vertical square functions and other operators associated with an elliptic operator abstract we study the vertical and conical square functions defined via elliptic operators in divergence form in general vertical and conical square functions are equivalent operators just in l 2 but when this square functions are defined through the heat or poisson semigroups that arise from an elliptic operator we are able to find open intervals containing 2 where the equivalence holds the intervals in question depend ultimately on the range where the semigroup is uniformly bounded or has off diagonal estimates as a consequence we obtain new boundedness results for some square functions besides we consider a non tangential maximal function associated with the poisson semigroup and extend the known range where that operator is bounded our methods are based on the use of extrapolation for muckenhoupt weights and change of angle estimates all our results are obtained in the general setting of a degenerate elliptic operator where the degeneracy is given by an a 2 weight in weighted lebesgue spaces of course they are valid in the unweighted and or non degenerate situations which can be seen as special cases and they provide new results even in those particular settings we also consider the square root of a degenerate elliptic operator in divergence form l w and improve the lower bound of the interval where this operator is known to be bounded on l p v d w finally we give unweighted boundedness results for the degenerate operators under consideration

vector coherent state constructions of u 3 symmetric tensors and their su 3 su 2 u 1 wigner coefficients generalized vector coherent state constructions of totally symmetric u 3 tensors are used to gain new expressions for the su 3 su 2 u 1 wigner coefficients for the coupling 1 1 20 3 3 these expressions show how the extremely simple formulas of le blanc and biedenharn involving a single 9 j coefficient arise as special cases of a general result that involves 12 j coefficients a simpler general result involving only 9 j coefficients and k normalization factors is derived in a way that can in principle be generalized to the generic coupling with multiplicity

boson realization of sp 4 r ii the generating kernel formulation in a previous paper of this series the matrix elements were discussed with respect to boson states of an operator k2 required for the boson realization of the sp 4 r lie algebra in the present paper it is shown that these matrix elements can be obtained from a generating kernel given by the overlap of sp 4 r coherent states the results have relevance for the determination of the matrix elements of the generators of the sp 4 r lie algebra with respect to the basis of irreps of the positive discrete series for the corresponding group and are in principle generalizable to symplectic algebras of higher dimensions

additive rank 1 preservers between hermitian matrix spaces and applications denote the set of n n complex hermitian matrices by hn we characterize the additive maps from hn into hm that preserve rank 1 matrices when the rank of the image of in is equal to n as applications the forms of the additive preserver of rank the additive preserver of rank additivity and the additive preserver of adjoint matrix between the spaces of hermitian matrices are also given

regularity theory for solutions to second order elliptic operators with complex coefficients and the l p dirichlet problem abstract we establish a new theory of regularity for elliptic complex valued second order equations of the form l div a when the coefficients of the matrix a satisfy a natural algebraic condition a strengthened version of a condition known in the literature as l p dissipativity precisely the regularity result is a reverse holder condition for l p averages of solutions on interior balls and serves as a replacement for the de giorgi nash moser regularity of solutions to real valued divergence form elliptic operators in a series of papers cialdea and maz ya studied necessary and sufficient conditions for l p dissipativity of second order complex coefficient operators and systems recently carbonaro and dragicevic introduced a condition they termed p ellipticity and showed that it had implications for boundedness of certain bilinear operators that arise from complex valued second order differential operators their p ellipticity condition is exactly our strengthened version of l p dissipativity the regularity results of the present paper are applied to solve l p dirichlet problems for l div a b when a and b satisfy a carleson measure condition which previously was known only in the real valued case we show solvability of the l 2 dirichlet problem as well as solvability of the l p dirichlet boundary value problem for p in the range where a is p elliptic

highest weight representations for gl m n and gl m n it is shown how vector coherent state vcs theory enlightens the simultaneous discussion of representation theory for classical lie algebras and superalgebras and provides an optimal framework for the explanation of the noted similarities and dissimilarities of their representations reducibility atypicality and the positive definitiveness of inner products in vcs representation spaces are discussed the discussion is exemplified through the parallel and explicit construction of highest weight ladder representations of the lie algebra gl m n and superalgebra gl m n in gl m gl n bases

