Eigenfunctions of Ordinary Differential Euler Operators
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EIGENFUNCTIONS OF ORDINARY DIFFERENTIAL EULER OPERATORS UDC 517.927, 517.923
Yu. Yu. Bagderina
Abstract. Asymptotic solutions of the eigenvalue problem for an Euler operator in a neighborhood of a regular singular point are considered. We find a condition under which the asymptotic expansion is free of logarithms. Eigenvalues expressed in terms of elementary functions in the form of a finite sum of quasi-polynomials are obtained for third-order Euler operators and also for commuting Euler operators of sixth and ninth orders. The problem on common eigenfunctions for commuting Euler operators is examined. In the case of operators of rank 2 and 3, it can be reduced to second- and third-order Bessel equations by differential substitutions. Keywords and phrases: eigenfunction, Euler operator, Fuchsian singularity. AMS Subject Classification: 47E05, 34L10, 34B30
1. Introduction. The main subject of the present paper is a linear ordinary differential equation (ODE), which can be called the Bessel equation of the nth order (see [1, 2]): En w = μn w,
μ = const = 0,
n ∈ N,
i.e., the eigenvalue problem for an Euler operator En = Dzn +
n k=1
Cnk
ak n−k D , zk z
ak ∈ C,
Cnk =
k , n
(1)
Dz =
d . dz
(2)
Equation (1) is a non-Fuchsian equation since only z = 0 is a regular singular point for it, while z = ∞ is an irregular singularity. The Frobenius method (see [5, 6]) is applicable to this equation. We outline it briefly in Sec. 2. It allows constructing a solution to Eq. (1) in a neighborhood of a regular singular point. In Sec. 3, conditions on the operator (2) are found, which imply that the series defining the solution of Eq. (1) do not contain logarithmic terms. In a particular case described in [1], the general solution of Eq. (1) can be represented as a finite sum of quasi-polynomials. In Sec. 3, all such solutions are found for the third-order Bessel equations and for Eqs. (1) of the sixth and ninth order, where the corresponding operators E6 and E9 commute. In Sec. 4, we consider the problem on common rank-l eigenfunction ψ(z) of commuting Euler operators Em and En , where l is the highest common factor of the numbers m and n. In the case of rank-2 and rank-3 operators, a differential substitution reduces the solution of the equations (Em − μm )ψ = 0,
(En − μn )ψ = 0
(3)
to integration of the second- and third-order Bessel equations, respectively. It seems that this is also valid for arbitrary l, when ψ(z) is found by integrating the lth order Bessel equation. 2.
Frobenius method. Singularities of solutions of the linear equation Ln w = 0,
Ln = Dzn + B1 (z)Dzn−1 + · · · + Bn−1 (z)Dz + Bn (z),
(4)
can arise only from singularities of the coefficients of the equation (see [6]). A point z0 is a singular point if the conditions of the existence are violated at z0 . Without loss of generality, we can assume that z0 = 0. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 152, Mathematical Physics, 2018. c 2021 Springer Science+Business Media
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