Investigation of the deficiency indices of the minimum two-term fourth-order differential operator with irregular coefficients

Abstract

This paper investigates the deficiency indices of symmetric singular Nonsemibounded minimum two - term diferential operator L₀ generated in L₂[x₀;∞), x₀ > 0, diferential expression ly = y⁽⁴⁾ + (q(x) + h(x))y; x [x₀; ∞); which q(x) is satisfies the following conditions: the function q(x) is twice continuously diferentiable function; q’(x) and q’’(x) do not change the sign for R > 0 a suficiently large x > R, as well as performed |q(x)|→+ ∞; x →+ ∞; |q’(x)| = o(|q^ζ (x)|) for x [x0; ∞], where 0 <ζ < 5/4. Taken together, these conditions are called the terms of the Titchmarsh-Levitan and h(x) is a fast oscillatory perturbation. It is well known that if the coeficients q(x) and h(x) have a regular behavior when x →+ ∞ then the equation ly = λy can be reduced to a system of linear diferential equations with almost a diagonal matrix, and then by using the well known Levinson theorem to construct the asymptotic behavior of solutions. In turn, the asymptotic formulas for the fundamental system of solutions of the equation ly = λy contain important information about the defect indices of the operator L₀ and on the quality of the spectral properties of self - adjoint extensions of the operator L₀. In [1] we obtained the asymptotic formulas for fundamental system of solutions of equations ly = λy with coeficients diferent from regular ones. In this work we used the asymptotic formulas obtained in [1] for investigating defect indices of the operator L₀.