collectivity and geometry iv sp 6 sp 2 o 3 basis states for open shells using generating function methods branching rules for sp 6 sp 2 o 3 are derived the branching rules suggest an integrity basis or set of elementary permissible diagrams in terms of which the subgroup basis states are defined they correspond to vibrational or bohr mottelson type states in the nuclear symplectic model

v admissibility poincare group and sturmian based vector coherent states we show that there exists a v admissible type subspace in the carrier space of the wigner representation of the poincar group in 1 3 dimensions this subspace is built up from an orthonormal set of sturmian type functions which verify naturally the assumption of rotational invariance under which relativistic coherent states frames were previously obtained then we propose an extension of the concept of v admissibility for the class of semi direct product groups which takes into account the relativity groups which were not covered by the existing approaches in the light of this extension the square integrability modulo an appropriate subgroup of the wigner representation of the group of poincar is revisited and some physical implications are discussed

weak type p p estimates for riesz transforms we consider the problem of l p boundedness of higher order riesz transforms associated to elliptic operators l of order 2m on as an application of the recently solved kato conjecture we show for all this generalizes the result of auscher and tchamitchian restricted to the case d 2m

the matrix representation of so7 in an su2 3 basis vector coherent state vcs theory is applied to so7 in an su2 3 basis some reduced matrix elements of the tensor operator that forms part of the so7 algebra are derived by using the k matrix technique

on classical adjoint commuting mappings between matrix algebras abstract let f be a field and let m and n be integers with m n 3 let m n denote the algebra of n n matrices over f in this note we characterize mappings m n m m that satisfy one of the following conditions 1 f 2 or f n 1 and adj a b adj a b for all a b m n and f with i n 0 2 is surjective and adj a b adj a b for every a b m n here adj a denotes the classical adjoint of the matrix a and i n is the identity matrix of order n we give examples showing the indispensability of the assumption i n 0 in our results

generalized coherent states as eigenstates of linearized quadratic casimir operators it is shown that the generalized coherent states for several groups and a supergroup are eigenstates of the linearized quadratic casimir operator of the associated algebra or superalgebra these operators may be useful in the study of geometrical phases

on some racah coefficients of u n some u n racah coefficients appearing in various applications of vector coherent state and k matrix theories are calculated for such a purpose use is made of their definition in terms of u n u n 1 reduced wigner coefficients and u n 1 racah coefficients by starting from known u 2 racah coefficients the recursion relations obtained are solved by induction over n

coherent states for the harmonic series irreducible representations of u p q i unitary operator coherent states unitary operator coherent states as defined by klauder 1963 perelomov 1972 and gilmore 1974 are considered for the harmonic series irreducible representations of u p q various properties of these states are reviewed including their reproducing kernel and measure explicit forms as well as the representation they lead to for the u p q generators starting from a contraction of the su p q lie algebra both dyson and holstein primakoff boson realisations of u p q are obtained in terms of pq pairs of boson creation annihilation operators and of the generators of a u p u q intrinsic group such boson realisations are applied to determine the matrix elements of the u p q generators between discrete bases classified according to the chain u p q contains implies u p u q finally the isomorphism between so 4 2 and su 2 2 is employed to derive the corresponding properties of the so 4 2 unitary operator coherent states for application in atomic physics

an algebraic algorithm for calculating clebsch gordan coefficients application to su 2 and su 3 a recent paper gave an explicit construction for inducing shift tensors of a compact reductive lie group from shift tensors of a suitably defined subgroup the shift tensors were defined on model spaces of holomorphic vector coherent state wave functions in this paper we use these shift tensors to obtain an algorithm for computing clebsch gordan coefficients the approach reproduces the known analytical results for su 2 and gives a simple algorithm for computing su 3 coefficients the algorithm is shown to yield analytical expressions for the multiplicity free su 3 couplings of type 20 10

