Abstract:
I attempt to give a pedagogical overview of the progress which has occurred during the past decade in the description of one-dimensional correlated fermions. Fermi liquid theory based on a quasi-particle picture, breaks down in one dimension because of the Peierls divergence and because of charge-spin separation. It is replaced by a Luttinger liquid whose elementary excitations are collective charge and spin modes, based on the exactly solvable Luttinger model. I review this model and various solutions with emphasis on bosonization (and its equivalence to conformal field theory), and its physical properties. The notion of a Luttinger liquid implies that all gapless 1D systems share these properties at low energies. Chapters 1 and 2 of the article contain an introduction and a discussion of the breakdown of Fermi liquid theory. Chapter 3 describes in detail the solution of the Luttinger model both by bosonization and by Green's functions methods and summarizes the properties of the model, expressed thorugh correlation functions. The relation to conformal field theory is discussed. Chapter 4 of the article introduces the notion of a Luttinger liquid. It describes in much detail the various mappings applied to realistic models of 1D correlated fermions, onto the Luttinger model, as well as important corrections to the Luttinger model properties discussed in Ch.3. Chapter 5 describes situations where the Luttinger liquid is not a stable fixed point, and where spin or charge gaps open in at least one channel. Chapter 6 discusses multi-band and multichain problems, in particular the stability of a Luttinger liquid with respect to interchain hopping. Ch. 7 gives a brief summary of experimental efforts to uncover Luttinger liquid correlations in quasi-1D materials.

Abstract:
We compute the spectral function rho(q,omega) of the one- dimensional Luttinger model. We discuss the distinct influences of charge-spin separation and of the anomalous dimensions of the fermion operators and their evolution with correlation strength. Charge-spin separation shows up in finite spectral weight at frequencies between v_sigma * q and v_rho * q where v_rho and v_sigma are the velocities of charge and spin fluctuations while spectral weight above v_rho * q and below -v_rho * q is generated by the hybridization of the Fermi surface at +/-k_F by interactions. There are nonuniversal power-law singularities at these special frequencies. We discuss the consistency of recent photoemission experiments on low-dimensional conductors with a Luttinger liquid picture which then would suggest very strong long- range interactions. It is pointed out that many-particle correlation functions in principle exhibit similar singularities but they probe different and complementary aspects of the Fermi surface interactions.

Abstract:
We calculate the spectral function of the Luther-Emery model which describes one-dimensional fermions with gapless charge and gapped spin degrees of freedom. We find a true singularity with interaction dependent exponents on the gapped spin dispersion and a finite maximum depending on the magnitude of the spin gap, on a shifted charge dispersion. We apply these results to photoemission experiments on charge density wave systems and discuss the spectral properties of a one-dimensional Mott insulator.

Abstract:
I discuss origin and possible experimental manifestations of charge-spin separation in 1D Luttinger and Luther-Emery liquids, the latter describing 1D Mott and Peierls insulators and superconductors. Emphasis is on photoemission where the spectral function generically shows two dispersing peaks associated with the collective charge and spin excitations, and on transport. I analyse the temperature dependences of the charge and spin conductivities of two organic conductors and conclude that most likely, charge-spin separation is not realized there and that they can be described as fluctuating Peierls insulators.

Abstract:
I give a brief introduction to Luttinger liquids. Luttinger liquids are paramagnetic one-dimensional metals without Landau quasi-particle excitations. The elementary excitations are collective charge and spin modes, leading to charge-spin separation. Correlation functions exhibit power-law behavior. All physical properties can be calculated, e.g. by bosonization, and depend on three parameters only: the renormalized coupling constant $K_{\rho}$, and the charge and spin velocities. I also discuss the stability of Luttinger liquids with respect to temperature, interchain coupling, lattice effects and phonons, and list important open problems.

Abstract:
I construct the spectral function of the Luther-Emery model which describes one-dimensional fermions with one gapless and one gapped degree of freedom, i.e. superconductors and Peierls and Mott insulators, by using symmetries, relations to other models, and known limits. Depending on the relative magnitudes of the charge and spin velocities, and on whether a charge or a spin gap is present, I find spectral functions differing in the number of singularities and presence or absence of anomalous dimensions of fermion operators. I find, for a Peierls system, one singularity with anomalous dimension and one finite maximum; for a superconductor two singularities with anomalous dimensions; and for a Mott insulator one or two singularities without anomalous dimension. In addition, there are strong shadow bands. I generalize the construction to arbitrary dynamical multi-particle correlation functions. The main aspects of this work are in agreement with numerical and Bethe Ansatz calculations by others. I also discuss the application to photoemission experiments on 1D Mott insulators and on the normal state of 1D Peierls systems, and propose the Luther-Emery model as the generic description of 1D charge density wave systems with important electronic correlations.

Abstract:
Bessel-type convolution algebras of bounded Borel measures on the matrix cones of positive semidefinite $q\times q$-matrices over $\mathbb R, \mathbb C, \mathbb H$ were introduced recently by R\"osler. These convolutions depend on some continuous parameter, generate commutative hypergroup structures and have Bessel functions of matrix argument as characters. Here, we first study the rich algebraic structure of these hypergroups. In particular, the subhypergroups and automorphisms are classified, and we show that each quotient by a subhypergroup carries a hypergroup structure of the same type. The algebraic properties are partially related to properties of random walks on matrix Bessel hypergroups. In particular, known properties of Wishart distributions, which form Gaussian convolution semigroups on these hypergroups, are put into a new light. Moreover, limit theorems for random walks on these hypergroups are presented. In particular, we obtain strong laws of large numbers and a central limit theorem with Wishart distributions as limits.

Abstract:
Let $\mathbb F=\mathbb R$ or $\mathbb C$ and $n\in\b N$. Let $(S_k)_{k\ge0}$ be a time-homogeneous random walk on $GL_n(\b F)$ associated with an $U_n(\b F)$-biinvariant measure $\nu\in M^1(GL_n(\b F))$. We derive a central limit theorem for the ordered singular spectrum $\sigma_{sing}(S_k)$ with a normal distribution as limit with explicit analytic formulas for the drift vector and the covariance matrix. The main ingredient for the proof will be a oscillatory result for the spherical functions $\phi_{i\rho+\lambda}$ of $(GL_n(\b F),U_n(\b F))$. More precisely, we present a necessarily unique mapping $m_{\bf 1}:G\to\b R^n$ such that for some constant $C$ and all $g\in G$, $\lambda\in\b R^n$, $$|\phi_{i\rho+\lambda}(g)- e^{i\lambda\cdot m_{\bf 1}(g)}|\le C\|\lambda\|^2.$$

Abstract:
In this paper we present explicit product formulas for a continuous two-parameter family of Heckman-Opdam hypergeometric functions of type BC on Weyl chambers $C_q\subset \mathbb R^q$ of type $B$. These formulas are related to continuous one-parameter families of probability-preserving convolution structures on $C_q\times\mathbb R$. These convolutions on $C_q\times\mathbb R$ are constructed via product formulas for the spherical functions of the symmetric spaces $U(p,q)/ (U(p)\times SU(q))$ and associated double coset convolutions on $C_q\times\mathbb T$ with the torus $\mathbb T$. We shall obtain positive product formulas for a restricted parameter set only, while the associated convolutions are always norm-decreasing. Our paper is related to recent positive product formulas of R\"osler for three series of Heckman-Opdam hypergeometric functions of type BC as well as to classical product formulas for Jacobi functions of Koornwinder and Trimeche for rank $q=1$.