Galoisian approach for a Sturm-Liouville problem on the infinite interval

Abstract

Abstract. We study a Sturm-Liouville type eigenvalue problem for second-order differential equations on the infinite interval (−∞, ∞). Here the eigenfunctions are nonzero solutions exponentially decaying at infinity. We prove that at any discrete eigenvalue the differential equations are integrable in the setting of differential Galois theory under general assumptions. Our result is illustrated with three examples for a stationary Schrödinger equation having a generalized Hulthén potential; a linear stability equation for a traveling front in the Allen-Cahn equation; and an eigenvalue problem related to the Lamé equation.

Integrability of the Zakharov-Shabat systems by quadrature

“…Some relations between nonintehgrability and chaotic dynamics in two-degree-of-freedom Hamiltonian systems were described in [17,22,24] based on the Morales-Ramis theory. Moreover, the differential Galois theory was used to discuss bifurcations of homoclinic orbits in four-dimensional ODEs [10] and a Sturm-Liouville problem of second-order ODEs on the infinite interval [11].…”

“…The ZS system (1.1) is transformed to a linear Schrödinger equation, as stated in Section 2. Its integrability in the meaning of differential Galois theory was also discussed in [7] for several classes of potentials which do not necessarily satisfy condition (A) and in [11] for a special potential which satisfies condition (A).…”

Liouvillian solutions for second order linear differential equations with polynomial coefficients

“…Also known as Liouvillian spectral set it is the set of eigenvalues for which the Schrödinger equation has Liouvillian eigenfunctions, see also [3,4]. In some scenarios it is known that bounded eigenfunctions of Schrödinger operator are necessarily Liouvillian, see [6]. Potentials with infinite countable algebraic spectrum are called algebraically solvable potentials and those with finite algebraic spectrum algebraically quasi-solvable potentials, for complete details see [4, §3.1, pp.…”

“…Equation(6) with M(x) = A(x)2 + B(x) has a polynomial-hyperexponential solution of polynomial degree d if and only if,Therefore ′ 2n,dis an algebraic subvariety of 2n contained in the union of the irreducible hypersurfaces of equations:Proof Note that, by definition of the sequence Δ d we have+ d = Δ d (A(x), B(x)). Analogously, the application of the AIM to Eq.…”

Liouvillian solutions for second order linear differential equations with polynomial coefficients

“…Also known as Liouvillian spectral set it is the set of eigenvalues for which the Schrödinger equation has Liouvillian eigenfunctions, see also [3,4]. In some scenarios it is known that bounded eigenfunctions of Schrödinger operator are necessarily Liouvillian, see [6]. Potentials with infinite countable algebraic spectrum are called algebraically solvable potentials and those with finite algebraic spectrum algebraically quasisolvable potentials, for complete details see [4, §3.1, pp.…”