coherent state theory of the noncompact symplectic group an extended coherent state theory is presented for the noncompact sp 3 r group which reveals a simple relationship between the sp 3 r algebra and its contracted u 3 boson limit the relationship is used to derive a remarkably accurate analytic expression for sp 3 r matrix elements for the generic lowest weight representations the expression is shown to be exact whenever the states involved are multiplicity free with respect to the u 3 subalgebra it is further shown how exact matrix elements are easily calculated in general dyson and holstein primakoff type u 3 boson expansions are given

boson realization of sp 4 i the matrix formulation holstein and primakoff derived long ago the boson realization of a su 2 lie algebra for an arbitrary irreducible representation irrep of the su 2 group the corresponding result for su 1 1 sp 2 is also well known this raises the question of whether it is possible to obtain in an explicit analytic and closed form and for any integer d the boson realization of a sp 2d lie algebra for an arbitrary irrep of the sp 2d group which is a problem of considerable physical interest the case d 2 already illustrates the problem in its full generality and thus in this paper we concentrate on sp 4 the dyson realization is well known and the passage to bosons satisfying the appropriate hermiticity conditions can be done by a similarity transformation through an operator k what we want though is an explicit boson realization for sp 2d similar to the one that exists for sp 2 in sec vi we show how we can get it for sp 4 if the operator k is known unfortunately while the matrix form of k2 can be ex

representations of quantum superalgebra uq gl 2 1 in a coherent state basis and generalization the coherent state method has proved to be useful in quantum physics and mathematics this method more precisely the vector coherent state method has been used by some authors to construct representations of superalgebras but almost to our knowledge it has not yet been extended to quantum superalgebras except uq osp 1 2 one of the smallest quantum superalgebras in this article the method is applied to a bigger quantum superalgebra namely uq gl 2 1 in constructing q boson fermion realizations and finite dimensional representations which when irreducible are classified into typical and nontypical representations this construction leads to a more general class of q boson fermion realizations and finite dimensional representations of uq gl 2 1 and thus at q 1 of gl 2 1 both gl 2 1 and uq gl 2 1 have found different physics applications therefore it is meaningful to construct their representations

vector coherent state representation theory a vector coherent state theory is formulated as a natural extension of standard coherent state theory it is shown that the godement representations and the coherent state representations of the sp n r groups of rowe and of deenen and quesne are special cases of this more general theory

analyticity of layer potentials and l 2 solvability of boundary value problems for divergence form elliptic equations with complex l infty coefficients we consider divergence form elliptic operators of the form l dv a x nabla defined in r n 1 x t in r n times r n geq 2 where the l infty coefficient matrix a is n 1

a reducible weil representation of sp 4 realized by differential operators in the space of smooth functions on h2 s1 this work presents a novel way to obtain the associated romanovski functions rn m x with n m in the three separate regions in terms of n and m we obtain the raising and lowering relations with respect to the both indices simultaneously in the three regions then a reducible weil representation of the real lie algebra sp 4 is realized in the space of complex valued smooth functions on h2 s1 by differential forms for the cartan weyl basis its invariant subspace is the second rare instance of the highest weight irreducible representation of sp 4 all whose weight spaces are one dimensional

classical adjoint commuting mappings on hermitian and symmetric matrices let m and n be integers such that m n 3 and let f and k be fields which possess involutions of f and of k respectively let hnf be the k linear space n n hermitian matrices over f in this note we address the general description of mappings satisfying one of the following conditions 1 hn f hm k with either k 2 or f k 3 and f and k of characteristic 2 if and are the identity maps and is surjective satisfying adj a b adj a b for every a b hn f 2 hn f hm f with either f 2 or f n 1 and adj a b adj a b for every a b hn k and f 3 hn f hm k is additive with adj a adja for every a hn f here f a f a a and k a k a a are the fixed fields with respect to the involutions of f and of k respectively and adja denotes the classical adjoint of the matrix a

some linear preserver problems on block triangular matrix algebras in this article we classify rank one nonincreasing maps on n square block triangular matrix algebras with the assumption that in is of rank n as applications we obtain complete classifications of adjugate commuting maps and compound commuting maps on block triangular matrix algebras

