conditionally solvable path integral problems ii natanzon potentials new classes of exactly solvable potentials are discussed within the path integral formalism they are constructed from the hypergeometric and confluent natanzon potentials respectively it is found that they allow incorporation of four free parameters which give rise to fractional power behaviour long range and strongly anharmonic terms we find six different classes of such potentials

many body diffusion and path integrals for identical particles for distinguishable particles it is well known that brownian motion and a feynman kac functional can be used to calculate the path integral for imaginary times for a general class of scalar potentials in order to treat identical particles we exploit the fact that this method separates the problem of the potential dealt with by the feynman kac functional from the process which gives sample paths of a noninteracting system for motion in one dimension we emphasize that the permutation symmetry of the identical particles completely determines the domain of brownian motion and the appropriate boundary conditions absorption for fermions reflection for bosons further analysis of the sample paths for motion in three dimensions allows us to decompose these paths into a superposition of one dimensional sample paths this reduction expresses the propagator and consequently the energy and other thermodynamical quantities in terms of well behaved one dimensional fermion and boson diffusion processes and the feynman kac functional

grothendieck standard conjectures morphic cohomology and hodge index theorem using morphic cohomology we produce a sequence of conjectures called morphic conjectures which terminates at the grothendieck standard conjecture a a refinement of hodge structures is given and with the assumption of morphic conjectures we prove a hodge index theorem we answer a question of friedlander and lawson by assuming the grothendieck standard conjecture b and prove that the topological filtration from morphic cohomology is equal to the grothendieck arithmetic filtration for some cases

self correspondences of the generic fibre of lefschetz pencils and the leray filtration abstract in this paper we give a detailed analysis of the interaction between homological self correspondences of the general fibre y k t of the lefschetz fibration x p 1 of a lefschetz pencil on a smooth projective variety x k and the leray filtration of we derive the result that if the standard conjecture b y holds then the operator x p x n 1 is algebraic where p x n 1 is defined as the inverse of l on l p n 1 x and 0 on l k p j x for 1 n 1 k j in the course of our proof we see that under the above assumption the kunneth projectors x i for i n 1 n n 1 are algebraic

l algebre de hopf et le groupe de galois motiviques d un corps de caracteristique nulle ii this is the second article of a series of two aiming at constructing and studying motivic galois groups in the context of triangulated motives these motivic galois groups were constructed in the first article and their algebras of regular functions were described concretely in terms of differential forms or algebraic cycles in the present article we gather important complements to the first one in the first part of this article we describe the link between the motivic galois group and the usual galois group of a subfield of in the second part we develop the basis of a theory of ramification for the motivic galois groups by constructing motivic versions of the decomposition and inertia groups associated to a geometric valuation we also introduce the relative motivic galois group of an extension k k which measures the difference between the motivic galois groups of k and k the latter is more accessible than its absolute counterpart in effect we show that it is a quotient of the pro algebraic completion of the topological fundamental pro group of a the pro variety homk k at least when the extension k k is of finite type

the functional analytic versus the functional integral approach to quantum hamiltonians the one dimensional hydrogen atoma the capabilities of the functional analytic and of the functional integral approach for the construction of the hamiltonian as a self adjoint operator on hilbert space are compared in the context of non relativistic quantum mechanics differences are worked out by taking the one dimensional hydrogen atom as an example that is a point mass on the euclidean line subjected to the inverse distance potential this particular choice is made with the intent to clarify a long lasting discussion about its spectral properties in fact for the four parameter family of possible hamiltonians the corresponding energy dependent green functions are derived in closed form the multiplicity of hamiltonians should be kept in mind when modeling certain experimental situations as for instance in quantum wires

remarks on the local time rescaling in path integration the local time transformation used in path integration is shown to be equivalent to standard procedures applied to schrodinger s equation

feynman integrals for a class of exponentially growing potentials we construct the feynman integrands for a class of exponentially growing time dependent potentials as white noise functionals we show that they solve the schrodinger equation the morse potential is considered as a special case

