This paper proposes an inductive method to construct bases for spaces of spherical harmonics over the unit sphere Ω2q of Cq. The bases are shown to have many interesting properties, among them orthogonality with respect to the inner product of L(Ω2q). As a bypass, we study the inner product [f, g] = f(D)(g(z))(0) over the space P(Cq) of polynomials in the variables z, z ∈ Cq, in which f(D) is t...

In this paper, we always assume that H is an N-dimensional complex Hilbert space. Let L(H) be the set of all linear operators from H to H . A state ρ of some quantum system, described by H , is a positive semi-definite operator of trace one, in particular, for each unit vector |ψ〉 ∈ H , the operator ρ = |ψ〉〈ψ| is said to be a pure state. The set of all states onH is denoted by D(H). If X,Y ∈ L(...

abstract: in the present paper, we give a necessary and sufficient condition for contact cr-warped product to be contact cr-product in kenmotsu space forms.

We define a new function-valued inner product on L2(G), called ?-bracket product, where G is a locally compact abelian group and ? is a topological isomorphism on G. We investigate the notion of ?-orthogonality, Bessel's Inequality and ?-orthonormal bases with respect to this inner product on L2(G).

The spectrum εn has a Brillouin-zone like structure with Ω the width of the first zone, i.e. for any solution |un(t)〉 of quasienergy εn, and any integer m, exp(imΩt)|un(t)〉 is also a solution, with eigenvalue εn + mΩ. The states |un(t)〉 form a Hilbert space of T-periodic functions. The inner product in this space can be defined starting from the standard bra-ket inner product 〈•|•〉 for time-ind...

It is shown that if a representation of a *-algebra on a vector space V is an irreducible *-representation with respect to some inner product on V then under appropriate technical conditions this property determines the inner product uniquely up to a constant factor. Ashtekar has suggested using the condition that a given representation of the algebra of quantum observables is a *-representatio...