relative index theorems and supersymmetric scattering theory we discuss supersymmetric scattering theory and employ krein s theory of spectral shift functions to investigate supersymmetric scattering systems this is the basis for the derivation of relative index theorems on some classes of open manifolds as an example we discuss the de rham complex for obstacles in n and asymptotically flat manifolds it is shown that the absolute or relative euler characteristic of an obstacle in n may be obtained from scattering data for the laplace operator on forms with absolute or relative boundary conditions respectively in the case of asymptotically flat manifolds we obtain the chern gauss bonnet theorem for thel2 euler characteristic

a fixed point formula and harish chandra s character formula the main result in this paper is a fixed point formula for equivariant indices of elliptic differential operators for proper actions by connected semisimple lie groups on possibly noncompact manifolds with compact quotients for compact groups and manifolds this reduces to the atiyah segal singer fixed point formula other special cases include an index theorem by connes and moscovici for homogeneous spaces and an earlier index theorem by the second author both in cases where the group acting is connected and semisimple as an application of this fixed point formula we give a new proof of harish chandra s character formula for discrete series representations

geometric zeta functions of locally symmetric spaces the theory of geometric zeta functions for locally symmetric spaces is generalized to the case of higher rank spaces we show that the zeta functions can be continued to meromorphic functions on the plane describe the divisor in terms of tangential cohomology and in terms of group cohomology which generalizes a conjecture of patterson we also extend the range of zeta functions in considering higher dimensional flats

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 topological trace formula the topological trace formula is a computation of the lefschetz number of a hecke correspondence c acting on the weighted cohomology groups defined in ghm of a locally symmetric space x it expresses this lefschetz number as a sum of contributions from fixed point components of c on the reductive borel serre compactifi cation of x the proof uses the lefschetz fixed point formula of gm2

a tower of coverings of quasi projective varieties abstract the main goal of this article is to relate asymptotic geometric properties on a tower of coverings of a non compact kahler manifold of finite volume with reasonable geometric assumptions to its universal covering examples to which our findings are applicable include moduli spaces of hyperbolic punctured riemann surfaces and hermitian locally symmetric spaces of finite volume

properties of zeros of orthogonal polynomials and related functions abstract there are many results in the literature on orthogonal polynomials concerning the way in which the zeros of a polynomial change as a parameter changes for the classical polynomials there are results due to markoff 1886 and stieltjes 1886 on the monotonic variation of zeros of jacobi and gegenbauer ultraspherical polynomials with respect to parameters our purpose here is to describe some recent results concerning formulas for the derivatives of zeros with respect to a parameter such formulas and their consequences are fairly well developed for bessel functions a limiting case of some of the classical orthogonal polynomials and have recently been explored for some orthogonal polynomials and related functions two general methods will be described the first use extensively by ismail the author and others is based on the hellmann feynman theorem which gives the derivative of an eigenvalue with respect to a parameter it can be applied in either a differential equations or a recurrence relations setting the second method developed in recent work of elbert and the author and applicable to suitable solutions of second order ordinary linear differential equations originates in work of richardson 1918 from this approach we obtain formulas for dc d where c c is a zero of the ultraspherical polynomial p n x these recover the known result that dc d 1 2 we get similar results for certain hermite functions these results are closely related to nicholson type formulas due to durand 1975 for sums of squares of solutions of second order linear ordinary differential equations nicholson type formulas are named after the corresponding formulas for bessel functions due to nicholson

the ψs polar decomposition when the cosquare of s is normal abstract let s m n c be nonsingular such that s t s is normal that is the cosquare of s is normal set s a s 1 a t s for a m n c we determine conditions on a so that a has a s polar decomposition we also find the possible jordan canonical forms of a s orthogonal matrix and of a s skew symmetric matrix in the cases a 1 s t s and b 1 s t s

