Research on singular Sturm–Liouville spectral problems with a weighted function

As early as 1910, Weyl gave a classification of the singular Sturm–Liouville equation, and divided it into the Limit Point Case and the Limit Circle Case at infinity. This led to the study of singular Sturm–Liouville spectrum theory. With the development of applications, the importance of singular Sturm–Liouville problems with a weighted function becomes more and more significant. This paper focuses on the study of singular Sturm–Liouville problems with a weighted function. Finally, an example of singular Sturm–Liouville problems with a weighted function is given.


Introduction
The Sturm-Liouville problems originated in the early 19th century by solving the heatconduction equation in partial differential equations, obtained by the method of separation of variables, and later found a wide range of applications in mathematics and physics. For example, the first eigenvalue of the regular Sturm-Liouville problems represents the first energy level in quantum mechanics and quantum chemistry (cf. [2,10,11,14]), and has been applied to the calculation of electron-cloud density, which is a powerful tool for understanding and explaining quantum phenomena. Since the mathematical models of many important issues in practical problems are defined on infinite intervals or on finite intervals with singularities at the endpoints of the coefficient functions, the singularity problem has been of interest to scholars of mathematics and physics. As the spectrum of the singular problem becomes more complicated, not only the pure point spectrum appears in the regular case, but also the absolutely continuous spectrum and the singular continuous spectrum. This leads to the fact that the spectral decomposition theorem for the regular case is no longer applicable (cf. [3,20,25]), and therefore more research on the spectral aspects of the singular problem is needed.
As early as 1910, Weyl gave a classification of the singular Sturm-Liouville equation by using the circle-set method, which divides it into the limiting point type and the limiting circle type at the infinity point. This led to the study of the singular Sturm-Liouville spectral theory. In 1937, Saks [15] proved De la Vallée Poussin's theorem using the Lesbgue decomposition of measures. In 1943, Loomis [9] proved Fatou's Lemma using the Poisson-Stieltjes integral. On the basis of these measure theories the Lebesgue decomposition of the spectral measures was performed to complete the classification of the spectrum of differential operators. In 1975, Levitan and Sargsjan [8] used the Lesbgue decomposition of the spectral measure to classify the spectrum into: the absolute continuous spectrum, the singular continuous spectrum, and the pure point spectrum. Regarding the absolute continuous spectrum, in 1957 Aronsajn [1] used the m(λ) function to prove that the absolute continuous spectrum of the Sturm-Liouville spectral problem is invariant under a rankone perturbation. In 1986, Simon and Wolff [18] gave equivalence conditions for the Borel transform of the measure and the spectral decomposition. In 1989, Simon and Spencer [17] proved that the potential function is a High Barrier function, i.e., the potential function tends to infinity, when the corresponding Sturm-Liouville differential operator has no absolutely continuous spectrum. The singular continuous spectrum is more complicated, in 1995, Simon [16] proved the existence of a purely singular continuous spectrum for the general operator, that is, there is an interval in which there is no point spectrum and no absolute continuous spectrum, but only a singular continuous spectrum.
With the deeper and deeper study of practical problems, the importance of singular Sturm-Liouville problems (cf. [4][5][6][7]19]) with a weighted function becomes more and more significant as the solution space expands from the L 2 space to the L 2 w space with a weighted function (cf. [12,13,[22][23][24]) and has more practical applications. This paper focuses on refining the definition of spectral measures for singular Sturm-Liouville problems with a weighted function. This paper finds several differences for the case of singular Sturm-Liouville problems with a weighted function based on the analysis of the spectral problem of general singular Sturm-Liouville problems. Finally, an example of singular Sturm-Liouville problems with a weighted function is given, and its expansion theorem and the expression of the support set of spectral measure are proved using the method of this paper.
Following this section, for extending the regular Sturm-Liouville boundary value problem to the singular problem, some preliminaries will be given in Sect. 2. In Sect. 3, Wely-Titchmarsh functions are introduced and the classification of the Limit Circle Case and the Limit Point Case is derived. In Sect. 4, some criteria of the Limit Point Case will be obtained. In Sect. 5, an example of the singular Sturm-Liouville problems with a weighted function is studied.
Take y(x, λ) as satisfying the Cauchy problem, which is the solution of equation (1). Denote λ n,b as the nth eigenvalue of the regular problem (1), (2), and (3). Then, the corresponding eigenfunction is y n,b (x) := y(x, λ n,b ), which satisfies the right boundary condition (3), By the Parseval Identity, for any f ∈ L 2 Now, we introduce the monotone nondecreasing function or measure ρ b (λ), asλ > 0.
By definition, ρ b (0) = 0. Then, the Parseval Identity can be rewritten as, where The above equation is called the generalized Fourier transform of f (x). In the following, let b → ∞, and we will prove that the Parseval Identity still holds.

