A Class of Generalized Differential Operators with Infinitely Many Discontinuous Points

Abstract

We define a class of differential operators 𝑇 which have infinitely many discontinuous points, and investigate the kernel of 𝑇 in terms of operator theory. It is shown that the solutions of 𝑇 have exponential behavior, and the dimension of the kernel is given.

1. Introduction and Motivations

In the study of ergodicity of billiard flows, a very important question concerns the hyperbolicity of these flows [1, 2]. Hyperbolicity is defined in the language of the linearized system; this is the situation where trajectories close to a given trajectory either diverge exponentially in time from it or converge in the same way. The differential operator 𝑇 we study is associated with the linearization of the flow; hence we ask whether solutions to 𝑇𝑓=0 have the corresponding property. However, this is a paper not about ergodic theory, but about the spectral theory [3, 4] of a class of generalized differential operators which comes up naturally in this context.

The first class of chaotic billiards was introduced by Sinai in [5]; he proved the ergodicity of plenary dispersing billiards. It took more than 30 years; until 2003, Bálint et al. were able to prove ergodicity for multidimensional dispersing billiard in [6]. However it remains an important and difficult question to study hyperbolicity for nondispersing chaotic billiards, as well as high dimensional Bunimovich type billiards. We plan to give a self-contained approach using spectral theory to study the asymptotic behavior of functions in the kernels of a class of ordinary differential operators, motivated by asymptotic behavior of Jacobian filed of certain dynamical systems with singularities, especially billiards.

In this paper we construct a new class of generalized differential operators associated with the impulsive equations. The differential operators we deal with are second order, matrix coefficient Schrödinger operators with infinitely many discontinuous points. These kinds of operators are more general than those occurring in billiard flow, but include these as a special case. In this case, the jump conditions correspond to the reflections. We investigate the exponential behavior of functions in the null-space of 𝑇 in terms of operator theory. On this basis, we obtain the relation between the minimal and maximal operators associated with the weighted operators and then characterize the dimension of the kernel. The results of this paper extend the result in the papers of Kauffman and Zhang [7] and Zhang and Lian [8] to more general case, which gives some hope that the structure of the differential operators may be used later to analyze some of the problems of greater interest in multidimensional billiards.

This paper is divided into five sections: the first section introduces the research background; the second section describes the main results derived in this paper; they are formulated in Theorems 7 and 8; in the third section, we studied the differential operator 𝑇; then, in the last two sections, we give the proof of the two main theorems.

2. Statement of the Main Results

Throughout this paper, we will let 0=𝑡₀<𝑡₁<⋯<𝑡_𝑖<⋯ be a partition 𝒫 of [0,∞), 𝑚≥1. Denote 𝐼_𝑖=[𝑡_𝑖,𝑡_𝑖+1), for all 𝑖=0,1,2,…. Denote 𝑢⋅𝑣 as the inner product of vectors 𝑢,𝑣∈ℂ^𝑚. Let 𝐼 be a real interval; denote by 𝐿²(𝐼) the Hilbert space of all measurable functions 𝑓 from 𝐼 to ℂ^𝑚, such that _𝐼𝑓²𝑑𝑡<∞, (1) with inner product 𝑓,𝑔=_𝐼𝑓⋅𝑔𝑑𝑡. (2)

Definition 1. Let 𝑙 be the differential expression on an interval 𝐼 defined by 𝑙𝑓=−𝑓^′′+𝐴(𝑡)𝑓, (3) where 𝐴(𝑡) is a positive semidefinite symmetric real 𝑚×𝑚 matrix for each 𝑡∈𝑅⁺ and a continuous function of 𝑡.

Definition 2. Let 𝑙_𝑝 be a set of all sequences of complex numbers {𝜉_𝑘}^∞_𝑘=1, which satisfy the condition ∑^∞_𝑘=1|𝜉_𝑘|^𝑝<∞. In this paper, every element of the sequence is a 𝑞-dimensional column vector; we denote 𝑙^𝑞_𝑝 as a set, which satisfy the following condition: 𝜉={𝜉_𝑘}^∞_𝑘=1∈𝑙^𝑞_𝑝; then 𝜉_𝑘=(𝜉¹_𝑘,𝜉²_𝑘,…,𝜉^𝑞_𝑘)^⊺ (𝑘=1,2,…) and {𝜉^𝑗_𝑘}^∞_𝑘=1∈𝑙_𝑝, for 𝑗=1,…,𝑞.