a new canonical form for complex symmetric matrices it is well known that every real symmetric matrix and every complex hermitian matrix is diagonalizable i e orthogonally similar to a diagonal matrix however a complex symmetric matrix with repeated eigenvalues may fail to be diagonalizable we present a block diagonal canonical form in which each block is quasi diagonal to which every complex symmetric matrix is orthogonally similar as far as applications are concerned complex symmetric matrices as opposed to hermitian matrices play an important role in theories of wave propagation in continuous media e g elasticity thermoelasticity

on the ϕj polar decomposition of matrices abstract we present new results on the j polar decomposition of matrices we show that every symplectic matrix may be written as a product of symplectic operation matrices we present a simple form attained under symplectic equivalence which makes it easy to determine if a matrix does not have a j polar decomposition we also determine the rank 4 matrices with j polar decomposition

eta invariants of dirac operators on locally symmetric manifolds fibres de dirac determinant de la transformee de cayley operateurs de dirac sur des espaces localement symetriques trace de l operateur de la chaleur impair interpretation cohomologique la formule de la fonction zeta invariants wistes et applications

analytic torsion of complete hyperbolic manifolds of finite volume abstract in this paper we define the analytic torsion for a complete oriented hyperbolic manifold of finite volume it depends on a representation of the fundamental group for manifolds of odd dimension we study the asymptotic behavior of the analytic torsion with respect to certain sequences of representations obtained by restriction of irreducible representations of the group of isometries of the hyperbolic space to the fundamental group

note on a theorem of h moscovici let be a discrete subgroup of a semisimple lie group g such that g has a finite volume using a theorem of moscovici we express the multiplicity of discrete series representations of g in the discrete spectrum of l2 g as the l2 index of a twisted dirac operator this result which extends a result of moscovici and of the author holds for all integrable discrete series and for infinitely many nonintegrable discrete series in particular up to computing l2 indices in the special rank one case it implies the osborne warner formula

contragredient equivalence a canonical form and some applications abstract let a and c be m by n complex matrices and let b and d be n by m complex matrices the pair a b is contragrediently equivalent to the pair c d if there are square nonsingular complex matrices x and y such that xay 1 c and ybx 1 d contragredient equivalence is a common generalization of four basic equivalence relations similarity consimilarity complex orthogonal equivalence and unitary equivalence we develop a complete set of invariants and an explicit canonical form for contragredient equivalence and show that a a t is contragrediently equivalent to c c t if and only if there are complex orthogonal matrices p and q such that c paq using this result we show that the following are equivalent for a given n by n complex matrix a 1 1 a qs for some complex orthogonal q and some complex symmetric s 2 2 a t a is similar to aa t 3 3 a a t is contragrediently equivalent to a t a 4 4 a q 1 a t q 2 for some complex orthogonal q 1 q 2 5 5 a pa t p for some complex orthogonal p we then consider a linear operator on n by n complex matrices that shares the following properties with transpose operators for every pair of n by n complex matrices a and b a preserves the spectrum of a b a a and c ab b a we show that a a is contragrediently similar to b b if and only if a x 1 bx 2 for some nonsingular x 1 x 2 that satisfy x 1 1 x 1 and x 1 2 x 2 we also consider a factorization of the form a xy where x 1 x and y y we use the canonical form for the contragredient equivalence relation to give a new proof of a theorem of flanders concerning the relative sizes of the nilpotent jordan blocks of ab and ba we present a sufficient condition for the existence of square roots of ab and ba and close with a canonical form for complex orthogonal equivalence

theory of recursive generation of systems of orthogonal polynomials an illustrative example abstract pending its publication in full detail the theory of recursive generation of systems of orthogonal polynomials is verbally described and illustrated by means of a typical example of practical application for this purpose the theory is applied to the polynomials of legendre chosen as initially given orthogonal sequence

the jordan canonical forms of complex orthogonal and skew symmetric matrices abstract we study the jordan canonical forms of complex orthogonal and skew symmetric matrices and consider some related results

