radiation conditions and scattering theory for n particle hamiltonians the correct form of the angular part of radiation conditions is found in scattering problem forn particle quantum systems the estimates obtained allow us to give an elementary proof of asymptotic completeness for such systems in the framework of the theory of smooth perturbations

the scattering matrix for the schrodinger operator with a long range electromagnetic potential we consider the schrodinger operator h i a 2 v in the space l2 rd with long range electrostatic v x and magnetic a x potentials using the scheme of smooth perturbations we give an elementary proof of the existence and completeness of modified wave operators for the pair h0 h our main goal is to study spectral properties of the corresponding scattering matrix s we obtain its stationary representation and show that its singular part up to compact terms is a pseudodifferential operator with an oscillating amplitude which is an explicit function of v and a finally we deduce from this result that in general for each 0 the spectrum of s covers the whole unit circle

stability estimates for an inverse scattering problem at high frequencies we consider an inverse scattering problem and its near field approximation at high frequencies we first prove for both problems lipschitz stability results for determining the low frequency component of the potential then we show that in the case of a radial potential supported sufficiently near the boundary infinite resolution can be achieved from measurements of the near field operator in the monotone case

structures of s matrices for three body schrodinger operators structures of the s matrix associated with the collision process from 2 clusters to 3 clusters are studied this s matrix is shown to have a continuous kernel except for 2 dimensional spheres on which 2 body subsystems have zero velocity on these spheres the s matrix has in general singularities whose existence arises from the zero eigenvalues and the zero resonances of the 2 body subsystems

applications of temperley lieb algebras to lorentz lattice gases motivated by the study of motion in a random environment we introduce and investigate a variant of the temperley lieb algebra this algebra is very rich providing us with three classes of solutions of the yang baxter equation this allows us to establish a theoretical framework to study the diffusive behaviour of a lorentz lattice gas exact results for the geometrical scaling behaviour of closed paths are also presented

consistent systems of correlators in non semisimple conformal field theory based on the modular functor associated with a not necessarily semisimple finite non degenerate ribbon category d we present a definition of a consistent system of bulk field correlators for

the geometrical approach to multidimensional inverse scattering we prove that in multidimensional short range potential scattering the high velocity limit of the scattering operator of an n body system determines uniquely the potential for a given long range potential the short range potential of the n body system is uniquely determined by the high velocity limit of the modified dollard scattering operator moreover we prove that any one of the dollard scattering operators determines uniquely the total potential we obtain as well a reconstruction formula with an error term our simple proof uses a geometrical time dependent method

lattice models of branched polymers statistics of uniform stars the authors investigate the statistical properties of uniform star polymers with f branches modelled on lattices in two and three dimensions it is shown that the growth constant exists and is equal to mu f where mu is the self avoiding walk limit the f dependence of the corresponding critical exponent gamma f is studied using exact enumeration and monte carlo techniques and the results are compared with the predictions of miyake and freed 1983 obtained using chain conformation space renormalisation group method

operateurs conjugues et proprietes de propagation we use an algebraic criteria based on local positivity of a commutator which asserts the existence of a direction of propagation for the flowe iht associated to a self adjoint operatorh

mourre s method and smoothing properties of dispersive equations abstract the purpose of this paper is to point out that mourre s method in spectral theory is useful and powerful in studying the smoothing effect of dispersive equations especially we will see that the smoothing effect is a quite general phenomenon even for operators with variable coefficients

on the mathematical theory of the aharonov bohm effect we consider the schr dinger operator h i a 2 in the space l2 2 with a magnetic potential a x a x2 x1 x 2 where a is an arbitrary function on the unit circle our goal is to study spectral properties of the corresponding scattering matrix s 0 we obtain its stationary representation and show that its singular part up to compact terms is a pseudodifferential operator of zero order whose symbol is an explicit function of a we deduce from this result that the essential spectrum of s does not depend on and consists of two complex conjugated and perhaps overlapping closed intervals of the unit circle finally we calculate the diagonal singularity of the scattering amplitude kernel of s considered as an integral operator in particular we show that for all these properties only the behaviour of a potential at infinity is essential the preceding papers on this subject treated the case a const and used the separation of variables in the schr dinger equation in the polar coordinates this technique does not of course work for arbitrary a from an analytical point of view our paper relies on some modern tools of scattering theory and well known properties of pseudodifferential operators

