Reviews in Mathematical Physics

Symmetry properties of PDE's are considered within a systematic and unifying scheme: particular attention is devoted to the notion of conditional symmetry, leading to the distinction and a precise characterization of the notions of "true" and "weak" conditional symmetry. Their relationship with exact and partial symmetries is also discussed. An extensive use of "symmetry-adapted" variables is made; several clarifying examples, including the case of Boussinesq equation, are also provided.

In this paper we propose a method to estimate the density matrix ρ of a d-level quantum system by measurements on the N-fold system in the joint state ρ^⊗N. The scheme is based on covariant observables and representation theory of unitary groups and it extends previous results concerning pure states and the estimation of the spectrum of ρ. We show that it is consistent (i.e. the original input state ρ is recovered with certainty if N → ∞), analyze its large deviation behavior, and calculate explicitly the corresponding rate function which describes the exponential decrease of error probabilities in the limit N → ∞. Finally, we discuss the question whether the proposed scheme provides the fastest possible decay of error probabilities.

A Gelfand triplet for the Hamiltonian H of the Friedrichs model on ℝ with multiplicity space , , is constructed such that exactly the resonances (poles of the inverse of the Livšic-matrix) are (generalized) eigenvalues of H. The corresponding eigen(anti)linear forms are calculated explicitly. Using the wave matrices for the wave (Möller) operators the corresponding eigen(anti)linear forms on the Schwartz space for the unperturbed Hamiltonian H₀ are also calculated. It turns out that they are of pure Dirac type and can be characterized by their corresponding Gamov vector λ → k/(ζ₀ - λ)^-1, ζ₀ resonance, , which is uniquely determined by restriction of to , where denotes the Hardy space of the upper half-plane. Simultaneously this restriction yields a truncation of the generalized evolution to the well-known decay semigroup for t ≥ 0 of the Toeplitz type on . That is: Exactly those pre-Gamov vectors λ → k/(ζ - λ)^-1, ζ from the lower half-plane, , have an extension to a generalized eigenvector of H if ζ is a resonance and if k is from that subspace of which is uniquely determined by its corresponding Dirac type antilinear form.

We show that the complicated ⋆-structure characterizing for positive q the U_qso(N)-covariant differential calculus on the noncommutative manifold boils down to similarity transformations involving the ribbon element of a central extension of U_qso(N) and its formal square root ṽ. Subspaces of the spaces of functions and of p-forms on are made into Hilbert spaces by introducing non-conventional "weights" in the integrals defining the corresponding scalar products, namely suitable positive-definite q-pseudodifferential operators ṽ′^±1 realizing the action of ṽ^±1; this serves to make the partial q-derivatives anti-hermitean and the exterior coderivative equal to the hermitean conjugate of the exterior derivative, as usual. There is a residual freedom in the choice of the weight m(r) along the "radial coordinate" r. Unless we choose a constant m, then the square-integrables functions/forms must fulfill an additional condition, namely, their analytic continuations to the complex r plane can have poles only on the sites of some special lattice. Among the functions naturally selected by this condition there are q-special functions with "quantized" free parameters.