brownian motion on the hyperbolic plane and selberg trace formula abstract we will show that the relation of the heat kernels for the schrodinger operators with uniform magnetic fields on the hyperbolic plane h 2 the maass laplacians and for the schrodinger operators with morse potentials on r is given by means of a one dimensional fourier transform in the framework of stochastic analysis where the brownian motion on h 2 plays an important role by using this relation we will give the explicit forms of the green functions as a typical related problem we will discuss the selberg trace formula the close relation of the trace formula with the corresponding classical mechanics will also be discussed

functional determinants in quantum field theory functional determinants of differential operators play a prominent role in theoretical and mathematical physics and in particular in quantum field theory they are however difficult to compute in non trivial cases for one dimensional problems a classical result of gel fand and yaglom dramatically simplifies the problem so that the functional determinant can be computed without computing the spectrum of eigenvalues here i report recent progress in extending this approach to higher dimensions i e functional determinants of partial differential operators with applications in quantum field theory

the selberg trace formula for bordered riemann surfaces a selberg trace formula is derived for the laplace beltrami operator on bordered riemann surfaces with dirichlet or neumann boundary conditions respectively using a construction via the compact double of the surface for which the standard trace formula is valid applications of the trace formula to spectral functions of the laplace beltrami operators are discussed and their functional determinants are explicitly expressed in terms of various selberg zeta functions for selberg s zeta function relevant to the dirichlet boundary value problem a representation as a dirichlet series is given for which we conjecture conditional convergence even within the critical strip for res 1 2

determinants of laplacians the determinant of the laplacian on spinor fields on a riemann surface is evaluated in terms of the value of the selberg zeta function at the middle of the critical strip a key role in deriving this relation is played by the barnes double gamma function

eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps we study eta invariants of dirac operators and regularized determinants of dirac lapla cians over hyperbolic manifolds with cusps and their relations with selberg zeta functions using the selberg trace formula and a detailed analysis of the unipotent orbital integral we show that the eta and zeta functions defined by the relative traces are regular at the origin so that we can define the eta invariant and the regularized determinant we also show that the selberg zeta function of odd type has a meromorphic extension over c prove a relation of the eta invariant and a certain value of the selberg zeta function of odd type and derive a corresponding functional equation these results generalize the earlier work of john millson to hyperbolic manifolds with cusps we also prove that the selberg zeta function of even type has a meromorphic extension over c relate it to the regularized determinant and obtain a corresponding functional equation

the selberg trace formula and selberg zeta function for cofinite kleinian groups with finite dimensional unitary representations for cofinite kleinian groups with finite dimensional unitary representations we derive the selberg trace formula as an application we define the corresponding selberg zeta function and compute its divisor thus generalizing results of elstrodt grunewald and mennicke to non trivial unitary representations we show that the presence of cuspidal elliptic elements sometimes adds ramification points to the zeta function in fact if open image in new window is the ring of eisenstein integers then the selberg zeta function of open image in new window contains ramification points and is the sixth root of a meromorphic function

analytic torsion for surfaces with cusps i compact perturbation theorem and anomaly formula we define the analytic torsion associated with a riemann surface endowed with a metric having poincare type singularities in the neighborhood of a finite number of points and a hermitian vector bundle with at most logarithmic singularities at those points coming from the metric on the negative power of the canonical line bundle twisted by the divisor of the points then we provide a relation between this analytic torsion and the ray singer analytic torsion of the compactified surface from this relation we then establish the anomaly formula which describes how the analytic torsion changes under the change of the metric on the surface and on the vector bundle

integrals involving a function associated with the euler maclaurin summation formula the authors first evaluate sums of certain series involving the riemann zeta function by using the theory of the double gamma function with or without the use of these results the authors then establish seberal integrals involving a certain periodic function which is associated for example with the euler maclaurin summation formula finally by introducing some new mathematical constants several families of definite integrals involving logarithms of gamma functions are considered relevant connections of the results presented here with those given in earlier works are also pointed out