smoothing effects and dispersion of singularities for the schrodinger evolution group we describe smoothing effects and dispersion of singularities for the schrodinger evolution group in the weighted sobolev spaces under a fairly general assumption on the potential it is shown that all singularities in the wavefunction vanish instantly whenever the initial state has sufficient decay we measure the regularity gained by the wavefunction by the decay property of the initial state no assumptions on the regularity of the initial state are imposed throughout the paper

operator spectrum and exact exponents of the fully packed loop model we develop a coulomb gas description of the critical fluctuations in the fully packed loop model on the honeycomb lattice we identify the complete operator spectrum of this model in terms of electric and magnetic vector charges and we calculate the scaling dimensions of these operators exactly we also study the geometrical properties of loops in this model and we derive exact results for the fractal dimension and the loop size distribution function a review of the many different representations of this model that have recently appeared in the literature is given

polymer coil globule transition by real space renormalisation a simple real space renormalisation scheme is proposed for the statistics of the polymer coil globule transition in two dimensions this approach describes the coil theta point and globular states of the macromolecule within a single two parameter transformation results from a small cell calculation include the theta point size exponent vtheta 0 494 and crossover exponent phi 0 486

spectral and scattering theory in perturbed stratified fluids ii transmission problems and exterior domains abstract the spectral and scattering theory of the acoustic propagator in perturbed stratified fluids is studied in the cases of transmission problems and of exterior domains

on the virial theorem in quantum mechanics we review the various assumptions under which abstract versions of the quantum mechanical virial theorem have been proved we point out a relationship between the virial theorem for a pair of operators h a and the regularity properties of the map we give an example showing that the statement of the virial theorem in cfks is incorrect

critical exponents of manhattan hamiltonian walks in two dimensions from potts and o n models we consider a set of hamiltonian circuits filling a manhattan lattice i e a square lattice with alternating traffic regulation we show that the generating function with fugacityz of this set is identical to the critical partition function of aq state potts model on an unoriented square lattice withq1 2 z the set of critical exponents governing correlations of hamiltonian circuits is derived using a coulomb gas technique these exponents are also found to be those of an o n vector model in the low temperature phase withn q1 2 z the critical exponents in the limitz 0 are then those of spanning trees q 0 and of dense polymers n 0 t tc corresponding to a conformal theory with central chargec 2 this shows that the manhattan orientation and the hamiltonian constraint of filling all the lattice are irrelevant for the infrared critical properties of hamiltonian walks

amplitude ratios for critical systems in the c 2 universality class we study the finite size corrections of the critical dense polymer cdp and the dimer models on n rectangular lattice we find that the finite size corrections in the cdp and dimer models depend in a crucial way on the parity of n and a change of the parity of n is equivalent to the change of boundary conditions we present a set of universal amplitude ratios for amplitudes in finite size correction terms of critical systems in the universality class with central charge c 2 the results are in perfect agreement with a perturbated conformal field theory under the assumption that all analytical corrections coming from the operators which belongs to the tower of the identity our results inspire many interesting problems for further studies

high energy and smoothness asymptotic expansion of the scattering amplitude abstract we find an explicit expression for the kernel of the scattering matrix for the schrodinger operator containing at high energies all terms of power order it turns out that the same expression gives a complete description of the diagonal singularities of the kernel in the angular variables the formula obtained is in some sense universal since it applies both to short and long range electric as well as magnetic potentials

einstein solvmanifolds attached to two step nilradicals a riemannian einstein solvmanifold possibly any noncompact homogeneous einstein space is almost completely determined by the nilradical of its lie algebra a nilpotent lie algebra which can serve as the nilradical of an einstein metric solvable lie algebra is called an einstein nilradical we give a classification of two step nilpotent einstein nilradicals with two dimensional center informally the defining matrix pencil must have no nilpotent blocks in the canonical form and no elementary divisors of a very high multiplicity we also show that the dual to a two step einstein nilradical is not in general an einstein nilradical