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation

path integral treatment of ring shaped topological defects explicit path integration is carried out in a space with a ring shaped topological defect using toroidal coordinates the toroidal aharonov bohm experiment is taken as an example

motives and filtrations on chow groups in this paper sequential to sa we discuss a philosophy of bloch and beilinson concerning filtrations on chow groups of algebraic vareties which provides us with a good understanding on the structure of at least certain portions of chow groups let k be a fixed ground field and let c be the category of smooth projective schemes over k with a fixed suitable weil cohomology theory cf notations x in mathcal c rightsquigarrow h x mathop oplus limits i in z h i x

motivation for hodge cycles given two smooth projective varieties x and y x is defined to motivate y if the motive of y is contained in the tensor category generated by x some techniques are given for checking this condition it is shown that in a number of cases moduli spaces of sheaves over curves or surfaces are motivated by the underlying curve or surface this is used to check the hodge and related conjectures for some of these examples

spectral functions of non essentially self adjoint operators fil falomir horacio alberto consejo nacional de investigaciones cientificas y tecnicas centro cientifico tecnologico conicet la plata instituto de fisica la plata universidad nacional de la plata facultad de ciencias exactas instituto de fisica la plata argentina

noncommutative motives numerical equivalence and semi simplicity making use of hochschild homology we introduce the correct category nnum k f of noncommutative numerical motives over a base ring k and with coefficients in a field f we prove that nnum k f is abelian semi simple and that grothendieck s category num k q of numerical motives embeds into nnum k q after being factored out by the action of the tate object as an application we obtain an alternative proof of jannsen s celebrate semi simplicity result which uses the noncommutative world instead of a classical weil cohomology

on the shafarevich and tate conjectures for hyperkahler varieties 1 problems and results 2 polarized hyperk ihler varieties and cubic fourfoids 3 the period mapping 4 the kuga satake construction 5 the kuga satake construction in the relative setting 6 proof of theorem 1 5 1 7 proof of theorem 1 6 1 8 descent 9 proof of theorem 1 3 1 appendix 1 spinorial shimura varieties appendix 2 a short proof of the hodge conjecture for cubic fourfolds 205 210 212 216 220 224 229 233 240 244 245

treatment of some singular potentials by change of variables in wiener integrals using a regularization of the classical kepler problem we show that a change of process and a change of time in stochastic differential equations allow a treatment of singular potentials in wiener integrals

how to solve path integrals in quantum mechanics a systematic classification of feynman path integrals in quantum mechanics is presented and a table of solvable path integrals is given which reflects the progress made during the last 15 years including of course the main contributions since the invention of the path integral by feynman in 1942 an outline of the general theory is given which will serve as a quick reference for solving path integrals explicit formulas for the so called basic path integrals are presented on which the general scheme to classify and calculate path integrals in quantum mechanics is based

conditionally solvable path integral problems some specific conditionally exactly solvable potentials are discussed within the path integral formalism they generalize the usually known potentials by the incorporation of fractional power behaviour and strongly anharmonic terms we find four different kinds of such potential the first is related to the coulomb potential the second is an anharmonic confinement potential and the third and fourth are related to the manning rosen potential

noncommutative motives in positive characteristic and their applications abstract let k be a base field of positive characteristic making use of topological periodic cyclic homology we start by proving that the category of noncommutative numerical motives over k is abelian semi simple as conjectured by kontsevich then we establish a far reaching noncommutative generalization of the weil conjectures originally proven by dwork and grothendieck in the same vein we establish a far reaching noncommutative generalization of the cohomological interpretations of the hasse weil zeta function originally proven by hesselholt as a third main result we prove that the numerical grothendieck group of every smooth proper dg category is a finitely generated free abelian group as claimed without proof by kuznetsov then we introduce the noncommutative motivic galois super groups and following an insight of kontsevich relate them to their classical commutative counterparts finally we explain how the motivic measure induced by berthelot s rigid cohomology can be recovered from the theory of noncommutative motives