analytic torsion of all vector bundles over an elliptic curve the spectrum of the hodge laplace operator of all holomorphic vector bundles with parallel curvature metrics over a flat elliptic curve is shown to be the union of the spectra of some infinite set of explicit finite rank matrices the analytic torsion is then deduced by direct computation another evaluation of the analytic torsion is then performed using bismut gillet and soule s comparison formula for short exact sequences of vector bundles this is a test for this method which could be useful when direct calculation is hopeless

zeta invariants for sequences of spectral type special functions and the lerch formula we discuss the spectral properties of a class of sequences of what we call spectral type we introduce an effective method to calculate the zeta invariants for this type of sequence such invariants are given in terms of some new and old special functions and we consider a number of examples in which we study the properties of these special functions

determinants of laplace like operators on riemann surfaces we calculate determinants of second order partial differential operators defined on riemann surfaces of genus greater than one using a relation between selberg s zeta function and functional determinants in addition we perform a calculation of these determinants directly using selberg s trace formula and compare our results with previous computations which followed the latter route

precise constants in bosonization formulas on riemann surfaces i a computation of the constant appearing in the spin 1 bosonization formula is given this constant relates faltings delta invariant to the zeta regularized determinant of the laplace operator with respect to the arakelov metric

quantum liouville theory in the background field formalism i compact riemann surfaces using polyakov s functional integral approach and the liouville action functional defined in zt87c and tt03a we formulate quantum liouville theory on a compact riemann surface x of genus g 1 for the partition function x and correlation functions with the stress energy tensor components langle prod i 1 n t z i prod k 1 l bar t bar w k x rangle we describe feynman rules in the background field formalism by expanding corresponding functional integrals around a classical solution the hyperbolic metric on x extending analysis in tak93 tak94 tak96a tak96b we define the regularization scheme for any choice of the global coordinate on x for the schottky and quasi fuchsian global coordinates we rigorously prove that one and two point correlation functions satisfy conformal ward identities in all orders of the perturbation theory obtained results are interpreted in terms of complex geometry of the projective line bundle fancyscript e c lambda h c 2 over the moduli space mathfrak m g where c is the central charge and h is the hodge line bundle and provide the friedan shenker fs87 complex geometry approach to cft with the first non trivial example besides rational models

some mathematical constants the theory of multiple gamma functions was studied in about 1900 and has been revived in the study of determinants of laplacians in the middle or so of 1980s there are some classes of mathematical constants involved naturally in the gamma and multiple gamma functions here we introduce those known mathematical constants associated with the gamma and multiple gamma functions and will show how they are involved if possible

a local families index formula for overline partial operators on punctured riemann surfaces using heat kernel methods developed by vaillant a local index formula is obtained for families of overline partial operators on the teichmuller universal curve of riemann surfaces of genus g with n punctures the formula also holds on the moduli space mathcal m g n in the sense of orbifolds where it can be written in terms of mumford morita miller classes the degree two part of the formula gives the curvature of the corresponding determinant line bundle equipped with the quillen connection a result originally obtained by takhtajan and zograf

on certain definite integrals which arise in automorphic lie theory calculates in closed form a family of definite integrals in b a c n 0 1 2 3 which arise in the calculation of regularized functional determinants associated with a compact space form x of a rank 1 riemannian symmetric space in the special case when n 0 b 1 2 a pi c 1 and x is a riemann surface the integral i0 1 2 pi 1 is known for example in the context of polyakov string theory