the zeta functions of ruelle and selberg for hyperbolic manifolds with cusps in this paper we study the ruelle zeta function and the selberg zeta functions attached to the fundamental representations for real hyperbolic manifolds with cusps in particular we show that they have meromorphic extensions to mathbb c and satisfy functional equations we also derive the order of the singularity of the ruelle zeta function at the origin to prove these results we completely analyze the weighted unipotent orbital integrals on the geometric side of the selberg trace formula when test functions are defined for the fundamental representations

analytic torsion and ruelle zeta functions for hyperbolic manifolds with cusps abstract in this paper we derive a relationship of the leading coefficient of the laurent expansion of the ruelle zeta function at s 0 and the analytic torsion for hyperbolic manifolds with cusps here the analytic torsion is defined by a certain regularized trace following melrose r b melrose the atiyah patodi singer index theorem res notes math vol 4 a k peters ltd wellesley ma 1993 this extends the result of fried which was proved for the compact case in d fried analytic torsion and closed geodesics on hyperbolic manifolds invent math 84 3 1986 523 540 to a noncompact case

the sums of symplectic hamiltonian and skew hamiltonian matrices abstract a complex 2 n 2 n matrix a is called skew hamiltonian hamiltonian and symplectic if a j a a j a and a j a 1 respectively in which j 0 i n i n 0 and a j j 1 a t j we prove that each 2 n 2 n matrix is a sum of type symplectic hamiltonian a 2 n 2 n matrix a is a sum of type symplectic symplectic if and only if a a j is similar to a j a a 2 n 2 n matrix a is a sum of type symplectic skew hamiltonian if and only if the jordan blocks of a a j with eigenvalue 2i and size k 2 come in pairs of the form j k 2 i j k 2 i and j k 2 i j k 1 2 i

stieltjes sums for zeros of orthogonal polynomials stieltjes considered sums of reciprocals of differences of zeros of a solution of a homogeneous second order linear differential equation here we re examine the derivation of these sums with a view to extending the class of differential equations to which the theory applies and including sums involving the zeros of the derivatives as well as those of the polynomials themselves

newton sum rules of zeros of semiclassical orthogonal polynomials abstract the distribution of zeros of the semiclassical orthogonal polynomials with weights w x x w c x wc x denoting a classical weight function and x being equal to x c or x2 c2 is investigated via the newton sum rules of zeros i e the sums of the rth powers of zeros recursion relations satisfied by these sum rules for semiclassical legendre laguerre and hermite polynomials are explicitly given extensions to other semiclassical polynomials are indicated

l2 index and the selberg trace formula abstract a method is developed for computing the l 2 index of a locally symmetric elliptic differential operator d acting on a locally symmetric manifold m g k with g semisimple of real rank one and of finite co volume based on applying the selberg trace formula to the difference of the two heat kernels associated to d the applications include an extension of the osborne warner multiplicity formula to certain non integrable discrete series derived from the l 2 spinor formula and showing the existence in some cases of non invariant l 2 cohomology classes in the middle dimension via the l 2 signature formula

on a theorem of osborne and warner multiplicities in the cuspidal spectrum osborne and warner have given a formula for the multiplicity of an integrable discrete series representation in l2 g when g is real rank one semi simple lie group and is a discrete subgroup of co finite volume we simplify and evaluate this formula to show that for most g as above the multiplicity is the formal degree of the representation times the volume of g when it is not we give a simple interpretation of the difference

unitary equivalence in an indefinite scalar product an analogue of singular value decomposition if h1 is an n n and h2 an m m invertible hermitian matrix and x and y are arbitrary complex m n matrices we call the last matrices equivalent if x u2yu1 for some h1 unitary matrix u1 and some h2 unitary matrix u2 it is well known that if h1 and h2 are positive definite then without loss of generality we can assume that they are identities and x and y are equivalent if and only if the diagonalizable matrices x x and y y have the same spectrum in the present paper we show that in general the jordan form of x x where x is the h1 h2 adjoint of x x h 11x h2 defines a finite number of nonequivalent classes of matrices and that each such class is defined by its integer matrix explicit formulas for all classes having the same jordan form of x x are presented the necessary and sufficient conditions for an operator to be presentable as uz with h unitary u and h self adjoint h consistent z h polar decomposition of the operator are found

