We obtain the parity invariant generating functional for the canonical transformation mapping the Liouville theory into a free scalar field and explain how it is related to the pseudoscalar transformation e-mail: [email protected] Present address: Department of Mathematical Sciences, University of Durham, UK 1 In this paper we consider the 2-dimensional Liouville theory, based on the Lagrangi...

This yields an irrationality criterion (which is the basic tool for proving the irrationality of specific numbers), and Liouville extended it into a transcendence criterion. The proof by Liouville involves the irreducible polynomial f ∈ Z[X] of the given irrational algebraic number α. Since α is algebraic, there exists an irreducible polynomial f ∈ Z[X] such that f(α) = 0. Let d be the degree o...

Eigenvalues in the essential spectrum of a weighted Sturm-Liouville operator are studied under the assumption that the weight function has one turning point. An abstract approach to the problem is given via a functional model for indefinite Sturm-Liouville operators. Algebraic multiplicities of eigenvalues are obtained. Also, operators with finite singular critical points are considered. MSC-cl...

We consider pointwise, box, and Hausdorff dimensions of invariant measures for circle diffeomorphisms. We discuss the cases of rational, Diophantine, and Liouville rotation numbers. Our main result is that for any Liouville number τ there exists a C∞ circle diffeomorphism with rotation number τ such that the pointwise and box dimensions of its unique invariant measure do not exist. Moreover, th...

We study the simplest examples of minimal string theory whose worldsheet description is the unitary (p, q) minimal model coupled to twodimensional gravity (Liouville field theory). In the Liouville sector, we show that four-point correlation functions of ‘tachyons’ exhibit logarithmic singularities, and that the theory turns out to be logarithmic. The relation with Zamolodchikov’s logarithmic d...