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On the expansions in eigenfunctions of Hill’s operator

In this paper we show how one can deduce the Titchmarsh expansion formula in eigenfunctions of Hill’s operator from the Gel’fand expansion formula. | Turk J Math 25 (2001) , 323 – 337. ¨ ITAK ˙ c TUB On the Expansions in Eigenfunctions of Hill’s Operator F. Aras and G. Sh. Guseinov Abstract In this paper we show how one can deduce the Titchmarsh expansion formula in eigenfunctions of Hill’s operator from the Gel’fand expansion formula. Key Words: Hill’s operator, spectrum, eigenvalues, eigenfunctions. 1. Introduction Consider the second-order differential equation 0 −[p(x)y0 ] + q(x)y = λρ(x)y (−∞ 0: p(x + ω) = p(x), q(x + ω) = q(x), ρ(x + ω) = ρ(x). In addition, we assume that p(x) > 0 and ρ(x) > 0 almost everywhere, and Z ω Z ω Z ω dx < ∞, |q(x)|dx < ∞, ρ(x)dx < ∞ . 0 p(x) 0 0 (2) Notice that we do not assume the differentiability and even the continuity of p(x). A function y = y(x) is called a solution of the equation (1) if its first derivative y0 (x) exists, p(x)y0 (x) is absolutely continuous and (1) is satisfied almost everywhere on (−∞, ∞). Let us set AMS Subject Classifications: 34L20, 47E05 323 ARAS, GUSEINOV y[1] (x) = p(x)y0 (x). This is the so-called quasi-derivative of y(x). For any solution y(x) and any point a ∈ (−∞, ∞) the value y(a) is finite, whereas the value y0 (a) may be infinite. However, the value y[1] (a) = limx→a p(x)y0 (x) certainly will be finite. Under condition (2) the existence and uniqueness theorem for solution y(x) of the equation (1) satisfying the initial conditions y(a) = c0 , y[1] (a) = c1 is valid (See, for example, [12, Kapitel 5].) For the results relating to eigenvalue and eigenfunction theory of periodic differential equations we refer to [1, 2, 11, 14]. Denote by − + − + − + µ+ 0 < µ2 ≤ µ2 < µ4 ≤ µ4 < . < µ2j ≤ µ2j < . the eigenvalues of the periodic boundary value problem (BVP) generated on the segment 0 ≤ x ≤ ω by Equation (1) and the boundary conditions y(0) = y(ω), y[1] (0) = y[1] (ω), (3) and by + − + − + µ− 1 ≤ µ1 < µ3 ≤ µ3 < . < µ2j+1 ≤ µ2j+1 < . the eigenvalues of the semi-periodic (or anti-periodic) BVP generated on the segment 0 ≤ x ≤