triangulated riemann surfaces with boundary and the weil petersson poisson structure given a hyperbolic surface with geodesic boundary s the lengths of a maximal system of disjoint simple geodesic arcs on s that start and end at s perpendicularly are coordinates on the teichmuller space t s we express the weil petersson poisson structure of t s in this system of coordinates and we prove that it limits pointwise to the piecewise linear poisson structure defined by kontsevich on the arc complex of s at the same time we obtain a formula for the first order variation of the distance between two closed geodesics under fenchel nielsen deformation

su 2 coherent state path integral the su 2 coherent state path integral is used to represent the matrix element of a propagator in the su 2 coherent state basis it is argued that the continuum representation of this integral is correct provided the necessary boundary term is taken into account in the case of the su 2 dynamical symmetry the path integral is explicitly computed by means of a change of variables the su 2 motion of the underlying phase space the correct stationary phase expansion for the propagator in terms of the total action including boundary term and classical trajectories is obtained

motives and siegel modular forms for a motive m over q we define the fundamental periods of m using invariant theory our definition generalizes deligne s periods we show that if a motive m is constructed from motives m 1 m 2 m n by a standard algebraic operation then the fundamental periods of m can be expressed as monomials of the fundamental periods of m 1 m 2 m n applying this theory we discuss two hypothetical motives attached to a siegel modular form we show that a siegel modular form of degree m has at most m 1 period invariants

quadratic algebras yang baxter equation and artin schelter regularity abstract we study two classes of quadratic algebras over a field k the class c n of all n generated pbw algebras with polynomial growth and finite global dimension and the class of quantum binomial algebras we show that a pbw algebra a is in c n iff its hilbert series is h a z 1 1 z n furthermore the class c n contains a unique up to isomorphism monomial algebra a k x 1 x n x j x i 1 i j n a surprising amount can be said when a is a quantum binomial algebra that is its defining relations are nondegenerate square free binomials x y c x y z t c x y k our main result shows that for an n generated quantum binomial algebra a the following conditions are equivalent i a is an artin schelter regular pbw algebra ii a is a yang baxter algebra that is the set of relations defines canonically a solution of the yang baxter equation iii a is a binomial skew polynomial ring with respect to an enumeration of x iv the koszul dual a is a quantum grassmann algebra

on the green function of linear evolution equations for a region with a boundary we derive a closed form expression for the green function of linear evolution equations with the dirichlet boundary condition for an arbitrary region based on the singular perturbation approach to boundary problems

motives with exceptional galois groups and the inverse galois problem we construct motivic l adic representations of textup gal overline mathbb q mathbb q into exceptional groups of type e 7 e 8 and g 2 whose image is zariski dense this answers a question of serre the construction is uniform for these groups and is inspired by the langlands correspondence for function fields as an application we solve new cases of the inverse galois problem the finite simple groups e 8 mathbb f ell are galois groups over mathbb q for large enough primes l

rigidity of polyhedral surfaces ii we study the rigidity of polyhedral surfaces using variational principles the action functionals are derived from the cosine laws the main focus of this paper is on the cosine law for a nontriangular region bounded by three possibly disjoint geodesics several of these cosine laws were first discovered and used by fenchel and nielsen by studying the derivative of the cosine laws we discover a uniform approach to several variational principles on polyhedral surfaces with or without boundary as a consequence the work of penner bobenko and springborn and thurston on rigidity of polyhedral surfaces and circle patterns are extended to a very general context

on tensor product l functions for quaternion unitary groups and gl 2 over totally real number fields mixed weight cases abstract we prove an algebraicity of special values of tensor product l functions for quaternion unitary groups and gl 2 in the mixed weight cases over any totally real number field as a consequence we can prove a period relation for automorphic forms of quaternion unitary groups of degree 2 and gsp 4 when these automorphic forms correspond under the conjectural jacquet langlands correspondence and a period relation for endoscopic liftings of gsp 4 in very general situations

emergent spacetime from modular motives the program of constructing spacetime geometry from string theoretic modular forms is extended to calabi yau varieties of dimensions three and four as well as higher rank motives modular forms on the worldsheet can be constructed from the geometry of spacetime by computing the l functions associated to omega motives of calabi yau varieties generated by their holomorphic n forms via galois representations the modular forms that emerge in this way are related to characters of the underlying rational conformal field theory the converse problem of constructing space from string theory proceeds in the class of diagonal theories by determining the motives associated to modular forms in the category of pure motives with complex multiplication the emerging picture suggests that the l function can be viewed as defining a map between the geometric category of motives and the category of conformal field theories on the worldsheet

