correspondence between berry s phase and lewis s phase for quadratic hamiltonians the authors show the existence of a relation between berry s phase and lewis s phase for the case of quadratic hamiltonians

a sampling theorem on homogeneous manifolds we consider a generalization of entire functions of spherical exponential type and lagrangian splines on manifolds an analog of the paleywiener theorem is given we also show that every spectral entire functionl on a manifold is uniquely determined by its values on some discrete sets of points the main result of the paper is a formula for reconstruction of spectral entire functions from their values on discrete sets using lagrangian splines

a m isometric operators in semi hilbertian spaces abstract in this work the concept of m isometry on a hilbert space are generalized when an additional semi inner product is considered this new concept is described by means of oblique projections

tensor product of n isometries abstract a hilbert space operator a is an m isometry if k 0 m 1 k m k a m k a m k 0 we prove that the tensor product of an m isometry a with an n isometry b is an m n 1 isometry this proves a conjecture of botelho and jamison 5 remark 5 1

banach strong novikov conjecture for polynomially contractible groups abstract we prove the banach strong novikov conjecture for groups having polynomially bounded higher order combinatorial functions this includes all automatic groups

ideal structure of uniform roe algebras of coarse spaces abstract let c u x e be the uniform roe algebra of a coarse space x e with uniformly locally finite coarse structure by a controlled truncation technique we show that the controlled propagation operators in an ideal i of c u x e are exactly the controlled truncations of elements in i it follows that the lattice of the ideals of the uniform roe algebra c u x e in which controlled propagation operators are dense the lattice of the invariant open subsets in the unit space of the groupoid g x introduced by skandalis tu and yu the lattice of the ideals of the coarse structure e and the lattice of the ideals of the coarse space x are mutually isomorphic these lattices also give rise to a type of classification for the ideals of c u x e

gauge transformation approach to the exact solution of a generalized harmonic oscillator the exact solution and the invariant operator of a generalized harmonic oscillator are obtained by performing three consecutive gauge transformations on the time dependent schrodinger equation in contrast to previous results they depend only on solutions of a linear differential equation for the associated classical harmonic oscillator non adiabatic and adiabatic berry phases for the system are calculated on the basis of the exact solution

a c algebra of geometric operators on self similar cw complexes novikov shubin and l2 betti numbers abstract a class of cw complexes called self similar complexes is introduced together with c algebras a j of operators endowed with a finite trace acting on square summable cellular j chains since the laplacian j belongs to a j l 2 betti numbers and novikov shubin numbers are defined for such complexes in terms of the trace in particular a relation involving the euler poincare characteristic is proved l 2 betti and novikov shubin numbers are computed for some self similar complexes arising from self similar fractals

coarse baum connes conjecture and rigidity for roe algebras abstract in this paper we connect the rigidity problem and the coarse baum connes conjecture for roe algebras in particular we show that if x and y are two uniformly locally finite metric spaces such that their roe algebras are isomorphic then x and y are coarsely equivalent provided either x or y satisfies the coarse baum connes conjecture with coefficients it is well known that coarse embeddability into a hilbert space implies the coarse baum connes conjecture with coefficients on the other hand we provide a new example of a finitely generated group satisfying the coarse baum connes conjecture with coefficients but which does not coarsely embed into a hilbert space

some mixed hodge structures on l 2 cohomology groups of coverings of kahler manifolds we give methods to compute l 2 cohomology groups of a covering manifold obtained by removing the pullback of a normal crossing divisor to a covering of a compact kahler manifold we prove that in suitable quotient categories these groups admit natural mixed hodge structure whose graded pieces are given by the expected gysin maps

berezin quantization and k homology the e theory defined by connes and higson provides a realization of k homology the generalized homology theory dual to k theory based on the notion of asymptotic homomorphisms with this realization it becomes possible to associate a k homology element to a quantization scheme in this article we associate an asymptotic homomorphism and k homology element to the berezin quantization of a bounded symmetric domain further we identify this element with the element of k homology defined by the dolbeault operator of the domain

the coarse baum connes conjecture for spaces which admit a uniform embedding into hilbert space corollary 1 2 let be a finitely generated group if as a metric space with a word length metric admits a uniform embedding into hilbert space and its classifying space b has the homotopy type of a finite cw complex then the strong novikov conjecture holds for i e the index map from k b to k c r is injective corollary 1 2 follows from theorem 1 1 and the descent principle 23 by index theory the strong novikov conjecture implies the novikov conjecture on the homotopy invariance of higher signatures cf 8 for an excellent