Definition 3. Let 𝐹([0,∞),ℂ^𝑚) denote the set of all measurable functions 𝑓 from [0,∞) to ℂ^𝑚, such that 𝑓 is differentiable (in distribution sense) almost everywhere. For any 𝑓∈𝐹([0,∞),ℂ^𝑚), and the partition 𝒫 of [0,∞), define the operator: Λ∶𝐹[0,∞),ℂ^𝑚⟶𝑙₀^4𝑚𝑓⟼𝑎, (4) where 𝑎={𝑎_𝑖}^∞_𝑖=1 and 𝑎_𝑖=(𝑓(𝑡_𝑖−0),𝑓^′(𝑡_𝑖−0),𝑓(𝑡_𝑖),𝑓^′(𝑡_𝑖))^𝑇 is a 4𝑚×1 matrix.

Definition 4. Let 𝑇∶𝐷(𝑇)⊂𝐿₂[0,∞),ℂ^𝑚⟶𝐿₂[0,∞),ℂ^𝑚×𝑙₀^2𝑚 (5) be a linear operator defined as 𝑇𝑓(𝑡)=𝑔(𝑡),𝑏,𝑔(𝑡)=𝑙𝑓(𝑡),𝑡∈𝐼_𝑖,𝑖=0,1,2,…,𝑏=𝑙𝑓̃ (6)(7)(8) with domain 𝐷(𝑇)=𝑓∈𝐿₂[0,∞),ℂ^𝑚∣𝑓^′∈𝐴𝐶𝐼_𝑖,^∞_𝑖=0𝑙𝑓²_2,𝐼𝑖<∞, (9) where 𝑙̃=(𝐵(𝑡_𝑖),−𝐼)Λ, and 𝐴𝐶(𝐼_𝑖) denotes the set of absolutely continuous complex valued functions on 𝐼_𝑖, and, for each 𝑖∈ℕ, 𝐵𝑡_𝑖=𝑈_𝑖0𝑂_𝑖𝑈_{𝑖2𝑚×2𝑚} (10) and the following holds:(1)𝑈^∗_𝑖𝑈_𝑖=𝐼, where 𝐼 is an identity 𝑚×𝑚 matrix;(2)𝑈^∗_𝑖𝑂_𝑖 is a symmetric operator on the ℝ^𝑚;(3)there exists a universal constant 𝐶₁>0, such that for any unit vector 𝑢∈ℝ^𝑚, any 𝑡>0, and any 𝑖∈ℕ, 𝐶₁≥𝑈^∗_𝑖𝑂_𝑖𝑢,𝑢≥0,‖𝐴(𝑡)‖<𝐶₁. (11)

Remark 5. In Definition 4, (8) represents the jump conditions at points {𝑡_𝑖}. The range of 𝑏 is in 𝑙₀^2𝑚, as one can check that 𝑏=𝑙𝑓̃=𝐵𝑡_𝑖,−𝐼Λ𝑓=𝐵𝑡_𝑖𝑓𝑡_𝑖𝑓−0_′𝑡_𝑖−𝑓𝑡−0𝑓^′𝑡^𝑖_𝑖. (12)

Definition 6 (Ω_𝜖,𝛿 condition). Let 𝜖>0, 𝛿>0. Ω_𝜖,𝑖 be a set of all vectors 𝑣∈ℝ^𝑚 such that 𝑈^∗_𝑖𝑂_𝑖𝑣⋅𝑣≥𝜖|𝑣|². (13) For any 𝑓∶[0,∞)→ℝ^𝑚, we say that 𝑓∈Ω_𝜖,𝛿 if there exists a subsequence {𝑖_𝑘}⊂ℕ with lim_𝑛→∞1inf𝑛_𝑘≤𝑛𝑡_𝑖𝑘−𝑡_𝑖𝑘−1≥𝛿, (14) such that 𝑓(𝑡_𝑖𝑘−0)∈Ω_{𝜖,𝑖𝑘} for all 𝑖_𝑘, 𝑘∈ℕ.

