| Abstract: | In this paper we study the WKB-Langer asymptotic expansion
of the eigenfunctions of a Schrödinger operator
 .
Applying these asymptotic formulae we prove that the exact
L2 eigenfunction ΨE(N,ℏ)
(and its derivative ℏΨ′E(N,ℏ))
of the Schrödinger operator with a well-shaped analytic potential
are approximated up to arbitrary order ℏm by the
semi-classical WKB-Langer approximate eigenfunction
ΨEm(N,ℏ),m (and its derivative
ℏΨ′Em(N,ℏ),m) respectively
in L2, i.e.
‖ΨE(N,ℏ)-ΨEm(N,ℏ),m‖L2=O(ℏm+1),‖ℏΨ′E(N,ℏ)-ℏΨ′Em(N,ℏ),m‖L2=O(ℏm+1)
uniformly for any N. Here Em(N,ℏ) approximates
E(N,ℏ) up to m-th order (in ℏ) and satisfies the
m-th order quantization condition. There are applications of this limit
to Burgers equations, turbulence and the large scale structure of the universe. |