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

a quasi local characterisation of lp roe algebras abstract very recently spakula and tikuisis provide a new characterisation of uniform roe algebras via quasi locality when the underlying metric spaces have straight finite decomposition complexity in this paper we improve their method to deal with the l p version of uniform roe algebras for any p 1 due to the lack of reflexivity on l 1 spaces some extra work is required for the case of p 1

decomposing algebraic m isometric tuples abstract we show that any m isometric tuple of commuting algebraic operators on a hilbert space can be decomposed as a sum of a spherical isometry and a commuting nilpotent tuple our approach applies as well to tuples of algebraic operators that are hereditary roots of polynomials in several variables

spectral invariant subalgebras of reduced crossed product c algebras abstractletg beadiscretegroupwithaproperlengthfunctionl inthispaper weprovethattheschwartz space s l2 g a is a spectral invariant dense subalgebra of the reduced crossedproduct c nr g a for all commutative c n algebra a with a g action if and only if g haspolynomialgrowthwithrespecttol r 2002elsevierscience usa allrightsreserved keywords reducedcrossedproductc n algebras spectralinvariantsubalgebras property mrd 0 introductionletg beadiscretegroupwithalengthfunctionl in 9 jolissaintprovedthatthe schwartzspace s l2 g is a spectralinvariant dense subalgebra of the reducedgroupc n algebrac n r g ifg haspolynomialgrowthorg isthefundamentalgroupofanegativelycurvedcompactmanifold in 4 delaharpegeneralizedjolissaint stheorem to all hyperbolic groups jolissaint and de la harpe s result plays animportantroleintheworkofconnes moscovicionthenovikovconjecture 2 andtheworkoflafforgueonthebaum connesconjecture 10 inthispaper wegiveanecessaryandsuf cientconditionfortheschwartzspaces

unitary transformation approach to the exact solution for the singular oscillator by performing unitary transformations the exact solution of the time dependent singular oscillator is obtained the invariant operator and the auxiliary equation are rigorously established the non adiabatic berry s phase is calculated

unitary transformation approach to the exact solutions of time dependent quantum systems with su 1 1 dynamical group a unitary transformation is introduced to determine the exact solution of the time dependent quantum system as examples two quantum systems with hamiltonian including the quadratic term of generators of su 1 1 are solved analytically

embeddings of uniform roe algebras in this paper we study embeddings of uniform roe algebras generally speaking given metric spaces x and y we are interested in which large scale geometric properties are stable under embedding of the uniform roe algebra of x into the uniform roe algebra of y

non adiabatic berry phase for periodic hamiltonians berry s 1984 formulation of the topological phase for periodic hamiltonians is extended to the case of non adiabatic evolution by proper choice of initial states this restores a full parallel with the non adiabatic formalism of aharonov and anandan 1987 for periodic orbits in a projective hilbert space the berry phase for the novel example of a single electron in a rigidly rotating octahedral environment is calculated using both this formalism and that used previously by aharonov and anandan and by page 1987

a m isometries on hilbert spaces abstract a bounded linear operator t on a hilbert space h is called an a m isometry for some positive operator a on h and integer m if k 0 m 1 m k m k t k a t k 0 we give some properties of a m isometries in particular we focus on spectral properties and the relation between a m isometries and m isometries also we obtain some dynamic properties of a m isometries as a negative answer to 22 question 1 with an example of an a isometric which is n supercyclic and sufficient conditions for an a m isometry to be not n supercyclic moreover we prove that the perturbation of a m isometry by a bigger class than nilpotent operators is not n supercyclic