strong limit multiplicity for arithmetic hyperbolic surfaces and 3 manifolds we show that every sequence of torsion free arithmetic congruence lattices in mathrm pgl 2 mathbb r or mathrm pgl 2 mathbb c satisfies a strong quantitative version of the limit multiplicity property we deduce that for r 0 in certain range growing linearly in the degree of the invariant trace field the volume of the r thin part of any congruence arithmetic hyperbolic surface or congruence arithmetic hyperbolic 3 manifold m is of order at most mathrm vol m 11 12 as an application we prove gelander s conjecture on homotopy type of arithmetic hyperbolic 3 manifolds we show that there are constants a b such that every such manifold m is homotopy equivalent to a simplicial complex with at most a mathrm vol m vertices all of degrees bounded by b

fourth order differential equations for numerator polynomials the author gives explicitly the fourth order differential equation satisfied by the numerator polynomials associated polynomials of the classical orthogonal polynomials the coefficients of the differential equation are at most a quadratic combination of the polynomials sigma and tau and their derivatives defined via the relation sigma rho te this differential equation is therefore valid for the jacobi laguerre hermite and bessel associated polynomials

on limit multiplicites of discrete series representations in spaces of automorphic forms of course the groups f are then arithmetic subgroups of g in this situation it seems natural to assume that as n the spectral decomposition of l2 f g will approximate that of g in particular let 3 be a discrete series representation of g if f is a subgroup of g let re f 6 be the multiplicity of 3 in the representation of g by right translations on l2 f g choose a haar measure dg on g if f is arithmetic let v f g be the finite volume of f g for the measure induced by dg let d 3 be the formal degree of 3 for the measure dg

spectral properties of the biconfluent heun differential equation abstract the spectrum of zeros of the polynomial solutions of the biconfluent heun differential equation is investigated by two different methods first the spectral newton sums i e the sums of the r th powers of the zeros are given in a rigorous and recurrent way second the density of zeros i e the number of zeros per unit interval is calculated in an explicit way within the so called semiclassical or bkw approximation this is done by using a general theorem which applies to a linear second order differential equation under certain conditions applications to the radial schrodinger equations associated to some quantum mechanical systems rotating harmonic oscillator confinement potentials as well as to the bessel differential equation are also shown

hamiltonian square roots of skew hamiltonian matrices revisited abstract recently h fassbender et al linear algebra appl 287 1999 125 proved the following theorem every real skew hamiltonian matrix w has a real hamiltonian square root h i e h 2 w we prove an analog of this theorem for complex matrices our approach may be of independent interest namely we use the polar decomposition of a nonsingular operator acting in a space with the symplectic inner product

