International Journal of Modern Physics B

We consider the current state of the quest for a quantum phase operator, which started in the earliest days of quantum mechanics.^4–7 Particular emphasis has been placed on analysis of the structure of the several distinct theories, both physical and mathematical, which has led us from classical function theory to the quantum theory of measurement.

It is shown that the well-known (1+1)-dimension equations of Korteweg-de Vries (KdV), Boussinesq (BSQ) and regulated long waves (RLW) have a “latent structure”, the latter connected with the and ℝLW sets. The elements of these sets are solition-like equations of the KdV(τ, α) and RLW(τ, α) type, where τ≥1 is the order of higher derivative with respect to t and the parameter α>0 defines the “modification” of the above input equations. For example, KdV(1, 1)=KdV, KdV(1, 1/2)=mKdV,…, KdV(2, 1)=BSQ, KdV(2, 1/2)=mBSQ,…;RLW(1,1)=RLW, RLW(1,1/2)=mRLW, etc.

A “DH-system” is defined as a multidimensional hydrogen atom (or its one-particle analogue), D≥1. Investigating many Coulomb problems in ℝ^D it is necessary to know exact analytical expressions of multipole matrix elements <q|r^k|q'>_D for DH-systems, where q=(N, µ) is a set of parameters, N —"principal” and µ — "orbital” quantum numbers. The paper deals with the new method for the evaluation of similar matrix elements using new properties of Appell’s function F₂(x, y) to the vicinity of the singular point (1, 1).

Such approach allows: 1) to get exact analytical expressions of these matrix elements (considering the selection rules) by means of Appell’s F₂ (or Clausen’s ₃F₂) functions; 2) to reveal “latent” symmetry of diagonal matrix elements with respect to the point k₀=−3/2, the above symmetry is connected with the property of Appell’s function F₂ (1,1) mirror-like symmetry; 3) to find (exact) asymptotics of the off-diagonal matrix elements in terms of Horn’s function ψ₁ (x, y); 4) to prove that the orthogonality of radial functions fN_µ (D, r) over N and μ for DH-systems is connected with the properties of Appell’s F₂ function to the vicinity of the singular point (1, 1), it generalizes the known result for 3H-atom by Pasternack and Sternheimer, J. Math. Phys.3, 1280 (1962).

Transport properties of mesoscopic rings confined between potential barriers are considered. Also considered is the double barrier structure with barriers fabricated of rings connected by a one-dimensional wire. Such structures we define as Aharonov-Bohm diodes with resonant tunneling (ABDRT) because for zero external magnetic flux their transport properties are quite similar to diodes with resonant tunneling (DRT). However, application of external magnetic flux gives rise to new resonant peaks of transmission determined by the quantization condition of the rings. Positions and widths of these novel flux induced resonant peaks extremely depend on the flux.