A General Approach of Quasi-Exactly Solvable Schroedinger Equations

We construct a general algorithm generating the analytic eigenfunctions as well as eigenvalues of one-dimensional stationary Schroedinger Hamiltonians. Both exact and quasi-exact Hamiltonians enter our formalism but we focus on quasi-exact interactions for which no such general approach has been considered before. In particular we concentrate on a generalized sextic oscillator but also on the Lame and the screened Coulomb potentials.Comment: 23 pages, no figur

SUSY Intertwining Relations of Third Order in Derivatives

The general solution of the intertwining relations between a pair of Schr\"odinger Hamiltonians by the supercharges of third order in derivatives is obtained. The solution is expressed in terms of one arbitrary function. Some properties of the spectrum of the Hamiltonian are derived, and wave functions for three energy levels are constructed. This construction can be interpreted as addition of three new levels to the spectrum of partner potential: a ground state and a pair of levels between successive excited states. Possible types of factorization of the third order supercharges are analysed, the connection with earlier known results is discussed.Comment: 17

On one-dimensional Schroedinger problems allowing polynomial solutions

We discuss the explicit construction of the Schroedinger equations admitting a representation through some family of general polynomials. Almost all solvable quantum potentials are shown to be generated by this approach. Some generalization has also been performed in higher-dimensional problems.Comment: 17 page

The nonius–monitoring vibrations of mechanic systems correlation method

Запропонований нонiусний алгоритм використання кореляцiйних методiв для виявлення i контролю схованих випадковими шумами вiбрацiй тiл обертання. Ефективнiсть такого пiдходу пiдтверджена чисельним моделюванням.The nonius algorithm of using correlation methods is proposed for detecting and monitoring of hidden by random noise vibrations of rotating bodies. Effectiveness of this approach is proved by multiple numerical simulations