implementation of the u n tensor operator calculus in a vector bargmann hilbert space using a vector coherent state framework and an associated vector bargmann hilbert space 1961 the structure of two classes of u n u n 1 unit projective operators is shown to be intimately related to 6 j and 9 j coefficients of the u n 1 subalgebra explicit verification of limit properties for these operators allows the unambiguous assignment of a set of canonical operator labels

vector coherent state theory of the generic representations of so 5 in an so 3 basis for applications of group theory in quantum mechanics one generally needs explicit matrix representations of the spectrum generating algebras that arise in bases that reduce the symmetry group of some hamiltonian of interest here we use vector coherent state techniques to develop an algorithm for constructing the matrices for arbitrary finite dimensional irreps of the so 5 lie algebra in an so 3 basis the so 3 subgroup of so 5 is defined by regarding so 5 as linear transformations of the five dimensional space of an so 3 irrep of angular momentum two a need for such irreps arises in the nuclear collective model of quadrupole vibrations and rotations the algorithm has been implemented in maple and some tables of results are presented

the dirichlet problem for higher order equations in composition form the present paper commences the study of higher order differential equations in composition form specifically we consider the equation lu divb adiva u 0 where a and b are elliptic matrices with complex valued bounded measurable coefficients and a is an accretive function elliptic operators of this type naturally arise for instance via a pull back of the bilaplacian 2 from a lipschitz domain to the upper half space more generally this form is preserved under a lipschitz change of variables contrary to the case of divergence form fourth order differential equations we establish well posedness of the dirichlet problem for the equation lu 0 with boundary data in l2 and with optimal estimates in terms of nontangential maximal functions and square functions

diameter preserving surjections in the geometry of matrices abstract we consider a class of graphs subject to certain restrictions including the finiteness of diameters any surjective mapping between graphs from this class is shown to be an isomorphism provided that the following holds any two points of are at a distance equal to the diameter of if and only if their images are at a distance equal to the diameter of this result is then applied to the graphs arising from the adjacency relations of spaces of rectangular matrices spaces of hermitian matrices and grassmann spaces projective spaces of rectangular matrices

linear operators preserving adjoint matrix between matrix spaces abstract we characterize the linear operators from the linear space of n n matrices into the linear space of m m matrices over any field f that preserve adjoint matrix where n 3

vector coherent states of non compact orthosymplectic lie supergroups vector coherent states are defined for the generic lowest weight ladder irreducible representations of the non compact orthosymplectic lie supergroups osp 2m 2n r and osp 2m 1 2n r their simultaneous use with k matrix theory is shown to provide a powerful tool for the study of such representations this is illustrated with the detailed example of osp 1 2n r

a new approach to absolute continuity of elliptic measure with applications to non symmetric equations in the late 1950s and early 1960s the work of de giorgi degi and nash n and then moser mo initiated the study of regularity of solutions to divergence form elliptic equations with merely bounded measurable coefficients weak solutions in a domain 0 a priori only in a sobolev space w 1 loc 0 were shown to be ho lder continuous of some order depending just on ellipticity and maximum principles and harnack inequalities were established the dirichlet problem for such operators with continuous data on the boundary was established in lsw this in turn paved the way for a more systematic and detailed study of the properties of the elliptic measures d l associated to l div a on a domain 0 the classical properties of existence of non tangential limits of solutions fatou type theorems and comparison principles appeared in cfms but owed a great deal to the earlier work of carleson ca and hunt and wheeden h w on harmonic functions in lipschitz domains all the results mentioned above were carried out for elliptic operators l div a where the matrix a aij has bounded measurable coefficients and is symmetric however it turns out that the symmetry of the matrix is not needed to get these results morrey mor first observed this in connection with the de giorgi nash moser theory for the results in cfms this fact has not been formally observed until now with appropriate reformulation in terms of adjoint operators and adjoint green s functions the results of cfms are valid without the symmetry assumption see section 1 doi 10 1006 aima 1999 1899 available online at http www idealibrary com on