resolvent smoothness and local decay at low energies for the standard model of non relativistic qed we consider an atom interacting with the quantized electromagnetic field in the standard model of non relativistic qed the nucleus is supposed to be fixed we prove smoothness of the resolvent and local decay of the photon dynamics for quantum states in a spectral interval i just above the ground state energy our results are uniform with respect to i their proofs are based on abstract mourre s theory a mourre inequality established by frohlich griesemer and sigal see frohlich et al 2008 14 hardy type estimates in fock space and a low energy dyadic decomposition

linear inviscid damping and enhanced viscous dissipation of shear flows by using the conjugate operator method abstract we study the large time behavior of solutions to two dimensional euler and navier stokes equations linearized about shear flows of the mixing layer type in the unbounded channel t r under a simple spectral stability assumption on a self adjoint operator we prove a local form of the linear inviscid damping that is uniform with respect to small viscosity we also prove a local form of the enhanced viscous dissipation that takes place at times of order 1 3 being the small viscosity to prove these results we use a hamiltonian approach following the conjugate operator method developed in the study of schrodinger operators combined with a hypocoercivity argument to handle the viscous case

absence of singular continuous spectrum for certain self adjoint operators we give a sufficient condition for a self adjoint operator to have the following properties in a neighborhood of a pointe of its spectrum a its point spectrum is finite b its singular continuous spectrum is empty c its resolvent satisfies a class of a priori estimates

maximal fermi walk configurations on the directed square lattice and standard young tableaux we consider configurations of n walkers each of which starts at the origin of a directed square lattice and makes the same number t of steps from node to node along the edges of the lattice bose walkers are not allowed to cross but can share edges fermi walk configurations must satisfy the additional constraint that no two walkers traverse the same path since for given t there are only a finite number of t step paths there is a limit nmax on the number of walkers allowed by the fermi condition the value of nmax is determined for six types of boundary conditions the number of fermi configurations of nmax walkers is also determined using a bijection to standard young tableaux in four cases there is no constraint on the endpoints of the walks and the relevant tableaux are shifted

boundary critical behavior of d 2 self avoiding walks on correlated and uncorrelated vacancies in this paper we present exact results for the critical exponents of interacting self avoiding walks with ends at a linear boundary effective interactions are mediated by vacancies correlated and uncorrelated on the dual lattice by choosing different boundary conditions several ordinary and special regimes can be described in terms of clusters geometry and of critical and lowtemperature properties of the mathcal o n 1 model in particular the problem of boundary exponents at the point is fully solved and implications for point universality are discussed the surface crossover exponent at the special transition of noninteracting self avoiding walks is also interpreted in terms of percolation dimensions

exact critical exponents for two dimensional dense polymers using conformal invariance and coulomb gas results the author gives the exact value in two dimensions of the eta exponent of l dense polymers attached by their extremities eta l l2 4 8 the value in two dimensions of the gamma exponent of a dense branched polymer of fixed topology with nl l leg vertices l or 1 is then deduced to be gamma sigma l or 1 nl 2 l l 18 32 these values correspond to a conformally invariant theory with central charge c 2

polynomial fusion rings of w extended logarithmic minimal models the countably infinite number of virasoro representations of the logarithmic minimal model lm p p can be reorganized into a finite number of w representations with respect to the extended virasoro algebra symmetry w using a lattice implementation of fusion we recently determined the fusion algebra of these representations and found that it closes albeit without an identity for p 1 here we provide a fusion matrix realization of this fusion algebra and identify a fusion ring isomorphic to it we also consider various extensions of it and quotients thereof and introduce and analyze commutative diagrams with morphisms between the involved fusion algebras and the corresponding quotient polynomial fusion rings one particular extension is reminiscent of the fundamental fusion algebra of lm p p and offers a natural way of introducing the missing identity for p 1 working out explicit fusion matrices is facilitated by a further enlargement based on a pair of mutual moore penrose inverses intertwining bet