transversal dirac families in riemannian foliations we describe a family of differential operators parametrized by the transversal vector potentials of a riemannian foliation relative to the clifford algebra of the foliation this family is non elliptic but in certain ways behaves like a standard dirac family in the absolute case as a result of its elliptic like regularity properties the analytic and topological indices of this family are defined as elements of k theory in the parameter space we indicate how the cohomology of the parameter space is described via suitable maps to fredholm operators we outline the proof of a theorem of vafa witten type on uniform bounds for the eigenvalues of this family using a spectral flow argument a determinant operator is also defined with the appropriate zeta function regularization dependent on the codimension of the foliation with respect to a generalized coupled dirac yang mills system we indicate how chiral anomalies are located relative to the foliation

on geometric phases and dynamical invariants a formalism is developed for calculating berry phases for nonadiabatic time periodic quantum systems when a dynamical invariant is known it is found that when the invariant is periodic and has a nondegenerate spectrum this method allows a convenient way to obtain generalized berry phases and the proper cyclic initial states the method is applied to the generalized harmonic oscillator and the two level system where the invariant operators are explicitly constructed formulae for the conventional berry s phases are readily obtained by taking the adiabatic limit of the exact results

analysis of berry s phase by the evolution operator method berry 1984 discovered that an eigenstate undergoing an adiabatic evolution in the parameter space will acquire a topological phase aharonov and anandan 1987 on the other hand showed that an eigenstate transporting round a closed circuit in the projective hilbert space suffices to generate the topological phase the authors employ the evolution operator method to study the propagation of an eigenstate they show that berry s phase can be represented as a closed path integral in the hilbert space and is independent of the choice for the base mod n t the treatments of berry and aharonov and anandan are shown to correspond to two different choices of the base therefore their two approaches are unified they have acquired a more general viewpoint on the origin of berry s phase

the longitudinal cocycle and the index of toeplitz operators abstract let d be a self adjoint leafwise elliptic operator on a foliated manifold compressing multiplication operators to the range of the positive spectral projection for d yields the class of leafwise toeplitz operators the extension generated by these operators is constructed a topological formula for the index of a toeplitz operator with invertible symbol is given this index can also be obtained by pairing the k theory class of the symbol with a certain cyclic cocycle if one lifts an elliptic operator on a closed manifold to a leafwise elliptic operator on an associated flat foliated principal bundle then this cocycle can be used to obtain refined invariants of the original operator

ghostbusting and property a abstract we show that the roe algebra of a bounded geometry metric space contains non compact ghosts if and only if the space does not have property a

uniform k theory and poincare duality for uniform k homology abstract we revisit spakula s uniform k homology construct the external product for it and use this to deduce homotopy invariance of uniform k homology we define uniform k theory and on manifolds of bounded geometry we give an interpretation of it via vector bundles of bounded geometry we further construct a cap product with uniform k homology and prove poincare duality between uniform k theory and uniform k homology on spinc manifolds of bounded geometry

an isometry plus a nilpotent operator is an m isometry applications abstract we prove that if an isometry a and a nilpotent operator q of order n commute then a q is a strict 2 n 1 isometry as an application of the main result we prove that a q cannot be n supercyclic for any n also we find an m isometry with prescribed spectrum k where k is the closed unit disk or a compact subset of the unit circle

analysis of quantum time dependent evolution by the averaged variational principle in this paper we give a general method to solve a time dependent schrodinger equation the formalism developed is based on the quantum variational principle conveniently averaged in addition we apply our method to study the time dependent systems in which the hamiltonians are linear functions of su 1 1 and su 2 lie algebras

correlation functions of the nonstationary quantum singular oscillator the exact density matrix elements and the correlation functions of the one dimensional non stationary singular oscillator the potential u x t omega 2 t x2 2 g x2 excited from the thermodynamic equilibrium states are obtained generating functions for rho nm and also for transition probabilities are constructed

hodge decomposition along the leaves of a riemannian foliation abstract a hodge decomposition theorem is proved for the leafwise laplacian of a riemannian foliation on a closed riemannian manifold

eta invariants of dirac operators on foliated manifolds we define the eta function of dirac operators on foliated manifolds we show that the eta functions are regular at the origin thus defining corresponding eta invariants of foliated manifolds under the hypothesis of invertibility of the operator in question we prove the atiyah singer relative index theorem for dirac operators on foliated manifolds then we discuss the homotopy invariance of the index with respect to secondary data