stability theorems for the continuous spectrum of a negatively curved manifold the spectrum of the laplacian a for a simply connected complete negatively curved riemannian manifold is studied the laplacian a0 of a simply connected constant curvature space mo is known up to unitary equivalence decay conditions are given on the metric g and curvature k of m which imply that the continuous part of a is unitarily equivalent to at introduction let m be a complete simply connected riemannian manifold having negative sectional curvatures since m is complete the laplacian a of m is a selfadjoint unbounded operator on l2m when m is the symmetric space mo of constant negative curvature 1 then a0 is known up to unitary equivalence as summarized in 2 in particular ao has purely absolutely continuous spectrum the present paper is concerned with stability of the continuous part of ao under perturbation of the metric g0 on mo theorem 4 12 gives decay conditions on the metric g and curvature k of m which guarantee that a has the same absolutely continuous part as ao much weaker decay conditions on k alone guarantee that a has no singular continuous spectrum as specified in theorem 5 7 combining these results one obtains criteria which ensure that a has the same continuous part as in our earlier paper 7 we established the above results for compactly supported perturbations of the metric on mo as well as obtaining stronger results the current paper provides a better method which may have other applications one technique used essentially is to transplant the heat kernel and resolvent kernel from mo to m as functions of the geodesic distance for background material on symmetric spaces and functional analysis the reader may consult 7 and the references given there 1 heat kernels for complete riemannian manifolds the classical construction of a fundamental solution for the heat equation as given in 1 pp 204 215 uses repeatedly the hypothesis that one is working on a compact riemannian manifold however as observed in 4 pp 7 8 and 5 pp 6 9 the usual method generalizes to any complete riemannian manifold m having bounded geometry here we say that m has bounded geometry if its injectivity radius 8 is bounded below and ii vr ii di where vwr is the ith covariant derivative of the curvature tensor r of m in this section we will show that the condition of bounded geometry can be received by the editors november 8 1979 and in revised form december 2 1979 and march 18 1980 ams mos subject classifications 1970 primary 53c99 58g99 1981 american mathematical society 0002 9947 81 oooo 0160 05 50 431 this content downloaded from 207 46 13 28 on wed 31 aug 2016 05 29 16 utc all use subject to http about jstor org terms

the weight functions generating functions and miscellaneous properties of the sequences of orthogonal polynomials of the second kind associated with the jacobi and the gegenbauer polynomials abstract in the introduction the main object of the paper namely the calculation and study of the weight functions and orthogonality relations corresponding to the sequences of polynomials of the second kind associated with the jacoby and the gegenbauer polynomials is briefly discussed section 2 contains the underlying theory restricted to the citation of a few general formulas regarding the orthogonality property of associated polynomials of the second kind section 3 which deals with the application to the case of the jacobi polynomials p n z n n starts with the construction of the orthogonality relation and the weight function 1 of the associated polynomials for arbitrary gt 1 the denominator of the fraction representing 1 comprises the principal value of an integral which is studied in detail in order to bring it in an explicit form most suited for practical use attention is also paid to the associated functions of the second kind and some of their properties next in a subsection 1 is expressed solely in terms of elementary functions in a number of special cases involving subsets of pairs when 1 1 n gt 1 1 1 gt0 and 1 0 gt 1 in another subsection the generating function for the associated polynomials is calculated and the analysis of the result comprises some related formulas in the final subsection the moments of the weight function 1 with respect to 0 1 and 1 are expressed in terms of the corresponding moments of the weight function of the jacobi polynomials both in finite determinantal form and by means of a recurrence relation in section 4 which concerns the application to the case of the gegenbauer polynomials c v n z n n the advantages of a new definition of the associated polynomials of the second kind are at first expounded then follow the orthogonality relation and the weight function 1 for arbitrary v gt 1 2 next 1 is calculated in terms of elementary functions for all integer and half odd integer values of v exceeding 1 2 in the former case a remarkable simplification occurs in the denominator of the fraction representing 1 after that follow three subsections devoted respectively to the functions of the second kind associated with the gegenbauer polynomials two different generating functions for the associated polynomials and related formulas and the moments of the weight function 1 the paper ends with two appendices in which certain types of sums are reduced to a one term expression

the l 2 index theorem for homogeneous spaces of lie groups the l2 index theorem for covering spaces of atiyah 3 and singer 20 asserts that given a discrete group f acting smoothly and freely on a manifold m with compact quotient m f m and an elliptic differential operator d on m which is f invariant and thus descends to an elliptic differential operator d on the compact manifold m then the p index indrd dimrkerd dimrkerd of d coincides with the ordinary index ind d dim ker d dim ker d of d here ker d is the space of l2 solutions of du 0 and dim r denotes the dimension function corresponding to the trace on the commutant of f acting on the hilbert space of l2 sections over m naturally associated to p the importance of this theorem lies in the fact that indd 0 implies the existence of nontrivial l2 solutions for the equation du 0 and as such it was used in a crucial way by atiyah and schmid 5 to construct explicit realizations of the discrete series representations for semisimple lie groups indeed if g is a lie group which possesses a discrete torsion free cocompact subgroup f and h is a compact subgroup of g then the l2 index theorem applied to the covering space m g h of m f g h combined with the index formula of atiyah singer 6 yields existence results for l2 solutions of g invariant elliptic equations on the homogeneous space g h it is relevant to note that with d denoting this time a g invariant elliptic differential operator on g h the ratio between the f index of d and the covolume of f gives a number independent of f this real number indg d can be in fact intrinsically defined as the difference of the two formal degrees corresponding to the representations of g on ker d and ker d respectively and hence makes sense for any unimodular lie group g even when it has no discrete cocompact subgroups which outside the semisimple case is the generic

