Nonlinear eigenvalue problems for generalized Painlev\'e equations
TL;DR: In this paper, it was shown that the differential equations for the first and second Painleve transcendents can be generalized to large classes of nonlinear differential equations, all of which have discrete eigenvalue spectra.
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Abstract: Eigenvalue problems for linear differential equations, such as time-independent Schrodinger equations, can be generalized to eigenvalue problems for nonlinear differential equations. In the nonlinear context a separatrix plays the role of an eigenfunction and the initial conditions that give rise to the separatrix play the role of eigenvalues. Previously studied examples of nonlinear differential equations that possess discrete eigenvalue spectra are the first-order equation $y'(x)=\cos[\pi xy(x)]$ and the first, second, and fourth Painleve transcendents. It is shown here that the differential equations for the first and second Painleve transcendents can be generalized to large classes of nonlinear differential equations, all of which have discrete eigenvalue spectra. The large-eigenvalue behavior is studied in detail, both analytically and numerically, and remarkable new features, such as hyperfine splitting of eigenvalues, are described quantitatively.
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Citations
Quantum Spectral Problems and Isomonodromic Deformations
TL;DR: In this article , a selfconsistent approach to study the spectral properties of a class of quantum mechanical operators by using the knowledge about monodromies of $2\times 2$ linear systems (Riemann-Hilbert correspondence) was developed.
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arXiv : Quantum spectral problems and isomonodromic deformations
TL;DR: In this paper, a selfconsistent approach to study the spectral properties of a class of quantum mechanical operators by using the knowledge about monodromies of $2\times 2$ linear systems (Riemann-Hilbert correspondence) was developed.
Addendum: Painlevé transcendents and PT -symmetric Hamiltonians (2015 J. Phys. A: Math. Theor. 48 475202)
Carl M. Bender,Javad Komijani +1 more
- 08 Mar 2022
TL;DR: Bender et al. as discussed by the authors showed that for a fixed initial value such as y(0) = 1 a discrete set of initial slopes y′(0), = b n give rise to separatrix solutions.
1
On the Connection Problem for the Second Painlevé Equation with Large Initial Data
19 Apr 2022
TL;DR: In this article , the authors considered two special cases of the connection problem for the homogenous second Painlevé equation (PII) using the method of uniform asymptotics proposed by Bassom et al.
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Exactly solvable nonlinear eigenvalue problems
TL;DR: The nonlinear eigenvalue problem of a class of second order semi-transcendental differential equations is studied in this paper, where the eigensolutions in the two problems are closely related.
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