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Some representative potentials of the anharmonic-oscillator type are constructed. Some corresponding spectra-shift operators are also constructed. These operators are a natural generalization of Fok creation and annihilation operators. The Schrodinger problem for these potentials leads to an equidistant energy spectrum for all excited states, which are separated from the ground state by an energy gap. The general properties of the dynamic system generated by spectral-shift operators of third degree are analyzed. Several examples of such anharmonic oscillators are discussed. The relationship between the eigenvectors of the Schrodinger problem and a certain type of nonclassical orthogonal polynomials is established.

Some aspects of the mechanical analogy recently proposed for the static and dynamic renormalisation group (RG) in the large-n limit are pointed out. In particular, starting from purely mechanical methods, a maximal set of the static RG invariants is obtained. Similar results also hold in the most complex dynamical case. Finally, by means of a canonical transformation, the original statistical mechanics problem is reduced to an equivalent anisotropic hyperbolic oscillator under stabilising conditions.

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