scattering theory for the dirac operator with a long range electromagnetic potential abstract we consider the dirac operator with a long range potential v x scalar pseudo scalar and vector components of v x may have arbitrary power like decay at infinity we introduce wave operators with time independent modifiers these modifiers are pseudo differential operators whose symbols are roughly speaking constructed in terms of approximate eigenfunctions of the stationary problem we derive and solve eikonal and transport equations for the corresponding phase and amplitude functions from an analytical point of view our proof of the existence and completeness of the wave operators relies on the limiting absorption principle and radiation estimates established in the paper this allows us to fit the long range scattering theory for the dirac operator into the framework of smooth perturbations finally we find the asymptotics for large times t of solutions u x t of the time dependent dirac equation

spectral distributions for long range perturbations abstract we study distributions which generalize the concept of spectral shift function for pseudo differential operators on r d we call such distributions spectral distributions relations between relative scattering determinants and spectral distributions are established they lead to the definition of regularized scattering phase these relations are analogous to the usual one for the standard spectral shift function we give several asymptotic properties in the high energy and semiclassical limits where both nontrapping and trapping cases are considered in particular we prove breit wigner formulae for the regularized scattering phases for semiclassical schrodinger operators with long range potentials

w extended fusion algebra of critical percolation two dimensional critical percolation is the member lm 2 3 of the infinite series of yang baxter integrable logarithmic minimal models lm p p we consider the continuum scaling limit of this lattice model as a rational logarithmic conformal field theory with extended w w2 3 symmetry and use a lattice approach on a strip to study the fundamental fusion rules in this extended picture we find that the representation content of the ensuing closed fusion algebra contains 26 w indecomposable representations with eight rank1 representations 14 rank 2 representations and four rank 3 representations weidentifytheserepresentationswithsuitablelimitsofyang baxterintegrable boundary conditions on the lattice and obtain their associated w extended characters the latter decompose as finite non negative sums of w irreducible characters of which 13 are required implementation of fusion on the lattice allows us to read off the fusion rules governing the fusion algebra of the 26 representations and to construct an explicit cayley table the closure of these representations among themselves under fusion is remarkable confirmation of the proposed extended symmetry

integrable boundary conditions and w extended fusion in the logarithmic minimal models lm 1 p we consider the logarithmic minimal models lm 1 p as rational logarithmic conformal field theories with extended w symmetry to make contact with the extended picture starting from the lattice we identify 4p 2 boundary conditions as specific limits of integrable boundary conditions of the underlying yang baxter integrable lattice models specifically we identify 2p integrable boundary conditions to match the 2p known irreducible w representations these 2p extended representations naturally decompose into infinite sums of the irreducible virasoro representations r s a further 2p 2 reducible yet indecomposable w representations of rank 2 are generated by fusion and these decompose as infinite sums of indecomposable rank 2 virasoro representations the fusion rules in the extended picture are deduced from the known fusion rules for the virasoro representations of lm 1 p and are found to be in agreement with previous works the closure of the fusion algebra on a finite number of representations in the extended picture is remarkable confirmation of the consistency of the lattice approach comment 15 pages

quantum scattering at low energies for a class of negative slowly decaying potentials including v x x with 0 2 we study the quantum mechanical scattering theory in the low energy regime using appropriate modifiers of the isozaki kitada type we show that scattering theory is well behaved on the whole continuous spectrum of the hamiltonian including the energy 0 we show that the modified scattering matrices s are welldefined and strongly continuous down to the zero energy threshold similarly we prove that the modified wave matrices and generalized eigenfunctions are norm continuous down to the zero energy if we use appropriate weighted spaces these results are used to derive oscillatory asymptotics of the standard short range and dollard type s matrices for the subclasses of potentials where both kinds of s matrices are defined for potentials whose leading part is x we show that the location of singularities of the kernel of s experiences an abrupt change from passing from positive energies to the limiting energy 0 this change corresponds to the behaviour of the classical orbits under stronger conditions one can extract the leading term of the asymptotics of the kernel of s at its singularities

