Sturm—Liouville and Dirac Operators

Let L bea linear operator defined on a certain set of elements. An element y ≠ 0 is called an eigenelement of L if Ly = λy, and A is called the corresponding eigenvalue of L.

As was noted in Section 1 of Chapter 1, a Sturm-Liouville problem is said to be singular if either the interval [a, b] is infinite or if the function q(x) on [a, b] is not summable (or both). Here, we obtain the expansion theorem for the singular problem, considering it as the limit of regular ones.

The spectrum associated with the problem for the Sturm-Liouville operator on the half-line [0, ∞) is the complement of the set of points in whose neighborhoods the spectral function ρ(λ) is constant. In the case of the whole line (-∞, ∞), the spectrum is complementary to the set of points in whose neighborhoods all functions ξ(λ), η (λ) and ζ(λ) are constant. It is obvious that the spectrum is a closed set.

It follows from the results of Chapter 3 that if the function q(x) of the Sturm-Liouville operator is bounded from below, and tends to +∞ as x → a or x → b (or both), then the spectrum of L is discrete (assuming that at least one of the endpoints is singular; furthermore, if at least one of them is regular, a boundary condition should be specified on it).

Consider the equation where q(x) is an infinitely differentiable function.

By inverse problems in spectral analysis, we mean those of reconstructing a linear operator from some of its spectral characteristics such as spectra (for different boundary conditions), a spectral function, scattering data, etc.

In a similar way to the case of the Sturm-Liouville operator, considered in Chapter 2, we prove the expansion theorem and the Parseval equation for the singular Dirac operator, considering the latter as the limit of regular problems.

Consider the problem on the half-axis [0, ∞), assuming that the coefficients q₁(x) and q₂(x) are in the class L(0,∞).