unitary representations branching rules and matrix elements for the non compact symplectic groups the complementarity of the symplectic and orthogonal groups is used to infer properties of the infinite dimensional unirreps of the former from the character theory of the latter the complete set of d series metaplectic unirreps of sp n r is identified and branching rules are given for their restrictions to the maximal compact subgroup u n developed in terms of the properties of schur functions a known algorithm for the evaluation of matrix elements of the sp 3 r lie algebra is extended to any sp n r and analytic expressions are given for important classes of unirreps and multiplicity free states

determinant preserving maps on matrix algebras abstract let m n be the algebra of all n n complex matrices if m n m n is a surjective mapping satisfying det a b det a b a b m n c then either is of the form a man a m n or is of the form a ma t n a m n where m n m n are nonsingular matrices with det mn 1

the solution of the kato square root problem for second order elliptic operators on rn we prove the kato conjecture for elliptic operators on jfin more precisely we establish that the domain of the square root of a uniformly complex elliptic operator l div av with bounded measurable coefficients in ietn is the

the matrix representations of g2 i representations in an so 4 basis irreducible representations of the real compact lie algebra g2 are given in g2 so 4 bases all missing labels are accounted for by the explicit construction of a g2 so 4 basis of vector holomorphic functions the general problem of missing internal labels is also briefly discussed

vector coherent state representations of so 5 in an so 3 basis vector coherent state theory is used to give an algorithm for constructing the so 5 irreps of type n 0 and n n in the physical so 3 basis the construction is remarkably simple and permits the matrices of the so 5 operators for irreps of low dimension and or small multiplicities to be computed by hand the construction is applied to the 2 0 and 2 2 irreps to illustrate the process

superfield and matrix realizations of highest weight representations for osp m 2n a differential representation of the classical lie superalgebra osp m 2n acting on superfield functions is given this representation is used to construct matrix representations of the finite dimensional irreducible representations of the algebra inner products on the irreducible spaces are discussed and classes of star and grade star equivalent representations are identified

collectivity and geometry iii the three dimensional case in the sp 6 sp 2 o 3 chain for closed shells as was indicated in previous papers general hamiltonians for systems of n particles in three dimensional space can be formulated in the enveloping algebra of the symplectic group sp 6n this group admits among others the subgroup sp 6 o n and as has been noticed by many authors collective hamiltonians can be formulated in the enveloping algebra of sp 6 so that their eigenstates can be characterized by a definite irreducible representation irrep of this group the mathematical problem is then to determine the matrix elements of the generators of sp 6 in a basis characterized by irreps of this group as well as of appropriate subgroups in the present series of papers the subgroups chosen where sp 2 o 3 as the casimir operator of sp 2 when n is formally related to the bohr mottelson vibrational hamiltonian bmvh while o 3 gives the angular momentum of the state we give an algorithm for determining these states that closely parallels the procedure followed for the bmvh programs are being developed to convert our algorithm into a computational tool for determining collective excitations in nuclei

vector coherent state theory of the semidirect sum lie algebras wsp 2n r positive discrete series irreducible representations of the semidirect sum lie algebras wsp 2n r w n sp 2n r are given in wsp 2n r contains implies sp 2n r contains implies u n bases the branching rule for the reduction of a wsp 2n r irreducible representation into a direct sum of multiplicity free sp 2n r irreducible representations is accounted for by the explicit construction of a wsp 2n r contains implies sp 2n r contains implies u n basis of vector valued holomorphic functions

vector coherent state realization of representations of the affine superalgebra a 1 0 1 the method of vector coherent states is generalized to study the representations of the affine superalgebra a 1 0 1 a new realization of all the irreducible highest weight representations of this affine superalgebra is obtained in terms of holomorphic superdifferential operators in infinitely many grassmannian and bosonic variables