einstein solvmanifolds and the pre einstein derivation an einstein nilradical is a nilpotent lie algebra which can be the nilradical of a metric einstein solvable lie algebra the classification of riemannian einstein solvmanifolds possibly of all noncompact homogeneous einstein spaces can be reduced to determining which nilpotent lie algebras are einstein nilradicals and to finding for every einstein nilradical its einstein metric solvable extension for every nilpotent lie algebra we construct an essentially unique derivation the pre einstein derivation the solvable extension by which may carry an einstein inner product using the pre einstein derivation we then give a variational characterization of einstein nilradicals as an application we prove an easy to check convex geometry condition for a nilpotent lie algebra with a nice basis to be an einstein nilradical and also show that a typical two step nilpotent lie algebra is an einstein nilradical the theory of riemannian homogeneous spaces with an einstein metric splits into three very different cases depending on the sign of the einstein constant the scalar curvature among them the picture is complete only in the ricci flat case by the result of ak every ricci flat homogeneous space is flat the major open conjecture in the case of negative scalar curvature is the alekseevski conjecture al1 asserting that a noncompact einstein homogeneous space admits a simply transitive solvable isometry group this is equivalent to saying that any such space is a solvmanifold a solvable lie group with a left invariant riemannian metric satisfying the einstein condition by a deep result of j lauret la5 any einstein solvmanifold is standard this means that the metric solvable lie algebra s of such a solvmanifold has the following property the orthogonal complement to the derived algebra of s is abelian the systematic study of standard einstein solvmanifolds and the term standard originated from the paper of j heber heb on the lie algebra level all the metric einstein solvable lie algebras can be obtained as the result of the following construction heb la1 la5 lw one starts with the three pieces of data a nilpotent lie algebra n a semisimple derivationof n and an inner product h i n on n with respect to which is symmetric an extension of n byis a solvable lie algebra s rh n as a linear space with adh n the inner product on s is defined by h h ni 0 k hk 2 tr and coincides with the existing one on n the resulting metric solvable lie algebra s h i is einstein provided n is nice and the derivationand the inner product h i n are chosen in the correct way note however that these conditions are expressed by a system of algebraic equations which could hardly be analyzed directly see section 2 metric einstein solvable lie algebras of higher rank with the codimension of the nilradical greater than one having the same nilradical n can be obtained from s via a known procedure by further adjoining to n semisimple derivation commuting with it turns out that the structure of an einstein metric solvable lie algebra is completely encoded in its nilradical in the following sense given a nilpotent lie algebra n there is no more than one possibly none choice ofand of h i n up to conjugation by aut n and scaling which may result in an einstein metric solvable lie algebra s h i definition 1 a nilpotent lie algebra is called an einstein nilradical if it is the nilradical of an einstein metric solvable lie algebra a derivationof an einstein nilradical n and an inner product h i n for which the metric solvable lie algebra s h i is einstein are called an einstein derivation and a nilsoliton inner product respectively

propagation estimates for schrodinger type operators on obtient des estimations de propagation pour un operateur de type schrodinger en utilisant des techniques de commutateur multiple on donne une nouvelle methode pour obtenir des estimations pour des puissances de la resolvante

bose and fermi walk configurations on planar graphs the number fcn h of n walk configurations of type c is investigated on certain two rooted directed planar graphs h which will be always realized as plane graphs in r2 c may be bose or fermi as defined by inui and katori both types of configuration are collections of non crossing walks which follow the directed paths between the roots of the plane graph h in the case of configurations of fermi type each walk may be included only once the number fbosen h is shown to be a polynomial in n of degree nmax 1 where nmax is the maximum number of walks in a fermi configuration the coefficient of the highest power of n in this polynomial is simply related to the number of maximal fermi walk configurations it is also shown that nmax c h 1 where c h is the number of finite faces on h extension of these results to multi rooted graphs is also discussed when h is the union of paths between two sites of the directed square lattice subject to various boundary conditions kreweras showed that the number of bose configurations is equal to the number of n element multi chains on segments of young s lattice he expressed this number as a determinant the elements of which are polynomials in n we evaluate this determinant by the method of lu decomposition in the case of watermelon configurations above a wall in this case the polynomial is a product of linear factors but on introducing a second wall the polynomial does not completely factorize but has a factor which is the number of watermelon configurations on the largest rectangular subgraph the number of two rooted star configurations is found to be the product of the numbers of watermelon configurations on the three rectangular subgraphs into which it may be partitioned