a new loop algebra and a corresponding integrable hierarchy as well as its integrable coupling a type of new interesting loop algebra gm m 1 2 with a simple commutation operation just like that in the loop algebra a1 is constructed with the help of the loop algebra gm a new multicomponent integrable system m akns kn hierarchy is worked out as reduction cases the m akns hierarchy and m kn hierarchy are engendered respectively in addition the system 1 akns kn which is a reduced case of the m akns kn hierarchy above is a unified expressing integrable model of the akns hierarchy and the kn hierarchy obviously the m akns kn hierarchy is again a united expressing integrable model of the multicomponent akns hierarchy m akns and the multicomponent kn hierarchy m kn this article provides a simple method for obtaining multicomponent integrable hierarchies of soliton equations finally we work out an integrable coupling of the m akns kn hierarchy

the selberg trace formula for dirac operators we examine spectra of dirac operators on compact hyperbolic surfaces particular attention is devoted to symmetry considerations leading to nontrivial multiplicities of eigenvalues the relation to spectra of maas laplace operators is also exploited our main result is a selberg trace formula for dirac operators on hyperbolic surfaces

simplified vacuum energy expressions for radial backgrounds and domain walls we extend our previous results of simplified expressions for functional determinants for radial schrodinger operators to the computation of vacuum energy or mass corrections for static but spatially radial backgrounds and for domain wall configurations our method is based on the zeta function approach to the gel fand yaglom theorem suitably extended to higher dimensional systems on separable manifolds we find new expressions that are easy to implement numerically for both zero and non zero temperatures

selberg trace formula for bordered riemann surfaces hyperbolic elliptic and parabolic conjugacy classes and determinants of maass laplacians the selberg trace formula for automorphic forms of weightme on bordered riemann surfaces is developed the trace formula is formulated for arbitrary fuchsian groups of the first kind with reflection symmetry which include hyperbolic elliptic and parabolic conjugacy classes in the case of compact bordered riemann surfaces we can explicitly evaluate determinants of maass laplacians for both dirichlet and neumann boundary conditions respectively some implications for the open bosonic string theory are mentioned

unstable periodic orbits and semiclassical quantisation the bohr sommerfeld quantisation condition has a meaningful extension to classically chaotic systems whose periodic unstable orbits are isolated it provides a semiclassical euler factorisation for the functional determinant of the quantal hamiltonian in contrast to the hadamard infinite product over the eigenvalues by which the exact determinant is defined

regularized products and determinants zeta regularized products are used to define determinants of operators in infinite dimensional spaces this article provides a general theory of regularized products and determinants which delivers a better approach to their existence and explicit determination

error bounds and exponential improvement for the asymptotic expansion of the barnes g function in this paper we establish new integral representations for the remainder term of the known asymptotic expansion of the logarithm of the barnes g function using these representations we obtain e

krull s theory for the double gamma function abstract the barnes g function g x 1 2 satisfies the functional equation log g x 1 log g x log x we complement w krull s work in bemerkungen zur differenzengleichung g x 1 g x x math nachrichten 1 1948 365 376 with additional results that yield a different characterization of the function g new expansions and sharp bounds for g on x 0 in terms of gamma and digamma functions a new expansion for the gamma function and summation formulae with polygamma functions

asymptotics of the spectrum and the selberg zeta function on the space of riemann surfaces letz s r be the selberg zeta function of a compact riemann surfacer we study the behavior ofz s r asr tends to infinity in the moduli space of stable curves the main result is an estimate forz s r valid fors in a neighborhood depending only on the genus ofs 1 our analysis gives an alternate proof of the belavin knizhnik double pole result 5

the asymptotics of the arakelov geen s function and faltings delta invariant we study the behavior of the arakelov green s function and faltings delta invariant on degenerating riemann surfaces

spectral analysis and zeta determinant on the deformed spheres we consider a class of singular riemannian manifolds the deformed spheres s n k defined as the classical spheres with a one parameter family g k of singular riemannian structures that reduces for k 1 to the classical metric after giving explicit formulas for the eigenvalues and eigenfunctions of the metric laplacian delta s n k we study the associated zeta functions zeta s delta s n k we introduce a general method to deal with some classes of simple and double abstract zeta functions generalizing the ones appearing in zeta s delta s n k an application of this method allows to obtain the main zeta invariants for these zeta functions in all dimensions and in particular zeta 0 delta s n k and zeta 0 delta s n k we give explicit formulas for the zeta regularized determinant in the low dimensional cases n 2 3 thus generalizing a result of dowker 25 and we compute the first coefficients in the expansion of these determinants in powers of the deformation parameter k