on a class of 6j coefficients with one multiplicity index for groups sp 2n so 2n and so 2n 1 an important class of 6j symbols for the groups sp 2n so 2n and so 2n 1 with one nontrivial multiplicity index is investigated an appropriate choice of a basis in the multiplicity space is made and the so called canonical form for 6j symbols is obtained their expressions depending on the roots of an nth order equation and explicit expressions for some simple class of representations are obtained

the neumann problem for elliptic equations with non smooth coefficients in this paper we study the neumann and regularity problems for divergence form symmetric elliptic operators l div av where a aifix is a matrix of bounded measurable functions and ellipticity is the existence of a 0 such that 2 114 2 ais ir 21412 for all i r we assume that l is defined in the unit ball b but the methods of proof are sufficiently general to allow b to be a bounded starlike lipschitz domain the neumann problem lu 0 in b avun q f

collectivity and geometry ii the two dimensional case in a previous paper of this series we showed that nuclear collective behavior can be related to the symplectic geometry of the many nucleon system when we introduce in it appropriate constraints we showed in that paper that a full discussion of collective behavior in the many body system requires in a space of d dimensions the basis for the irreducible representations of the sp 2d group in the chain sp 2d sp 2 o d as well as the matrix elements of the generators of sp 2d in this basis in the present paper we implement this program fully for d 2 both because of the mathematical interest of the chain sp 4 sp 2 o 2 and as a stepping stone to the chain sp 6 sp 2 o 3 which is central to the discussion of collective motions

a double rotor coherent state construction of the wigner supermultiplet su 4 contains implies su 2 su 2 a double rotor vector coherent state construction of the su 4 3 su 2 x sup wigner supermultiplet leads to a simple matrix representation of this scheme matrix elements of the vector coherent state realizations of the group generators are given by very simple expressions in tem of ordinary spin s and isospin t wigner coefficients with intrinsic projection labels ks and kg the k mawix technique is used to effectively elevate these labels lo the status of good quantum numbers the kkt matrices bre given in analytic form for a number of impomt irreducible representations

vectorlike coherent states with noncompact stability group an investigation is made of coherent states that differ from the usual ones in two ways a they are connected with the coset space g h where the stability subgroup h may be noncompact and b the notion of an h invariant ray is replaced by the more general notion of an h invariant subspace a general framework is given for vectorlike coherent states with the help of the nonlinear realization technique as well as with the rigged hilbert space theory the vectorlike coherent states are found for the poincare group and the hyperbolic coherent states are found for the su 1 1 group

the matrix representations of g2 ii representations in an su 3 basis irreducible representations of the real compact lie algebra g2 are given in g2 su 3 bases a missing label is accounted for by the explicit construction of a g2 su 3 basis of vector holomorphic functions analytical results are given for two multiplicity free classes of irreps it is also shown how vector coherent state vcs theory accommodates the decomposition of the nilpotent raising operator subalgebra of an arbitrary lie algebra into a finite but arbitrary number of irreducible tensorial sets under transformations generated by a stability algebra

linear preservers on matrices abstract let u denote either the vector space of n n matrices or the vector space of n n symmetric matrices over an infinite field f in this paper we characterize linear mappings l on u that satisfy one of the following properties i l adja adjl a for all a in u ii l preserves idempotent matrices and l in in where f is the real field r or the complex field c iii l ea el a for all a in u where f r or c

square integrability modulo a subgroup we prove a weak form of the frobenius reciprocity theorem for locally compact groups as a consequence we propose a definition of square integrable representation modulo a subgroup that clarifies the relations between coherent states wavelet transforms and covariant localisation observables a self contained proof of the imprimitivity theorem for covariant positive operator valued measures is given

a new generalization of vector coherent state theory for the so 5 u 2 proton neutron quasispin group the introduction of a set of intrinsic coordinates to give an explicit construction of the intrinsic states of vector coherent state theory has greatly simplified earlier attempts to generalize this theory to include operators lying outside the group algebra very explicit vector coherent state construction of such operators can now be given in terms of vector coupled combinations of intrinsic and collective operators when organized into tensors which induce specific shifts in irreducible representations these lead to the reduced wigner coefficients needed in practical calculations the so 5 u 2 proton neutron quasispin algebra is used as an example to give further simplifications of earlier results all wigner coefficients needed to give the n t dependence of matrix elements in the seniority scheme can now be given through a few terms expressed solely through angular momentum recoupling coefficients and the k matrix elements of vector coherent state theory