einstein solvmanifolds are standard we study einstein manifolds admitting a transitive solvable lie group of isometries solvmanifolds it is conjectured that these exhaust the class of noncompact homogeneous einstein manifolds j heber has shown that under a simple algebraic condition he calls such a solvmanifold standard einstein solvmanifolds have many remarkable structural and uniqueness properties in this paper we prove that any einstein solvmanifold is standard by applying a stratification procedure adapted from one in geometric invariant theory due to f kirwan

propagation of singularities in many body scattering in the presence of bound states abstract in this paper we describe the propagation of singularities of tempered distributional solutions u s of h u 0 where h is a many body hamiltonian h v 0 v a v a and is not a threshold of h under the assumption that the inter particle e g two body interactions v a are real valued polyhomogeneous symbols of order 1 e g coulomb type with the singularity at the origin removed here the term singularity refers to a microlocal description of the lack of decay at infinity thus we prove that the set of singularities of u is a union of maximally extended broken bicharacteristics of h these are curves in the characteristic variety of h which can be quite complicated due to the existence of bound states we use this result to describe the wave front relation of the s matrices we also analyze lagrangian properties of this relation which shows that the relation is not too large in terms of its dimension

on solvable lie groups of negative ricci curvature we study solvable lie groups which admit a left invariant metric of strictly negative ricci curvature we obtain necessary and sufficient conditions of the existence of such a metric for lie groups the nilradical of whose lie algebra is either abelian or heisenberg or standard filiform and discuss some open questions

spanning forests and the q state potts model in the limit q 0 we study the q state potts model with nearest neighbor coupling v e j 1 in the limit q v 0 with the ratio w v q held fixed combinatorially this limit gives rise to the generating polynomial of spanning forests physically it provides information about the potts model phase diagram in the neighborhood of q v 0 0 we have studied this model on the square and triangular lattices using a transfer matrix approach at both real and complex values of w for both lattices we have computed the symbolic transfer matrices for cylindrical strips of widths 2 l 10 as well as the limiting curves b of partition function zeros in the complex w plane for real w we find two distinct phases separated by a transition point w w0 where w0 1 4 resp w0 0 1753 0 0002 for the square resp triangular lattice for w w0 we find a non critical disordered phase that is compatible with the predicted asymptotic freedom as w for w w0 our results are compatible with a massless berker kadanoff phase with central charge c 2 and leading thermal scaling dimension xt 1 2 marginally irrelevant operator at w w0 we find a first order critical point the first derivative of the free energy is discontinuous at w0 while the correlation length diverges as w w0 and is infinite at w w0 the critical ehavior at w w0 seems to be the same for both lattices and it differs from that of the berker kadanoff phase our results suggest that the central charge is c 1 the leading thermal scaling dimension is xt 1 0 and the critical exponents are 1 d 1 2 and 1

critical behaviour of the fully packed loop model on the square lattice we investigate the critical behaviour of the fully packed o n loop model on the square lattice in which each vertex is visited once by a loop a transfer matrix analysis shows that this model can be interpreted as a superposition of a low temperature o n model and an solid on solid sos model as for the fully packed model on the honeycomb lattice however not all of the critical exponents are the same for both lattices in contrast the fully packed model on the triangular lattice appears to behave as a pure low temperature o n model

spectral and scattering theory for differential and hankel operators abstract we consider a class of hankel operators h realized in the space l 2 r as integral operators with kernels h t s where h t p ln t t 1 and p x x n p n 1 x n 1 p 0 is an arbitrary real polynomial of degree n this class contains the classical carleman operator when n 0 we show that a hankel operator h in this class can be reduced by an explicit unitary transformation essentially by the mellin transform to a differential operator a v q d v in the space l 2 r here q x x n q n 1 x n 1 q 0 is a polynomial determined by p x and v 1 2 cosh 1 2 is the universal function then the operator a v q d v reduces by the generalized liouville transform to the standard differential operator b d n b n 1 x d n 1 b 0 x with the coefficients b m x m 0 n 1 decaying sufficiently rapidly as x this allows us to use the results of spectral theory of differential operators for the study of spectral properties of generalized carleman operators in particular we show that the absolutely continuous spectrum of h is simple and coincides with r if n is odd and it has multiplicity 2 and coincides with 0 if n 2 is even the singular continuous spectrum of h is empty and its eigenvalues may accumulate to the point 0 only as a by product of our considerations we develop spectral theory of a new class of degenerate differential operators a v q d v where q x is an arbitrary real polynomial and v is an almost arbitrary real function decaying at infinity