quantum canonical transformations and exact solution of the schrodinger equation time dependent unitary transformations are used to study the schrodinger equation for explicitly time dependent hamiltonians of the form h t r t j where r is an arbitrary real vector valued function of time and j is the angular momentum operator the solution of the schrodinger equation for the most general hamiltonian of this form is shown to be equivalent to the special case r 1 0 t this corresponds to the problem of a driven two level atom for the spin half representation of j it is also shown that by requiring the magnitude of r to depend on its direction in a particular way one can solve the schrodinger equation exactly in particular it is shown that for every hamiltonian of the form h t r t j there is another hamiltonian with the same eigenstates for which the schrodinger equation is exactly solved the application of the results to the exact solution of the parallel transport equation and exact holomony calculation for su 2 principal bundles yang mills gauge theory is also pointed

maximal and reduced roe algebras of coarsely embeddable spaces in 7 gong wang and yu introduced a maximal or universal version of the roe c algebra associated to a metric space we study the relationship between this maximal roe algebra and the usual version in both the uniform and non uniform cases the main result is that if a uniformly discrete bounded geometry metric space x coarsely embeds into a hilbert space then the canonical map between the maximal and usual uniform roe algebras induces an isomorphism on k theory we also give a simple proof that if x has property a then the maximal and usual uniform roe algebras are the same these two results are natural coarse geometric analogues of certain well known implications of a t menability and amenability for group c algebras the techniques used are e theoretic building on work of higson kasparov and trout 11 12 and yu 28

characteristic classes and distortion of diffeomorphisms let m and n be manifolds of bounded geometry that is the absolute value of their curvatures as well as their covariant derivatives are bounded from above and their injectivity radii are bounded from below a map f m n has bounded d stortion if there is an upper bound on the norm of the derivative d f a bounded distortion diffeomorphism cb d d is assumed to have an inverse which also has bounded distortion similarly homotopies are assumed to have bounded distortion note that this implies that homotopic maps are a finite distance apart we prove the following theorem that describes when characteristic classes of a compact manifold obstruct the existence of bounded distortion diffeomorphisms of universal covers theorem there are compact manifolds a and b with fundamental group r and a map f a b with jcpcb i pca for some rational pontrjagin class p whose universal cover is b d homotopic to a b d diffeomorphism if and only if r is nonamenable

classification of hereditary matrices abstract a classical approach used to obtain basic facts in the theory of square matrices involves an analysis of the relationship between polynomials p in one variable and square matrices t such that p t 0 we consider matrices and operators which satisfy a different type of polynomial constraint let h be a complex hilbert space t be a bounded linear transformation of h t be the adjoint of t and c x y be the algebra of polynomials in x and y with complex coefficients for a polynomial p c x y in two variables with complex coefficients define p t m n 0 p m n t n t m where p m n is the coefficient of ynxm in the expansion of p in a power series about the point 0 0 t is called a root of p if and only if p t 0 note that if p c x y is a polynomial in the single variable x then the definition of p t given here agrees with the classical definition in this paper we study the relationships which p t 0 forces between p and t when t is an algebraic operator i e there exists n 1 and complex numbers a0 an 1 such that 0 a0 a1t an 1tn 1 tn the classification starts with the following observation suppose p c x y and an algebraic operator t l h satisfy p t 0 then certain subspaces of h which are invariant for t must be orthogonal or certain coefficients of p must vanish this leads to the notions of a graph attached to each p c x y and a graph attached to each square matrix t for diagonalizable t a necessary and sufficient graph theoretic condition for solving p t 0 is given for nondiagonalizable t this condition is necessary but not sufficient the use of these graphs does however reduce the problem to the problem of solving the equation p t 0 for t with exactly one or two eigenvalues for t with one eigenvalue we give a necessary and sufficient condition for solving p t 0 this leaves the case of solving p t 0 when t has exactly two eigenvalues this problem mixes algebra involving polynomials with matrix theory we show that it is equivalent to the purely algebraic problem of determining if equations of the form i j eci jxi r j s 0 have solutions of finite support with certain nonvanishing properties we call these equations bi hankel equations subordinate to a given subset e of the lattice of integer pairs i j 0 i m 1 0 j n 1 it turns out that there is an algorithm which uses grobner bases for determining if the type of solution we seek exists and for computing it

