First-order nonlinear eigenvalue problems involving functions of a general oscillatory behavior

TL;DR

This paper introduces a first-order nonlinear eigenvalue problem involving functions with general oscillatory behavior, analyzing asymptotic eigenvalue growth for special functions like Bessel, Airy, gamma, and zeta functions.

Contribution

It formulates a new class of nonlinear eigenvalue problems and derives asymptotic eigenvalue behaviors, extending the concept of eigenfunctions to nonlinear differential equations.

Findings

Eigenvalues grow as n^{1/4} for Bessel functions.

Eigenvalues grow factorially for reciprocal gamma function.

Large eigenvalue limits reduce to a random walk problem on a half-line.

Abstract

Eigenvalue problems arise in many areas of physics, from solving a classical electromagnetic problem to calculating the quantum bound states of the hydrogen atom. In textbooks, eigenvalue problems are defined for linear problems, particularly linear differential equations such as time-independent Schr\"odinger equations. Eigenfunctions of such problems exhibit several standard features independent of the form of the underlying equations. As discussed in Bender \emph{et al} [\href{http://dx.doi.org/10.1088/1751-8113/47/23/235204}{J.~Phys.~A 47, 235204 (2014)}], separatrices of nonlinear differential equations share some of these features. In this sense, they can be considered eigenfunctions of nonlinear differential equations, and the quantized initial conditions that give rise to the separatrices can be interpreted as eigenvalues. We introduce a first-order nonlinear eigenvalue problem involving a general class of functions and obtain the large-eigenvalue limit by reducing it to a random walk problem on a half-line. The introduced general class of functions covers many special functions such as the Bessel and Airy functions, which are themselves solutions of second-order differential equations. For instance, in a special case involving the Bessel functions of the first kind, i.e., for $y'(x)=J_\nu(xy)$, we show that the eigenvalues asymptotically grow as $2^{41/42} n^{1/4}$. We also introduce and discuss nonlinear eigenvalue problems involving the reciprocal gamma and the Riemann zeta functions, which are not solutions to simple differential equations. With the reciprocal gamma function, i.e., for $y'(x)=1/\Gamma(-xy)$, we show that the $n$th eigenvalue grows factorially fast as $\sqrt{(1-2n)/\Gamma(r_{2n-1})}$, where $r_k$ is the $k$th root of the digamma function.