necessary and sufficient conditions for absolute continuity of elliptic harmonic measure our purpose in this paper is to give a necessary and sufficient condition on the modulus of continuity of the coefficients of an elliptic operator in divergence form in order that the corresponding poisson kernel for the dirichlet problem exist this condition on the coefficients is global continuity together with the additional property that the modulus of continuity along some nontangential direction at each boundary point be bounded uniformly in these points and directions by a function 7 t satisfying the dini type condition

the connections between dirichlet regularity and neumann problems for second order elliptic operators with complex bounded measurable coefficients abstract the present paper discusses relations between regularity dirichlet and neumann problems we investigate the boundary problems for block operators and prove in particular that the solvability of the regularity problem does not imply the solvability of the dual dirichlet problem for general elliptic operators with complex bounded measurable coefficients this is strikingly different from the case of real operators for which such an implication was established in 1993 by c kenig j pipher invent math 113 3 1993 447 509 and since then has served as an integral part of many results

hardy and bmo spaces associated to divergence form elliptic operators consider a second order divergence form elliptic operator l with complex bounded measurable coefficients in general operators based on l such as the riesz transform or square function may lie beyond the scope of the calderon zygmund theory they need not be bounded in the classical hardy bmo and even some l p spaces in this work we develop a theory of hardy and bmo spaces associated to l which includes in particular a molecular decomposition maximal and square function characterizations duality of hardy and bmo spaces and a john nirenberg inequality

a general theorem on square integrability vector coherent states we derive a generalization of the well known theorem for the square integrability of a unitary irreducible representation of a locally compact group the generalization covers the case of representations admitting vector coherent states the result is illustrated by an example drawn from the isochronous galilei group the construction yields a wide variety of coherent states labeled by phase space points which satisfy a resolution of the identity condition and incorporate spin degrees of freedom

vector coherent state representations induced representations and geometric quantization i scalar coherent state representations coherent state theory is shown to reproduce three categories of representations of the spectrum generating algebra for an algebraic model i classical realizations which are the starting point for geometric quantization ii induced unitary representations corresponding to prequantization and iii irreducible unitary representations obtained in geometric quantization by choice of a polarization these representations establish an intimate relation between coherent state theory and geometric quantization in the context of induced representations

new perspective on the u n wigner racah calculus i vector coherent state theory and construction of gel fand bases using a vector coherent state theory it is shown that the construction of gel fand bases for the unitary group is particularly simple the very specific rules for construction of the states greatly facilitate the subsequent computation of matrix elements of the generators of the u n lie algebra

heisenberg weyl algebras of symmetric and antisymmetric bosons all the polynomials in symmetric 2 and antisymmetric 11 u n bosons are constructed in u n bases and u n reduced matrix elements for the bosons between polynomial basis states are computed applications to representation theory of lie groups paired fermion and boson physics are briefly discussed

coherent state triplets and their inner products it is shown that if h is a hilbert space for a representation of a group g then there are triplets of spaces fh h fh in which fh is a space of coherent state or vector coherent state wave functions and fh is its dual relative to a conveniently defined measure it is shown also that there is a sequence of maps fh h fh which facilitates the construction of the corresponding inner products after completion if necessary the spaces fh h fh become isomorphic hilbert spaces it is shown that the inner product for h is often easier to evaluate in fh than in fh thus we obtain integral expressions for the inner products of coherent state and vector coherent state representations these expressions are equivalent to the algebraic expressions of k matrix theory but they are frequently more efficient to apply the construction is illustrated by many examples