algebraic properties of operator roots of polynomials abstract properties of m selfadjoint and m isometric operators have been investigated by several researchers particularly interesting to us are algebraic properties of nilpotent perturbations of such operators mccullough and rodman showed in the nineties that if q n 0 and a is a selfadjoint operator commuting with q then the sum a q is a 2 n 1 selfadjoint operator very recently bermudez martinon and noda proved a similar result for nilpotent perturbations of isometries via a new approach we obtain simple proofs of these results and other generalizations to operator roots of polynomials

metric sparsification and operator norm localization abstract we study an operator norm localization property and its applications to the coarse novikov conjecture in operator k theory in particular we introduce a sufficient geometric condition called metric sparsification for the operator norm localization property this is used to give many examples of finitely generated groups with infinite asymptotic dimension and the operator norm localization property we also show that a sequence of expanding graphs does not possess the operator norm localization property

some results on higher order isometries and symmetries products and sums with a nilpotent operator abstract we study the sum of an m isometry or an m symmetric operator with a nilpotent operator we also study the product of two m isometries or two n symmetries we obtain several theorems generalizing previous work our method is straightforward and thus offers direct and simple insights into why these theorems are true

on tuples of commuting operators in positive semidefinite inner product spaces abstract this paper is concerned with the study of certain properties of operator tuples on a complex hilbert space h when a semi inner product induced by a positive operator a on h is considered in particular we show that r a t a t for every commuting operator tuple t t 1 t d such that each t k admits an a adjoint operator where r a t and a t denote respectively the a joint spectral radius and the a joint numerical radius of t this study allows to establish that r a t a t t a for every a normal commuting tuple of operators t where t a is denoted to be the a joint operator seminorm of t in addition the a joint spectral radius of an a m isometric tuple of commuting operators is studied

fibred coarse embeddings a t menability and the coarse analogue of the novikov conjecture abstract the connection between the coarse geometry of metric spaces and analytic properties of topological groupoids is well known one of the main results of skandalis tu and yu is that a space admits a coarse embedding into hilbert space if and only if a certain associated topological groupoid is a t menable this groupoid characterisation then reduces the proof that the coarse baum connes conjecture holds for a coarsely embeddable space to known results for a t menable groupoids the property of admitting a fibred coarse embedding into hilbert space was introduced by chen wang and yu to provide a property that is sufficient for the maximal analogue to the coarse baum connes conjecture and in this paper we connect this property to the traditional coarse baum connes conjecture using a restriction of the coarse groupoid and homological algebra additionally we use this results to give a characterisation of the a t menability for residually finite discrete groups

the nuclear dimension of c algebras abstract we introduce the nuclear dimension of a c algebra this is a noncommutative version of topological covering dimension based on a modification of the earlier concept of decomposition rank our notion behaves well with respect to inductive limits tensor products hereditary subalgebras hence ideals quotients and even extensions it can be computed for many examples in particular it is finite for all uct kirchberg algebras in fact all classes of nuclear c algebras which have so far been successfully classified consist of examples with finite nuclear dimension and it turns out that finite nuclear dimension implies many properties relevant for the classification program surprisingly the concept is also linked to coarse geometry since for a discrete metric space of bounded geometry the nuclear dimension of the associated uniform roe algebra is dominated by the asymptotic dimension of the underlying space

a trace on fractal graphs and the ihara zeta function starting with ihara s work in 1968 there has been a growing interest in the study of zeta functions of finite graphs by sunada hashimoto bass stark and terras mizuno and sato to name just a few authors then clair and mokhtari sharghi studied zeta functions for infinite graphs acted upon by a discrete group of automorphisms the main formula in all these treatments establishes a connection between the zeta function originally defined as an infinite product and the laplacian of the graph in this article we consider a different class of infinite graphs they are fractal graphs i e they enjoy a self similarity property we de ne a zeta function for these graphs and using the machinery of operator algebras we prove a determinant formula which relates the zeta function with the laplacian of the graph we also prove functional equations and a formula which allows approximation of the zeta function by the zeta functions of finite subgraphs