polarizations and grothendieck s standard conjectures we prove that grothendieck s hodge standard conjecture holds for abelian varieties in arbitrary characteristic if the hodge conjecture holds for complex abelian varieties of cm type for abelian varieties with no exotic algebraic classes we prove the hodge standard conjecture unconditionally

hodge classes on abelian varieties of low dimension in this paper we study hodge classes on complex abelian varieties x if dimx then it is wellknown that every hodge class on x is a linear combination of products of divisor classes in the authors showed that if x is simple of dimension then every hodge class is a linear combination of products of divisor classes and weil classesif there are any the notion of a weil class shall be briey reviewed in the aim of this note is to extend this to arbitrary abelian varieties of dimension in order to state our main results let us describe some special cases a the abelian variety x is isogenous to a product x x where x is an elliptic curve with complex multiplication by an imaginary quadratic eld k and where x is a simple abelian threefold such that there exists an embedding k endx b the abelian variety x is simple of dimension such that endx is a eld containing an imaginary quadratic eld k which acts on the tangent space tx with multiplicities see x for further explanation c the abelian variety x is simple of dimension with d endx a denite quaternion algebra over q type iii in the albert classication note that for every d n q the subalgebra q d is an imaginary quadratic eld d the abelian variety x is simple of dimension with endx q

integral motives and special values of zeta functions for each field k we define a category of rationally decomposed mixed motives with z coefficients when k is finite we show that the category is tannakian and we prove formulas relating the behaviour of zeta functions near integers to certain ext groups

pole structure of the hamiltonian ζ function for a singular potential we study the pole structure of the function associated with the hamiltonian h of a quantum mechanical particle living in the half line subject to the singular potential gx 2 x2 we show that h admits nontrivial self adjoint extensions sae in a given range of values of the parameter g the functions of these operators present poles which depend on g and in general do not coincide with half an integer they can even be irrational the corresponding residues depend on the sae considered

changing dimension and time two well founded and practical techniques for path integration in quantum physics in a reasonably self contained presentation mathematical rigour is supplied to the important ideas of solving certain non gaussian path integrals by changes of dimension and or path dependent time transformations the resulting genuine path integral calculus neither requires discretization prescriptions nor sophisticated methods from the theory of stochastic differential equations the power of the calculus is illustrated by two standard quantum physics applications first the calculation of the time dependent propagator corresponding to a particle on the half line in a harmonic plus inverse square potential is shown to be a simple exercise second the first rigorous derivation of the energy dependent green function of the one dimensional morse system is given

on differential operators attached to certain representations of classical groups classification et construction de sous espaces irreductibles de sr integrales contenant les applications polynomiales l operateur differentiel d operateurs differentiels de type arithmetique applications aux series de eisenstein

on special values of certain l functions we prove an algebraicity result concerning special values at critical points in the sense of deligne of tensor product l functions associated to automorphic representations of special orthogonal groups for quadratic forms which are totally definite and cuspidal representations of rm gl 2 corresponding to primitive cusp forms over totally real number fields we also prove the reciprocity law i e the equivariance under the action of rm gal overline bbb q bbb q for the special values in the appendix the second author calculates the deligne periods for such l functions assuming the existence of corresponding motives and the automorphic transfer to a quasi split form of the special orthogonal group our result conforms with the celebrated conjecture of deligne on special values of motivic l functions

the general besselian and legendrian path integrals the general besselian and legendrian path integrals based on the confluent and hypergeometric natanzon potentials are calculated these two solutions cover all other path integral representations which are related to the radial harmonic oscillator and the modified poschl teller path integral

