integrable evolution equations on the n dimensional sphere the problem of a classification of integrable evolution equations on the break n dimensional sphere is considered we modify the main notions of the symmetry approach such as the formal symmetry and the canonical series of conserved densities to the case of such equations using these theoretical results we solve several special classification problems the main result is a complete classification of integrable isotropic evolution equations of third order on the sphere an important class of anisotropic equations is also considered

holomorphic extensions of representations i automorphic functions let g be a connected real semisimple lie group contained in its complexification gc and let k be a maximal compact subgroup of g we construct a kc g double coset domain in gc and we show that the action of g on the k finite vectors of any irreducible unitary representation of g has a holomorphic extension to this domain for the resultant holomorphic extension of k finite matrix coefficients we obtain estimates of the singularities at the boundary as well as majorant minorant estimates along the boundary we obtain l bounds on holomorphically extended automorphic functions on g k in terms of sobolev norms and we use these to estimate the fourier coefficients of combinations of automorphic functions in a number of cases e g of triple products of maas forms

an inversion formula for the segal bargmann transform on a symmetric space of non compact type abstract we prove analogs of the heat kernel transform inversion formulae of golse leichtnam and the author e leichtnam f golse m stenzel intrinsic microlocal analysis and inversion formulae for the heat equation on compact real analytic riemannian manifolds ann sci ecole norm sup 4 29 6 1996 669 736 mr1422988 97h 58153 theorems 0 3 0 4 in the setting of a riemannian symmetric space of helgason s non compact type

the segal bargmann coherent state transform for compact lie groups abstract let k be an arbitrary compact connected lie group we describe on k an analog of the segal bargmann coherent state transform and we prove theorem 1 that this generalized coherent state transform maps l 2 k isometrically onto the space of holomorphic functions in l 2 g where g is the complexification of k and where is an appropriate heat kernel measure on g the generalized coherent state transform is defined in terms of the heat kernel on the compact group k and its analytic continuation to the complex group g we also define a k averaged version of the coherent state transform and we prove a similar result for it theorem 2 in the appendix we describe the coherent states which motivate the definitions of these transforms in section 9 we obtain inversion formulas for both the coherent state transform and the k averaged coherent state transform theorems 3 4 as a consequence we obtain formulas for certain bergman kernels on the complex group g theorems 5 6 in section 10 we derive a general paley wiener theorem for k and we exhibit a family of convolution transforms each of which is an isometric isomorphism of l 2 k onto a certain l 2 space of holomorphic functions on g these transforms include the k averaged coherent state transform as a special case we also discuss the analogous results for r 1 the results of section 10 are theorems 7 10 finally in section 11 we discuss the case of quotient spaces we prove theorem 11 that both the coherent state transform and the k averaged coherent state transform pass in a natural way from k to k h where h is a closed connected subgroup of k

classification of integrable polynomial vector evolution equations several classes of systems of evolution equations with one or two vector unknowns are considered we also investigate systems with one vector and one scalar unknown for these classes all equations having the simplest higher symmetry are listed

resonance expansions of propagators in the presence of potential barriers abstract we show that the schrodinger propagator can be expanded in terms of resonances at energy levels at which a barrier separates the interaction region from infinity the expansions hold for all times with errors small in the semi classical parameter as a byproduct we obtain a result on the approximation of clusters of resonant states by clusters of eigenfunctions of a self adjoint reference operator

some qualitative questions on the equation div a x u u f x u abstract in this article we establish several applications of picone s identity for the operator of the form d i v a x u u such as hardy type inequality sturmian comparison theorem monotonicity property of the first eigenvalue nonexistence of positive supersolutions and caccioppoli inequality under certain conditions on a

the inverse segal bargmann transform for compact lie groups in this paper i give a new inversion formula theorem 1 for the generalized segal bargmann transform introduced in b c hall j funct anal 122 1994 103 151 the inversion formula may be viewed as a formula for theinverseheat operator for a compact lie group i then use this formula to give a new direct proof of the unitary of thek invariant form of the segal bargmann transform theorem 2 the proof of the inversion formula relies on an identity theorem 5 which relates the laplacian for a compact lie groupkto the laplacian for the non compact dual symmetric space kc k

on the principal eigenvalue of a second order linear elliptic problem on considere le probleme de dirichlet 7b lu mu dans u 0 sur dans un domaine borne r n a frontiere lisse 7b l est un operateur differentiel uniformement elliptique sur de la forme 7b lu i n j 1 a ij 2 u x j x i i n 1 a i u x i a 0 u a coefficients a valeurs reels appartenant a c 0 1

the segal bargmann transform on a symmetric space of compact type abstract we study the segal bargmann transform on a symmetric space x of compact type mapping l 2 x into holomorphic functions on the complexification x c we invert this transform by integrating against a dual heat kernel measure in the fibers of a natural fibration of x c over x we prove that the segal bargmann transform is an isometry from l 2 x onto the space of holomorphic functions on x c which are square integrable with respect to a natural measure these results extend those of b hall in the compact group case

the segal bargmann transform for noncompact symmetric spaces of the complex type we consider the generalized segal bargmann transform defined in terms of the heat operator for a noncompact symmetric space of the complex type for radial functions we show that the segal bargmann transform is a unitary map onto a certain l 2 space of meromorphic functions for general functions we give an inversion formula for the segal bargmann transform involving integration against an unwrapped version of the heat kernel for the dual compact symmetric space both results involve delicate cancellations of singularities

isometry theorem for the segal bargmann transform on a noncompact symmetric space of the complex type we consider the segal bargmann transform on a noncompact symmetric space of the complex type we establish isometry and surjectivity theorems for the transform in a form as parallel as possible to the results in the dual compact case the isometry theorem involves integration over a tube of radius r in the complexification followed by analytic continuation with respect to r a cancellation of singularities allows the relevant integral to have a nonsingular extension to large r even though the function being integrated has singularities

auto backlund transformations and superposition formulas for solutions of drinfeld sokolov systems abstract the paper is devoted to constructing auto backlund transformations abt and superposition formulas for the solutions of the drinfeld sokolov ds systems the transformations are derived from pairs of differential substitutions relating different systems of the ds type the nonlinear superposition formulas for solutions of the ds systems are obtained from the assumption of commutativity of the bianchi diagram we indicate a seed solution for each system which can be used to generate multi soliton solutions as an application of the superposition formulas we construct two soliton solutions for each of the ds systems

on the principal eigenvalues of indefinite elliptic problems on considere les deux systemes d equations lu u 2 b j d j u au wu lv wv k v dans un domaine borne lisse r n on suppose que a w sont des matrices m m ueh 1 2 on donne des conditions suffisantes pour l existence de vecteurs propres positifs et des bornes superieures et inferieures sur les valeurs propres correspondantes