on rigidity of roe algebras roe algebras are c algebras built using large scale or coarse aspects of a metric space x d in the special case that x is a finitely generated group and d is a word metric the simplest roe algebra associated to d is isomorphic to the crossed product c algebra l 1 r roe algebras are coarse invariants in the sense that if x and y are coarsely equivalent metric spaces then their roe algebras are isomorphic motivated in part by the coarse baum connes conjecture we ask if there is a converse to the above statement that is if x and y are metric spaces with isomorphic roe algebras must x and y be coarsely equivalent we show that for very large classes of spaces the answer to this question and some related questions is yes this can be thought of as a c rigidity result it shows that the roe algebra construction preserves a large amount of information about the space and is thus surprisingly rigid as an example of our results in the group case we have that if and are finitely generated elementary amenable hyperbolic or linear groups

time evolution of quantum systems with time dependent hamiltonian and the invariant hermitian operator we study the time evolution of a class of exactly solvable time dependent quantum systems with a time dependent hamiltonian given by a linear combination of su 1 1 and su 2 generators with the help of the invariant hermitian operator the exact common solutions of the schrodinger equations for both the su 1 1 and su 2 systems are obtained in terms of eigenstates of the invariant operator the adiabatic and non adiabatic berry phases are calculated with the exact solutions moreover we derive an explicit time evolution operator which is used to investigate the time dependent two photon squeezing states and su 2 squeezing states the squeezing properties of the time dependent su 1 1 coherent states are also discussed

a renormalized index theorem for some complete asymptotically regular metrics the gauss bonnet theorem abstract we study the gauss bonnet theorem as a renormalized index theorem for edge metrics these metrics include the poincare einstein metrics of the ads cft correspondence and the asymptotically cylindrical metrics of the atiyah patodi singer index theorem we use renormalization to make sense of the curvature integral and the dimensions of the l 2 cohomology spaces as well as to carry out the heat equation proof of the index theorem for conformally compact metrics even mod x m we show that the finite time supertrace of the heat kernel on conformally compact manifolds renormalizes independently of the choice of special boundary defining function

an index for gauge invariant operators and the dixmier douady invariant let g b be a bundle of compact lie groups acting on a fiber bundle y b in this paper we introduce and study gauge equivariant k theory groups k i g y these groups satisfy the usual properties of the equivariant k theory groups but also some new phenomena arise due to the topological non triviality of the bundle g b as an application we define a gauge equivariant index for a family of elliptic operators p b b b invariant with respect to the action of g b which in this approach is an element of k 0 g b we then give another definition of the gauge equivariant index as an element of k 0 c g the k theory group of the banach algebra c g we prove that k 0 c g k 0 g g and that the two definitions of the gauge equivariant index are equivalent the algebra c g is the algebra of continuous sections of a certain field of c algebras with non trivial dixmier douady invariant the gauge equivariant k theory groups are thus examples of twisted k theory groups which have recently turned out to be useful in the study of ramond ramond fields

a serre swan theorem for bundles of bounded geometry abstract the serre swan theorem in differential geometry establishes an equivalence between the category of smooth vector bundles over a smooth compact manifold and the category of finitely generated projective modules over the unital ring of smooth functions this theorem is here generalized to manifolds of bounded geometry in this context it states that the category of hilbert bundles of bounded geometry is equivalent to the category of operator modules over the operator algebra of continuously differentiable functions which vanish at infinity operator modules are generalizations of hilbert c modules where the category of c algebras has been replaced by a more flexible category of involutive algebras of bounded operators the operator algebras operator modules play an important role in the study of the unbounded kasparov product

uniform k homology theory we define a uniform version of analytic k homology theory for separable proper metric spaces furthermore we define an index map from this theory into the k theory of uniform roe c algebras analogous to the coarse assembly map from analytic k homology into the k theory of roe c algebras we show that our theory has a mayer vietoris sequence we prove that for a torsion free countable discrete group the direct limit of the uniform k homology of the rips complexes of limd k u pd is isomorphic to k top l the left hand side of the baum connes conjecture with coefficients in l in particular this provides a computation of the uniform k homology groups for some torsion free groups as an application of uniform k homology we prove a criterion for amenability in terms of vanishing of a fundamental class in spirit of similar criteria in uniformly finite homology and k theory of uniform roe algebras

