On One Integral Equation in the Theory of Transform Operators
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L DIFFERENTIAL EQUATIONS
On One Integral Equation in the Theory of Transform Operators S. M. Sitnika,* a
Belgorod State University, Belgorod, 308002 Russia *e-mail: [email protected]
Received February 15, 2020; revised February 15, 2020; accepted April 9, 2020
Abstract—Integral representations of solutions of one differential equation with singularities in the coefficients, containing the Bessel operator perturbed by some potential, are considered. The existence of integral representations of a certain type for such solutions is proved by the method of successive approximations using transform operators. Potentials with strong singularities at the origin are allowed. As compared with the known results, the Riemann function is expressed not via the general hypergeometric function, but, more specifically, via the Legendre function, which helps to avoid unknown constants in the estimates. Keywords: transform operator, Riemann function, Gauss hypergeometric function, Legendre function, singular potential DOI: 10.1134/S096554252008014X
1. INTRODUCTION. FORMULATION OF THE PROBLEM Consider the problem of constructing an integral representation of a certain kind for solutions of the differential equation
Bαu( x) − q( x )u( x) = 0,
x > 0,
(1)
where Bα is the Bessel operator of the form (2) Bαu = u''( x) + 2α u'( x), α > 0, x > 0. x This problem is solved by the method of transform operators. For this, it suffices to construct a Poissontype transform operator Pα of the form ∞
Pαu( x) = u( x) + P ( x, t )u(t )dt
(3)
x
with some kernel P ( x, t ) that interwines the operators Bα and Bα − q( x) according to the formula
BαPαu = Pα (Bα − q( x))u
(4)
on functions u ∈ C 2(0, ∞). As a result, we obtain a formula expressing the solutions of Eq. (1) with a spectral parameter, which has the form
Bαu( x) − q( x )u( x) = λ 2u( x),
x > 0,
via the solutions of the unperturbed equation, i.e., via Bessel functions. In this case, the spectral parameter λ does not affect the form of linear transform operators whose kernels do not depend on it. This approach reflects one of the main applications of transform operators: the expression of solutions of more complex differential equations via similar simpler ones. The theory of transform operators is an important branch of modern mathematics, which has numerous applications (see [1–6]). The possibility of representation (3) with a sufficiently “good” kernel P for a wide class of potentials q( x ) underlies the classical methods for solving inverse problems in the quantum scattering theory [7, 8]. For Sturm–Liouville equations, transform operators of form (3) were first constructed by B.Ya. Levin (see [1–6]). 1381
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SITNIK
The transform operators for the Bessel operator of the Sonin and Poisson’s type were introduced by Delsart; for the first time, the theory of them was presented in Russian and developed in the famous work by Levitan [9]. Later, in a number of works, transform operators with property (4) for variable potentials were also considered, simultaneously with So
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