse examples to show ...