a notion of geometric complexity and its application to topological rigidity we introduce a geometric invariant called finite decomposition complexity fdc to study topological rigidity of manifolds we prove for instance that if the fundamental group of a compact aspherical manifold m has fdc and if n is homotopy equivalent to m then m n is homeomorphic to n n for n large enough this statement is known as the stable borel conjecture on the other hand we show that the class of fdc groups includes all countable subgroups of gl n k for any field k

aperiodic tilings positive scalar curvature and amenability of spaces the object of this paper is to begin a geometric study of noncompact spaces whose local structure has bounded complexity manifolds of this sort arise as leaves of foliations of compact manifolds and as their universal covers we shall introduce a coarse homology theory using chains of bounded complexity and study some of its first properties the most interesting result characterizes when h uf x vanishes as an analogue and strengthening of f0lner s amenability criterion for groups in terms of isoperimetric inequalities see 4 one can view this result as producing a successful infinite ponzi scheme on any nonamenable space each point with only finite resources gives to some of its neighbors some of these resources yet receives more from the remaining neighbors as one can imagine this is useful for eliminating obstructions on noncompact spaces this has a number of applications we present two of them the first produces tilings that are unbalanced on any nonamenable polyhedron unbalanced tilings are automatically aperiodic and this gives many examples of sets of tiles that tile only aperiodically unfortunately imbalance is a particularly unsubtle reason for aperiodicity so that the aperiodic tilings of euclidean space penrose tilings are necessarily not accessible to our method on the other hand most other simply connected noncompact symmetric spaces even have unbalanced tilings using our criterion the second application regards characteristic numbers of manifolds whose universal covers have positive scalar curvature we prove a converse to a theorem of roe we show that for any nonamenable group f one can find a spin manifold with fundamental group f with nonzero a genus whose universal cover has a uniformly positive scalar curvature metric of bounded geometry in the natural strict quasi isometry class

an index theorem on open manifolds ii on decrit et on applique un nouveau theoreme d indice pour des operateurs elliptiques sur certaines varietes non compactes on calcule un indice a valeur reelle pour des operateurs de type dirac sur des varietes non compactes de geometrie bornee

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

unitary transformation approach to the cyclic evolution of su 1 1 and su 2 time dependent systems and geometrical phases using the time dependent unitary transformation instead of the invariant operator the solutions of su 1 1 and su 2 time dependent quantum systems are obtained it is shown that the evolution operator is decomposed into the product form of two unitary operators in such a way that one of them has the same periodicity as the hamiltonian and the other correspond to the floquet operator which gives the cyclic states and their associated phases over one period of the evolution the non adiabatic berry s or aharonov anandan phases are determined totally by such a unitary transformation

noncommutative riemann integration and novikov shubin invariants for open manifolds given a c algebra a with a semicontinuous semifinite trace acting on the hilbert space h we define the family ar of bounded riemann measurable elements w r t as a suitable closure a la dedekind of a in analogy with one of the classical characterizations of riemann measurable functions and show that ar is a c algebra and extends to a semicontinuous semifinite trace on ar then unbounded riemann measurable operators are defined as the closed operators on h which are affiliated to a and can be approximated in measure by operators in ar in analogy with unbounded riemann integration unbounded riemann measurable operators form a a e bimodule on ar denoted by ar and such a bimodule contains the functional calculi of selfadjoint elements of ar under unbounded riemann measurable functions besides extends to a bimodule trace on ar type ii1 singular traces for c algebras can be defined on the bimodule of unbounded riemann measurable operators noncommutative riemann integration and singular traces for c algebras are then used to define novikov shubin numbers for amenable open manifolds to show their invariance under quasi isometries and to prove that they are noncommutative asymptotic dimensions

hadamard states for the klein gordon equation on lorentzian manifolds of bounded geometry we consider the klein gordon equation on a class of lorentzian manifolds with cauchy surface of bounded geometry which is shown to include examples such as exterior kerr kerr de sitter spacetime and the maximal globally hyperbolic extension of the kerr outer region in this setup we give an approximate diagonalization and a microlocal decomposition of the cauchy evolution using a time dependent version of the pseudodifferential calculus on riemannian manifolds of bounded geometry we apply this result to construct all pure regular hadamard states and associated feynman inverses where regular refers to the state s two point function having cauchy data given by pseudodifferential operators this allows us to conclude that there is a one parameter family of elliptic pseudodifferential operators that encodes both the choice of pure regular hadamard state and the underlying spacetime metric

