We will in this note show that it is possible to diagonalise the Lund
Fragmentation Model. We show that the basic original result, the Lund Area
law, can be factorised into a product of transition operators, each describing
the production of a single particle and the two adjacent breakup points
(vertex positions) of the string field. The transition operator has a discrete
spectrum of (orthonormal) eigenfunctions, describing the vertex positions
(which in a dual way corresponds to the momentum transfers between the
produced particles) and discrete eigenvalues, which only depend upon the
particle produced. The eigenfunctions turn out to be the well-known two-
dimensional harmonic oscillator functions and the eigenvalues are the analytic
continuations of these functions to time-like values (corresponding to the
particle mass). In this way all observables in the model can be expressed in
terms of analytical formulas. In this note only the 1+1-dimensional version
of the model is treated but we end with remarks on the extensions to gluonic
radiation, transverse momentum generation etc, to be performed in future
papers.