Title: Nonlinear eigenvalue problems
Carl M. Bender
Department of Physics, Washington University, St. Louis, MO63130, USAandDepartment of Mathematical Science, City University London,Northampton Square, London EC1V 0HB, UK
Abstract This talk presents an asymptotic study of the differential equation y'(x)=cos[pi xy(x)] subject to the initial condition y(0)=a. This differential equation is nonlinear, but its solutions bear a striking resemblance to solutions to the linear time-independent Schroedinger eigenvalue problem in quantum mechanics: As x increases from 0, y(x) oscillates and thus resembles a quantum wave function in a classically allowed region. At a critical value x=x_{crit}, where x_{crit} depends on a, the solution y(x) undergoes a transition; the oscillations abruptly cease and y(x) decays to 0 monotonically as x–>infty. This transition resembles the transition in a quantum wave function at a turning point as one enters the classically forbidden region. Furthermore, the initial condition a falls into discrete classes; in the nth class of initial conditions a_{n-1} < a < a_n (n=1,2,3,…), y(x) exhibits exactly n maxima in the oscillatory region. The boundaries a_n of these classes are the analogs of quantum-mechanical eigenvalues. An asymptotic calculation of a_n for large n is analogous to a high-energy semiclassical (WKB) calculation of eigenvalues in quantum mechanics. The principal result is that as n–>infty, a_n is asymptotic to Asqrt{n}, where A=2^{5/6}. Numerical analysis reveals that the first Painleve transcendent has an eigenvalue structure that is quite similar to that of the equation y'(x)=cos[pi xy(x)] and that the nth eigenvalue grows with n like a constant times n^{3/5}. Finally, it is noted that the constant A is numerically close to the lower bound on the power-series constant P in the theory of complex variables, which is associated with the asymptotic behavior of zeros of partial sums of Taylor series. This work was done in collaboration with A. Fring and D. W. Hook. ———–