First-order nonlinear eigenvalue problems involving functions of a general oscillatory behavior
نویسندگان
چکیده
Eigenvalue problems arise in many areas of physics, from solving a classical electromagnetic problem to calculating the quantum bound states hydrogen atom. In textbooks, eigenvalue are defined for linear problems, particularly differential equations such as time-independent Schr\"odinger equations. Eigenfunctions exhibit several standard features independent form underlying As discussed Bender \emph{et al} [\href{http://dx.doi.org/10.1088/1751-8113/47/23/235204}{J.~Phys.~A 47, 235204 (2014)}], separatrices nonlinear share some these features. this sense, they can be considered eigenfunctions equations, and quantized initial conditions that give rise interpreted eigenvalues. We introduce first-order involving general class functions obtain large-eigenvalue limit by reducing it random walk on half-line. The introduced covers special Bessel Airy functions, which themselves solutions second-order For instance, case first kind, i.e., $y'(x)=J_\nu(xy)$, we show eigenvalues asymptotically grow $2^{41/42} n^{1/4}$. also discuss reciprocal gamma Riemann zeta not simple With function, $y'(x)=1/\Gamma(-xy)$, $n$th grows factorially fast $\sqrt{(1-2n)/\Gamma(r_{2n-1})}$, where $r_k$ is $k$th root digamma function.
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ژورنال
عنوان ژورنال: Journal of Physics A
سال: 2021
ISSN: ['1751-8113', '1751-8121']
DOI: https://doi.org/10.1088/1751-8121/ac2e29