quasi locality and property a abstract let x be a metric space with bounded geometry p 0 1 and let e be a banach space the main result of this paper is that either if x has yu s property a and p 1 or without any condition on x when p 0 1 then quasi local operators on l p x e belong to the appropriate variant of the roe algebra of x this generalises the existing results of this type by lange and rabinovich engel tikuisis and the first author and li wang and the second author as consequences we obtain that uniform l p roe algebras of spaces with property a are closed under taking inverses and another condition characterising property a akin to the operator norm localisation for quasi local operators

joint numerical ranges of operators in semi hilbertian spaces abstract in this paper we aim to investigate the concept of numerical range and maximal numerical range relative to a positive operator of a d tuple of bounded linear operators on a hilbert space some properties and applications of these sets are studied mainly it is proved that they are convex for d 1 this generalizes the well known toeplitz hausdorff theorem 24 16 and stampfli s result 23 moreover under additional hypotheses we show that these sets are convex for d 2

exotic cohomology and index theory the main result of this paper is a version of the atiyah singer index theorem for dirac type operators on noncompact complete riemannian manifolds the statement of the theorem involves a novel cohomology theory for such manifolds this theory called exotic cohomology depends on the structure at infinity of a space more precisely it depends on the way that large bounded sets fit together for each cohomology class in this theory we define a higher index of a dirac type operator enjoying the stability and vanishing properties of the usual atiyah singer index these higher indices are analogous to the novikov higher signatures our main theorem will compute these higher indices in terms of standard topological invariants applications of this index theorem include a different approach to some of the results of gromov and lawson 10 on topological obstructions to positive scalar curvature the concept of index that we will use involves the a theory functors k0 and kx for operator algebras 3 14 suppose that b is an ideal in a unital algebra c and let t e c be invertible modulo b in the classical atiyah singer index theorem one takes c to be the bounded operators on the l space of some compact manifold b the compact operators and t an elliptic pseudodifferential operator of order zero then t has an index in the theory group kq b in the classical case this is just z and one recovers the usual fredholm index now let m be a complete riemannian manifold possibly noncompact in 18 i introduced an algebra m which is defined as follows sf m consists of all bounded operators a on l m that have a kernel representation

so 2 1 invariant systems and the berry phase the general quadratic time dependent quantum hamiltonian is analysed using a time dependent realisation of its so 2 1 invariance lie algebra several interesting features of this procedure are pointed out as an example the authors easily calculate dynamical and geometrical berry phases using only an algebraic procedure an incorrect result previously published by another author is also pointed out

exact wavefunctions for a time dependent coulomb potential the one dimensional schrodinger equation associated with a time dependent coulomb potential is studied the invariant operator method lewis and riesenfeld and unitary transformation approach are employed to derive quantum solutions of the system we obtain an ordinary second order differential equation whose analytical exact solution has been unknown it is confirmed that the form of this equation is similar to the radial schrodinger equation for the hydrogen atom in a arbitrary strong magnetic field the qualitative properties for the eigenstates spectrum are described separately for the different values of the parameter 0 appearing in the x2 term x being the position i e 0 0 0 0 and 0 0 for the 0 0 case the eigenvalue equation of invariant operator reduces to a solvable form and consequently we have provided exact eigenstates of the time dependent hamiltonian system

c0 coarse geometry and scalar curvature in this paper we introduce an alternative form of coarse geometry on proper metric spaces which is more delicate at infinity than the standard metric coarse structure there is an assembly map from the k homology of a space to the k theory of the c algebra associated to the new coarse structure which factors through the coarse k homology of the space with the new coarse structure a dirac type operator on a complete riemannian manifold m gives rise to a class in k homology and its image under assembly gives a higher index in the k theory group the main result of this paper is a vanishing theorem for the index of the dirac operator on an open spin manifold for which the scalar curvature x tends to infinity as x tends to infinity this is derived from a spectral vanishing theorem for any dirac type operator with discrete spectrum and finite dimensional eigenspaces