critical values of l functions on gsp 2 gl 2 this paper deals with deligne s conjecture on the critical values of l functions let z g h s denote the tensor product l function attached to a siegel modular form g of weight k and an elliptic cusp form h of weight l we assume that the first fourier jacobi coefficient of g is not identically zero then deligne s conjecture is fully proven for z g h s when l 2k 2 and partly for the remaining case

pullback of the lifting of elliptic cusp forms and miyawaki s conjecture we construct a lifting from siegel cusp forms of degree r to siegel cusp forms of degree r 2n for r n 1 our result is a partial solution of a conjecture made by miyawaki 27 page 307 in 1992 in particular we can calculate the standard l function of a cusp form of degree 3 and weight 12 which is in accordance with miyawaki s conjecture we give a conjecture on the petersson inner product of the lifting in terms of certain l values

delta function perturbations and neumann boundary conditions by path integration delta function perturbations and neumann boundary conditions are incorporated into the path integral formalism the starting point is consideration of the path integral representation for the one dimensional dirac particle together with a relativistic point interaction the nonrelativistic limit yields either the usual delta function or a delta function perturbation making their strengths infinitely repulsive one respectively obtains either dirichlet or neumann boundary conditions in the path integral

new aspects of the path integrational treatment of the coulomb potential the well known treatment of the path integral for the coulomb potential by means of the kustaanheimo stiefel transformation and a time transformation is made more transparent by reducing the problem to the equality of two measures on the space of paths for this equality two proofs are given an elementary computational one and a short one recurring to general features of stochastic processes it is shown that the time transformation is a special case of the well known time change for a continuous local martingale by which the process is changed to a brownian motion and which is determined by its quadratic variation

motives and admissible representations of automorphism groups of fields some of basic properties of the groups of automorphisms of algebraically closed fields and of their smooth representations are studied in characteristic zero grothendieck motives modulo numerical equivalence are identified with a full subcategory in the category of graded smooth representations of certain automorphism groups of algebraically closed fields

noncommutative geometry and motives the thermodynamics of endomotives we combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of l functions the analogue in characteristic zero of the action of the frobenius on l adic cohomology is the action of the scaling group on the cyclic homology of the cokernel in a suitable category of motives of a restriction map of noncommutative spaces the latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive a noncommutative generalization of artin motives semigroups of endomorphisms of algebraic varieties give rise canonically to such endomotives with an action of the absolute galois group the semigroup of endomorphisms of the multiplicative group yields the bost connes system from which one obtains through the above procedure the desired cohomological interpretation of the zeros of the riemann zeta function in the last section we also give a lefschetz formula for the archimedean local l factors of arithmetic varieties

on motivic decompositions arising from the method of bialynicki birula recently v chernousov s gille and a merkurjev have obtained a decomposition of the motive of an isotropic smooth projective homogeneous variety analogous to the bruhat decomposition using the method of a bialynicki birula and a corollary which is essentially due to s del bano i generalize this decomposition to the case of a possibly anisotropic smooth projective variety homogeneous under the action of an isotropic reductive group this answers a question of n karpenko

hodge classes on certain types of abelian varieties the hodge conjecture for a states that such classes are algebraic in the sense that they are q linear combinations of cohomology classes of algebraic subvarieties of codimension p on a this statement is well known for p 1 lefschetz and it is trivially true for p 0 hence for a given a it holds for allp if the ring formed by the hodge classes under cup product is generated by the hodge classes of codimension 1 we thus have a simple criterion for the hodge conjecture to be true for a which we call the 1 1 criterion responding to a question posed by tate 14 p 107 mumford found that the 1 1 criterion while true for all a such that dim a c 3 is not necessarily satisfied by abelian varieties of dimension 4 although mumford s counterexample involved an abelian variety of cm type 51 weil 15 has stressed that its most important feature is the action of an imaginary quadratic field k on a in such a way that lie a becomes a free k x c module abelian varieties of precisely this type play an important role in deligne s proof 2 that hodge cycles on an arbitrary abelian variety are absolutely hodge there is speculation that the hodge conjecture is false at least in general for such abelian varieties but no method is available at present for settling this question either way indeed one knows no example of an algebraic variety for which the hodge conjecture is false and only scattered examples shioda 12 of