higher selberg zeta functions in the paper kw2 we introduced a new type of selberg zeta function for establishing a certain identity among the non trivial zeroes of the selberg zeta function and of the riemann zeta function we shall call this zeta function a higher selberg zeta function the purpose of this paper is to study the analytic properties of the higher selberg zeta function z s especially to obtain the functional equation we also describe the gamma factor of z s in terms of the triple sine function explicitly and further determine the complete higher selberg zeta function with having a discussion of a certain generalized zeta regularization

extremal metrics for spectral functions of dirac operators in even and odd dimensions abstract let m n g be a closed smooth riemannian spin manifold and denote by its atiyah singer dirac operator we study the variation of riemannian metrics for the zeta function and functional determinant of 2 and prove finiteness of the morse index at stationary metrics and local extremality at such metrics under general i e not only conformal change of metrics in even dimensions which is also a new case for the conformal laplacian the relevant stability operator is of log polyhomogeneous pseudodifferential type and we prove new results of independent interest on the spectrum for such operators we use this to prove local extremality under variation of the riemannian metric which in the important example when m n g is the round n sphere gives a partial verification of branson s conjecture on the pattern of extremals thus det 2 has a local max max min min when the dimension is 4 k 4 k 1 4 k 2 4 k 3 respectively

determinants of laplacians and isopolar metrics on surfaces of infinite area we construct a determinant of the laplacian for infinite area surfaces which are hyperbolic near infinity and without cusps in the case of a convex co compact hyperbolic metric the determinant can be related to the selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the laplacian in the hyperbolic near infinity case the determinant is analyzed through the zeta regularized relative determinant for a conformal metric perturbation we establish that this relative determinant is a ratio of entire functions of order two with divisor corresponding to eigenvalues and resonances of the perturbed and unperturbed metrics these results are applied to the problem of compactness in the smooth topology for the class of metrics with a given set of eigenvalues and resonances

eta invariant and selberg zeta function of odd type over convex co compact hyperbolic manifolds abstract we show meromorphic extension and give a complete description of the divisors of a selberg zeta function of odd type z o associated to the spinor bundle on an odd dimensional convex co compact hyperbolic manifold h 2 n 1 as a byproduct we do a full analysis of the spectral and scattering theory of the dirac operator on asymptotically hyperbolic manifolds we show that there is a natural eta invariant d associated to the dirac operator d over a convex co compact hyperbolic manifold h 2 n 1 and that exp i d z o 0 thus extending millson s formula to this setting under some assumption on the exponent of convergence of poincare series for the group we also define an eta invariant for the odd signature operator and we show that for schottky 3 dimensional hyperbolic manifolds it gives the argument of a holomorphic function which appears in the zograf factorization formula relating two natural kahler potentials for weil petersson metric on schottky space

regularized determinants of the laplacian for cofinite kleinian groups with finite dimensional unitary representations for cofinite kleinian groups or equivalently finite volume three dimensional hyperbolic orbifolds with finite dimensional unitary representations we evaluate the regularized determinant of the laplacian using w muller s regularization we give an explicit formula relating the determinant to the selberg zeta function

heat kernels and super determinants of laplace operators on super riemann surfaces heat kernels of laplacians on superfields of arbitrary tensor weight on super riemann surfaces are constructed and are used to compute the super determinants of these operators in terms of the selberg super zeta function

asymptotic expansions of the double zeta function abstract the double zeta function of barnes 2 v z w is considered for large and small values of z and w with w 0 arg z r v 2 and r z 0 to the whole complex z plane and complex v plane with arg z

