The Schrödinger equation with potential V(x) = 2λ1cos x + 2λ2cos αx is considered. Its solution reduces to the problem of finding the eigenvectors of a matrix. The eigenvalues of this matrix show a band structure which is very sensitive to the value of the parameter α. The solutions of the Schrödinger equation are presented, and their physical meaning is discussed. The potential V(x) has a complex multiwell structure, and quantum tunneling occurs. The accuracy of all approximations is carefully studied.

One-dimensional systems with the multiwell potential V (x) = 2λ 1 cos x + 2λ 2 cos αx are considered. Localized wavefunctions, in the case when the energy is greater than the maximum value of the potential, are presented. Wigner and Weyl functions, corresponding to the wavefunctions of these systems, are also studied. It is shown that they are very sensitive to the value of the parameter α.

No abstract

This paper is a sequel to our previous work in which we found a elementary arithmetic operators of simple continued fractions. We consider the continued fractions (C.F.). We found the multiplication operation of two continued fractions with positive non integer numerators. We apply our finding to many examples of continued fractions.

Simple continued fractions for (rational and irrational) are considered. The power of the simple continued fractions are discovered. On other hand we discover how to calculate as a simple continued fractions. The most important that we did in this paper, we prove by theorem any two simple continued fractions the fraction is  or  (for any n, m). Many definitions and examples that we used of that low and theorem are presented.