eta invariants and their adiabatic limits 0 introduction 1 algebraic preliminaries 2 the finite dimensional case superconnections a odd dimensional base spaces b even dimensional base spaces appendix 1 coupling to general superconnections 3 auxiliary grassman variables and auxiliary adiabatic limits 4 the infinite dimensional case fibrations a elementary geometry of fibrations b odd dimensional base spaces c even dimensional base spaces d the case in which the dirac operator on the fiber is not invertible appendix 2 multifibrations

the unitary spectrum for real rank one groups let g be a connected real semi simple lie group with finite center one of the main problems in harmonic analysis is to determine the unitary spectrum of g in this paper we treat this question in the case when real rank of g is i although the answer was known for g of classical type previously 2 5 7 15 we have redone this work sometimes giving simpler arguments with the arbitrary rank case in mind we have tried to deal with the problem of classifying unitary representations in as systematic a way as possible pro ceeding by induction on the dimension of g we have concentrated on the case when g is linear and has a compact cartan subgroup the other cases being known already as an application we also give a list of the unitary representations that contribute to the l2 index formula for the dirac operator with coefficients as is apparent from the calculations in 19 this is intimately connected with one of our main techniques for determining the unitary spectrum the dirac in equality we will deal with some related problems in a future paper

polar decompositions in finite dimensional indefinite scalar product spaces general theory abstract several classes of polar decompositions of real and complex matrices with respect to a given indefinite scalar product are studied matrices that admit such polar decompositions are described in various ways in particular a full description of all polar decompositions of a given matrix up to the natural similarity between polar decompositions is given

on the growth of l 2 invariants for sequences of lattices in lie groups we study the asymptotic behaviour of betti numbers twisted torsion and other spectral invariants of sequences of locally symmetric spaces our main results are uniform versions of the degeorge wallach theorem of a theorem of delorme and various other limit multiplicity theorems a basic idea is to adapt the notion of benjamini schramm convergence bs convergence originally introduced for sequences of finite graphs of bounded degree to sequences of riemannian manifolds and analyze the possible limits we show that bs convergence of locally symmetric spaces g k implies convergence in an appropriate sense of the normalized relative plancherel measures associated to l2 g this then yields convergence of normalized multiplicities of unitary representations betti numbers and other spectral invariants on the other hand when the corresponding lie group g is simple and of real rank at least two we prove that there is only one possible bs limit i e when the volume tends to infinity locally symmetric spaces always bs converge to their universal cover g k this leads to various general uniform results when restricting to arbitrary sequences of congruence covers of a fixed arithmetic manifold we prove a strong quantitative version of bs convergence which in turn implies upper estimates on the rate of convergence of normalized betti numbers in the spirit of sarnak xue an important role in our approach is played by the notion of invariant random subgroups for higher rank simple lie groups g we exploit rigidity theory and in particular the nevo stuck zimmer theorem and kazhdan s property t to obtain a complete understanding of the space of irs s of g 2017 department of mathematics princeton university

skew ϕ polar decompositions abstract we present necessary and sufficient conditions so that a square complex matrix a can be written as a q s with orthogonal q and skew symmetric s we also present necessary and sufficient conditions so that a 2n by 2n complex matrix b can be written as b w r with symplectic w and hamiltonian r