Semiclassical analysis of edge state energies in the integer quantum Hall effect

Analysis of edge-state energies in the integer quantum Hall effect is carried out within the semiclassical approximation. When the system is wide so that each edge can be considered separately, this problem is equivalent to that of a one dimensional harmonic oscillator centered at x = x(c) and an infinite wall at x = 0, and appears in numerous physical contexts. The eigenvalues E-n(x(c)) for a given quantum number n are solutions of the equation S( E, x(c)) = pi[n + gamma(E, x(c))] where S is the WKB action and 0 < gamma < 1 encodes all the information on the connection procedure at the turning points. A careful implication of the WKB connection formulae results in an excellent approximation to the exact energy eigenvalues. The dependence of gamma[En(x(c)), x(c)] equivalent to gamma(n)(x(c)) on x(c) is analyzed between its two extreme values 1/2 as x(c) -> -infinity far inside the sample and 3/4 as x(c) -> infinity far outside the sample. The edge-state energies E-n(x(c)) obey an almost exact scaling law of the form E-n(x(c)) = 4[n + gamma(n)(x(c))]f (x(c)/root 4n+3) and the scaling function f(y) is explicitly elucidated.