determinant expression of selberg zeta functions ii we will prove that for psl 2 r and its cofinite subgroup the selberg zeta function is expressed by the determinant of the laplacian we will also give an explicit calculation in case of congruence subgroups and deduce that the part of the determinant of the laplacian composed of the continuous spectrum is expressed by dirichlet functions the first discovery of the relation between the selberg zeta function and the determinant of the laplacian was by physicists 3 4 7 sarnak 14 and voros 15 obtained the determinant expression of selberg zeta functions for compact riemann surfaces with torsionfree fundamental groups in those cases all the spectrum of the laplacians are discrete the determinant was defined via the holomorphy at the origin of the spectral zeta function of minakshisundaram and pleijel 13 for noncompact but finite riemann surfaces with torsion free fundamental groups these results are generalized by efrat 5 in this case there exist both discrete and continuous spectrum he constructs the spectral zeta function composed not only of eigenvalues but some values concerning continuous spectrum which are decided by all the poles of the scattering de terminant in the selberg trace formula the determinant of the laplacian is defined by the standard method with the holomorphy of the spectral zeta func tion at the origin the aim of the present paper is to generalize his results to the case with any fundamental group y 3 and to give some arithmetic examples of the determinant of the laplacians 4 in 4 we restrict ourselves to the case when y is a congruence subgroup of psl 2 z in this case the partial spectral zeta function composed of only eigenvalues is also holomorphic at the origin 11 theorem 3 3 hence we have a decomposition of the determinant into parts corresponding to the discrete and continuous spectrum the scatter ing determinant is expressed very explicitly by huxley 8 in terms of dirichlet 7 functions i x almost all the poles of the scattering determinant are de scribed by nontrivial zeros of l s x the continuous part of the determinant

eigenvalue and eigenfunction error estimates for finite element formulations of linear hydroelasticity convergence of an approximate method for determining vibrational eigenpairs of an elastic solid containing an incompressible fluid is examined the field variables are solid displacement and fluid pressure we show that in suitable sobolev spaces a variational formulation exists whose solution eigenvalues and eigenfunctions are identified with those of a compact operator a nonconforming finite element approximation of this variational problem is described and optimal a priori error estimates are obtained for both the eigenvalues and eigenfunctions

series involving the zeta function and multiple gamma functions the theory of multiple gamma functions which was recently revived in the study of the determinants of the laplacians was applied in several earlier works in order to evaluate some families of series involving the riemann zeta function as well as to compute the determinants of the laplacians here in the present paper the authors address the converse problem and apply various known or new formulas for series associated with the zeta and related functions with a view to developing the corresponding theory of multiple gamma functions and then using these series to compute the determinants of the laplacians on the n dimensional unit sphere s n n 5 6 7 explicitly

spectral functions special functions and the selberg zeta function the functional determinant of an eigenvalue sequence as defined by zeta regularization can be simply evaluated by quadratures we apply this procedure to the selberg trace formula for a compact riemann surface to find a factorization of the selberg zeta function into two functional determinants respectively related to the laplacian on the compact surface itself and on the sphere we also apply our formalism to various explicit eigenvalue sequences reproducing in a simpler way classical results about the gamma function and the barnesg function concerning the latter our method explains its connection to the selberg zeta function and evaluates the related glaisher kinkelin constanta

spectral functions for regular sturm liouville problems in this paper we provide a detailed analysis of the analytic continuation of the spectral zeta function associated with one dimensional regular sturm liouville problems endowed with self adjoint separated and coupled boundary conditions the spectral zeta function is represented in terms of a complex integral and the analytic continuation in the entire complex plane is achieved by using the well known liouville green or wkb asymptotic expansion of the eigenfunctions associated with the problem the analytically continued expression of the spectral zeta function is then used to compute the functional determinant of the sturm liouville operator and the coefficients of the asymptotic expansion of the associated heat kernel