asymptotic completeness for 3 particle long range systems we prove asymptotic completeness for 3 particle systems with pair potentials satisfying the virial condition and decreasing at infinity as for 1 2

w extended kac representations and integrable boundary conditions in the logarithmic minimal models wlm 1 p we construct new yang baxter integrable boundary conditions in the lattice approach to the logarithmic minimal model giving rise to reducible yet indecomposable representations of rank 1 in the continuum scaling limit we interpret these extended kac representations as finitely generated extended feigin fuchs modules over the triplet w algebra the extended fusion rules of these representations are inferred from the recently conjectured virasoro fusion rules of the kac representations in the underlying logarithmic minimal model we also introduce the modules contragredient to the extended kac modules and work out the correspondingly extended fusion algebra our results are in accordance with the kazhdan lusztig dual of tensor products of modules over the restricted quantum universal enveloping algebra at q e i p finally polynomial fusion rings isomorphic with the various fusion algebras are determined and the corresponding grothendieck ring of characters is identified

conformally parallel g2 structures on a class of solvmanifolds starting from a 6 dimensional nilpotent lie group n endowed with an invariant su 3 structure we construct a homogeneous conformally parallel g 2 metric on an associated solvmanifold we classify all half flat su 3 structures that endow the rank one solvable extension of n with a conformally parallel g 2 structure by suitably deforming the su 3 structures obtained we are able to describe the corresponding non homogeneous ricci flat metrics with holonomy contained in g 2 in the process we also find a new metric with exceptional holonomy

negative ricci curvature on some non solvable lie groups ii we construct many examples of lie groups admitting a left invariant metric of negative ricci curvature we study lie algebras which are semidirect products mathfrak l mathfrak a oplus mathfrak u imes mathfrak n and we obtain examples where mathfrak u is any semisimple compact real lie algebra mathfrak a is one dimensional and mathfrak n is a representation of mathfrak u which satisfies some conditions in particular when mathfrak u mathfrak s mathfrak u m mathfrak s mathfrak o m or mathfrak s mathfrak p m and mathfrak n is a representation of mathfrak u in some space of homogeneous polynomials we show that these conditions are indeed satisfied in the case mathfrak u mathfrak s mathfrak u 2 we get a more general construction where mathfrak n can be any nilpotent lie algebra where mathfrak s mathfrak u 2 acts by derivations we also prove a general result in the case when mathfrak u is a semisimple lie algebra of non compact type

long range scattering for higher order schrodinger operators abstract this paper is concerned with the scattering problem for the higher order schrodinger operator m v x where v x is a long range potential at this moment the classical wave operators do not exist so we provide one kind of modified wave operators and show the existence and the asymptotic completeness of such wave operators as well as the expression of the corresponding scattering operator

micro local resolvent estimates for 2 body schrodinger operators on considere l operateur de schrodinger h v x dans l 2 r n en relation avec des localisations dans l espace de phase r x n r n

absolute continuity of the floquet spectrum for a nonlinearly forced harmonic oscillator we prove that the floquet spectrum of the time periodic schrodinger equation corresponding to a mildly nonlinear resonant forcing is purely absolutely continuous for suitably small

the scattering amplitude for the schrodinger equation with a long range potential we consider the schrodinger operator with a long range potential v x in the space our goal is to study spectral properties of the corresponding scattering matrix and a diagonal singularity of its kernel the scattering amplitude it turns out that in contrast to the short range case the dirac function singularity of at the diagonal disappears and the spectrum of the scattering matrix covers the whole unit circle for an asymptotically homogeneous function v x of order we show that typically where the module w and the phase are asymptotically homogeneous functions as of orders and respectively leading terms of asymptotics of w and at are calculated in the case 1 our results generalize in the limit the well known formula of gordon and mott