Lemma 2.1 For any positive integer N , there exists a positive constant A(N, w), such that
where A(N, w) only depends on N and w, and is independent of b.
Proof Let Since w(x) ∈ L 1 loc [0, ∞) and w(x) > 0, a.e., it follows that c h > 0, for any h > 0. In the case sin α = 0, there exists sufficiently small positive numbers h, such that Using (6) and the Parseval Identity (4), we can obtain that Hence, Note that c h only depends on N and w, and is independent of b. Thus, (5) has been proved.
In the case sin α = 0, |y (0, λ)| = | cos α| = 1. Hence, there exists a sufficiently small num- In this case, define function f h (x) as Similar to the case where sin α = 0, we can obtain that Hence, Similarly, we know that c h only depends on N and w, and is independent of b. The proposition has been proved.
In the following proof, Helly's selection theorem is needed. By Helly's selection theorem, we can use the regular Sturm-Liouville problem (1) to approximate the singular problem and study the properties of the spectrum.
Suppose f m ∈ D satisfying the left boundary condition (2) and is a compactly supported function, i.e., Then, the Parseval Identity tells us, where F m is the Fourier transform of f m , i.e., Using the Green formula, we can obtain Substituting (10) into F m (λ), we have Using the Green formula again, we obtain By Lemma 2.1, we know that the monotone function family {ρ b (λ), λ ∈ (-N, N)} is bounded by A(N, w), which only depends on N and w, and is independent of b. Then, by Helly's Selection Theorem Lemma 2.2, there exists a subsequence b k , such that ρ b k weakly convergent to measure ρ. Hence, for any F ∈ L 2 (-N, N), we have By (13) and (12), taking the limit, we can obtain So far, we have proved the Parseval Identity when f is a function that is compactly supported. In the following, we will prove the general case. For any f ∈ L 2 w (0, ∞), there exist a sequence of compactly supported functions {f m (x)}, such that Hence, Using the Parseval Identity of compact support functions, the Fourier transform of sequence f m is a Cauchy sequence in L 2 dρ (-∞, ∞), i.e., By the Completeness of L 2 dρ (-∞, ∞), there exists a function F(λ) ∈ L 2 dρ (-∞, ∞) that satisfies The proposition has been proved w (0, ∞), and its generalized Fourier transform G, we have , Subtracting the two formulas, we can obtain and this identity is called the generalized Parseval Identity.
By the differentiable dependence of solutions on parameters, we know that ψ and ϕ are both analytic functions of λ and their Wronskian satisfies All solutions of equation (1) except ψ can be expressed as For any b ∈ (0, ∞) and β ∈ [0, π), let m = m(b, λ, β), such that χ satisfies the right boundary condition (3), We can obtain the expression of m (λ, b), which is a fractional linear mapping. Hence, as β goes through (0, π), the graph of m forms a circle C(λ, b). After calculation, we can obtain that the center of circle C(λ, b) is Now, consider two solutions of equation (1), f (x) and g(x), and they are satisfied if τ f = λf and τ g =λg. Then, we can obtain where In particular, let f = ψ, g =ψ andλ =λ, then we have and hence we can deduce that and the radius is Using these facts, we can obtain the following summaries. The circles satisfy that for any 0 < b 1 < b 2 < ∞, C(λ, b 2 ) ⊂ C(λ, b 1 ). Hence, for any mλ = 0, C(λ, b) → a circle C(λ) or a point m(λ), as b → ∞. Furthermore, if we set χ = ϕ(λ) + mψ(λ), then we have that b 0 χ(t) 2 w(t) dt < mm mλ By (1), we know that there exists at least one L 2 w solution at +∞. As C(λ, b) → a circle C(λ), (19) tells us that all solutions of (15) belong to L 2 w at +∞, and at this time, (15) is called the Limit Circle Case at +∞. As C(λ, b) → a point m(λ), there is only one linear independence solution of (15) belonging to L 2 w at +∞, and at this time, (15) is called the Limit Point Case at +∞. In the next section, we will study some judgments about the two classifications at infinity.

A criterion of the limit point case
Now, we consider the formal differential operator ly :=py + qy = λwy, y = y(x), x ∈ (0, ∞), with the nature domain D := y, py ∈ L 2 w [0, ∞) : y, py ∈ AC[0, ∞) and where y ∈ AC[0, ∞) means y is an absolute continuous function on [0, ∞). The formal differential l and τ in (1) satisfy the relationship l = wτ . In this section, we will use the coefficient functions p, q, and w in equation (21) to describe whether the differential equation is the Limit Point Case or the Limit Circle Case (see [21]). In the following, some preliminaries will be given. For any complex number λ ∈ C, let χ(t, λ) be any solution of ly = λwy. Then, χ(t, λ) is the solution of the differential equation lyλ 0 wy = (λλ 0 )wy.
Using the above two inequalities and (22) we can obtain the next estimate and hence The right-hand side of the inequality is independent of t, therefore let t → ∞, which can give χ ∈ L 2 w [0, ∞).
In particular, when the weighted function w ≡ 1, w / ∈ L 1 [0, ∞), Therefore, it follows from the above theorem that in this case (21) is a limiting point type at the infinity point. However, if w ∈ L 1 [0, ∞), this theorem does not necessarily hold, see the following example.