asymptotic behavior of the selberg zeta functions for degenerating families of hyperbolic manifolds let mk be a degenerating sequence of finite volume hyperbolic manifolds of dimension d with d 2 or d 3 with finite volume limit m let z m k s be the associated sequence of selberg zeta functions and let mathcal z k s be the product of local factors in the euler product expansion of z m k s corresponding to the pinching geodesics on mk the main result in this article is to prove that z m k s mathcal z k s converges to z m infty s for all s in mathbf c with re s d 1 2 the significant feature of our analysis is that the convergence of z m k s mathcal z k s to z m infty s is obtained up to the critical line including the right half of the critical strip a region where the euler product definition of the selberg zeta function does not converge in the case d 2 our result reproves by different means the main theorem in schulze j funct anal 236 120 160 2006

determinant of the laplacian on a non compact three dimensional hyperbolic manifold with finite volume the functional determinant of laplace type operators on a three dimensional non compact hyperbolic manifold with invariant fundamental domain of finite volume is expressed via the selberg zeta function related to the picard group

horizontal lift of the brownian motion on the hyperbolic plane and the selberg trace formula by using an explicit representation for the horizontal lift of the brownian motion on the poincare upper half plane h2 we show an expression for the heat kernel for the de rham kodaira laplacian on h2 we apply the result to a study on the selberg trace formula

semiclassical determinants of schrodinger operators on bundles with curvature of low rank the semiclassical k limit of a schrodinger operator k acting on the kth tensor power of a line bundle is studied an inductive argument is given to show that a previously derived infinite series representation for the leading term in the large k expansion of ln det k reduces under certain circumstances to simple expressions involving the riemann zeta function these results hold when the curvature of the bundle is confined to a single plane at each point

an exact trace formula and zeta functions for an infinite quantum graph with a non standard weyl asymptotics we study a quantum hamiltonian that is given by the negative laplacian and an infinite chain of like potentials with strength 0 on the half line and which is equivalent to a one parameter family of laplacians on an infinite metric graph this graph consists of an infinite chain of edges with the metric structure defined by assigning an interval in 0 ln to each edge with length we show that the one parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each 0 and prove that the dirichlet laplacian is the limit of the one parameter family in the strong resolvent sense the spectrum of the resulting dirichlet quantum graph is also purely discrete the eigenvalues are given by n n2 with multiplicities d n where d n denotes the divisor function we can thus relate the spectral problem of this infinite quantum graph to dirichlet s famous divisor problem and infer the non standard weyl asymptotics for the eigenvalue counting function based on an exact trace formula the voronoi summation formula we derive explicit formulae for the trace of the wave group the heat kernel the resolvent and for various spectral zeta functions these results enable us to establish a well defined renormalized secular equation and a selberg like zeta function defined in terms of the classical periodic orbits of the graph for which we derive an exact functional equation and prove that the analogue of the riemann hypothesis is true

on determinants of laplacians on riemann surfaces determinants of laplacians on tensors and spinors of arbitrary weights on compact hyperbolic riemann surfaces are computed in terms of values of selberg zeta functions at half integer points

determinants and conformal anomalies of gjms operators on spheres the conformal anomalies and functional determinants of the branson gjms operators p2k on the d dimensional sphere are evaluated in explicit terms for any d and k such that k d 2 if d is even the determinants are given in terms of multiple gamma functions and a rational multiplicative anomaly which vanishes for odd d taking the mode system on the sphere as the union of neumann and dirichlet ones on the hemisphere is a basic part of the method and leads to a heuristic explanation of the non existence of super critical operators 2k d for even d significant use is made of the barnes zeta function the results are given in terms of ratios of determinants of operators on a d 1 dimensional bulk dual sphere for odd dimensions the log determinant is written in terms of multiple sine functions and agreement is found with holographic computations yielding an integral over a plancherel measure the n d determinant ratio is also found explicitly for even dimensions ehrhart polynomials are encountered