collectivity and geometry i general approach in the last decade an extensive literature appeared in which the microscopic collective behavior of nuclei was associated with definite irreducible representations irreps of either the o n or sp 6 groups where n a 1 and a is the number of nucleons it became clear that the two approaches are equivalent as problems with 3n degrees of freedom are characterized by a definite irrep of the group sp 6n and for its subgroup sp 6 o n the irrep of o n determines that of sp 6 and vice versa thus one can consider that collective effects appear when one introduces the constraint that in the many body system the states are restricted to a definite irrep of o n and thus also of sp 6 and the hamiltonians are in the enveloping algebra of sp 6 rather than in that of sp 6n once sp 6 becomes the paramount group of collective motions the problem is to determine the matrix elements of the generators of sp 6 in a basis characterized by irreps of its subgroups what subgroups to choose rowe and rosensteel

vector coherent state theory and its application to the orthogonal groups vector coherent state theory is developed and presented in a form that explicitly exhibits its general applicability to the ladder representations of all semisimple lie groups and their lie algebras it is shown that in a suitable basis the vector coherent state inner product can be inferred algebraically by k matrix theory and changed to a simpler bargmann inner product thereby facilitating the explicit calculation of the matrix representaions of lie algebras applications are made to the even and odd orthogonal lie algebras

partially coherent states of the real symplectic group in the present paper we introduce partially coherent states for the positive discrete series irreducible representations d n 2 1 n 2 of sp 2d r encountered in physical applications these states are characterized by both continuous and discrete labels the latter specify the row of the irreducible representation 1 2 d of the maximal compact subgroup u d while the former parametrize an element of the factor space sp 2d r h where h is the sp 2d r subgroup leaving the 1 2 d representation space invariant we consider three classes of partially coherent states respectively generalizing the perelomov and barut girardello coherent states as well as some recently introduced intermediate coherent states we prove that each family of partially coherent states forms an overcomplete set in the representation space of d n 2 1 n 2 and study its generating function properties we show that it leads to a representation of the sp 2d r generators in the form of differential operato

the regularity problem for second order elliptic operators with complex valued bounded measurable coefficients the present paper establishes a certain duality between the dirichlet and regularity problems for elliptic operators with t independent complex bounded measurable coefficients t being the transversal direction to the boundary to be precise we show that the dirichlet boundary value problem is solvable in l p subject to the square function and non tangential maximal function estimates if and only if the corresponding regularity problem is solvable in l p moreover the solutions admit layer potential representations in particular we prove that for any elliptic operator with t independent real possibly non symmetric coefficients there exists a p 1 such that the regularity problem is well posed in l p

application of poisson maps on coadjoint orbits of sp 6 r group to many body dynamics the canonical transformation approach generated by the semisimple subgroup gcm 3 sp 6 r is applied to reduction of the lie poisson bracket on coadjoint orbits of the sp 6 r group and the poisson coalgebra spo 6 is determined investigating the construction of the n particle phase induced by this reduction we identify the poisson coalgebra spo 6 as the algebra of quadratic o k k n 1 invariant forms on symplectic 6 k 12 dimensional phase space t o k 3 o k k 3 the general classification scheme of poisson orbits for spo 6 is found and applied to the classification of coadjoint orbits of the sp 6 r group occurring in the decomposition of n particle phase spaces we show that the spo k k 4 6 poisson action on some class of surfaces determined by casimir invariants is not transitive the poisson maps for all classes of orbits spo 4 and spo 6 are found the quantum unitary irreducible representations of spo 4 are obtained

analytical expressions for the matrix elements of the non compact symplectic algebra analytic matrix elements are derived for all the lowest weight representations of the noncompact symplectic algebra sp 3 r it is shown that the expressions are exact for representations of the type sigma 1 sigma 2 sigma 3 and for all states of an arbitrary representation that are multiplicity free with respect to the u 3 subalgebra furthermore they are remarkably accurate in general

new perspective on the u n wigner racah calculus iii applications to u 2 and u 3 for pt ii see ibid vol 20 no 14 p 4613 35 1987 using the framework introduced in parts i and ii the author derives an interesting relationship between u 3 symmetric tensors and u 2 racah coefficients closed formulae for generic u 2 wigner and racah coefficients are given in terms of alternative parametrisations for such coefficients