Note that any function 𝑓∈Ω_𝜖,𝛿 has a certain nice property that 𝑈^∗_𝑖𝑘𝑂_𝑖𝑘𝑓²𝑡_𝑖𝑘−0≥𝜖𝑓𝑡_𝑖𝑘−0² (15) for a subsequence 𝑡_𝑖𝑘 which are not too close to each other.

The main results we derived are Theorems 7 and 8.

Theorem 7. Let 𝑓∈ker(𝑇), and there exist 𝛿>0, 𝜖>0 such that 𝑓∈Ω_𝜖,𝛿 for interval 𝐼. Then one has the following.(1)If |𝑓| is nonincreasing, then there exists 𝛼₁>0, such that, for all 𝜆∈(0,𝛼₁) and 𝑡∈𝐼, 𝑓(𝑡)≤𝑒^−𝜆𝑡𝑓(0). (16)(2)If |𝑓| is nondecreasing, then there exists 𝛼₂>0, such that, for all 𝜆∈(0,𝛼₂) and 𝑡∈𝐼, 𝑓(𝑡)≥𝑒^𝜆𝑡𝑓(0). (17)

Next theorem characterizes the dimension of the kernel of 𝑇.

Theorem 8. There exist exactly 𝑚 linearly independent bounded solutions to the equation 𝑇𝑓=0.

Example 9. Consider the one-dimensional case; that is, 𝑚=1. Let 𝑈_𝑖=1, 𝑂_𝑖 be a real number, and let 𝐴(𝑡)=𝑎(𝑡) be a continuous function of 𝑡 and satisfy the following conditions: 𝑎(𝑡)≥0, ∃𝐶₁>0, s.t., ‖𝑎(𝑡)‖<𝐶₁; then the equation 𝑇𝑓=0 becomes 𝑓^′′(𝑡)(𝑡)𝑓𝑡=𝑎𝑓,_𝑖−0−𝑓𝑡_𝑖𝑓=0,_′𝑡_𝑖−0+𝑂_𝑖𝑓𝑡_𝑖−0−𝑓^′𝑡_𝑖=0. (18) It describes the movement of the billiard particle in a smooth table; when it reaches the border, its position is not changed, but the direction is changed. Theorems 7 and 8 hold for this special equation; that is, the solutions of (18) have exponential behavior, and there exists exactly one linearly independent bounded solution.

3. Characterization of the Generalized Differential Operator 𝑇

In this section, we give a characterization on the generalized differential operator 𝑇.

Definition 10. Define an operator 𝑊∶𝐷(𝑊)⊂𝑙₂^2𝑚→𝑙₂^2𝑚, such that 𝑊(𝑎)=𝑎,∀𝑎∈𝑙₂^2𝑚,𝑎_𝑖=𝑄_𝑖𝑎_𝑖−𝑋_𝑖𝑎_𝑖+1, (19) where, for 𝑖=1,2,…,𝑄_𝑖 and 𝑋_𝑖 are 2𝑚×2𝑚 nonsingular matrices with ‖𝑄_𝑖‖_∞≤𝐶, ‖𝑋_𝑖‖_∞≤𝐶 for some 𝐶>0, with 𝐷(𝑊)=𝑎∈𝑙₂^2𝑚∣𝑊(𝑎)∈𝑙₂^2𝑚. (20)

Lemma 11. 𝑊 is surjective.

Proof. 𝑊 can be written as 𝑄𝑊=¹−𝑋₁00⋯0𝑄₂−𝑋₂0⋯00𝑄₃−𝑋₃⋯.⋮⋮⋮⋮⋱ (21) Since 𝑄_𝑖 and 𝑋_𝑖 are nonsingular matrices, so 𝑊 is an invertible matrix. For any 𝑎∈𝑙₂^2𝑚, let 𝑎=𝑊⁻¹𝑎. Then we can check that 𝑊(𝑎)=𝑎 and 𝑎∈𝑙₂^2𝑚.

In order to study the operator 𝑇, we first introduce an operator 𝐿 which was studied in [7, 9] and review some properties of 𝐿.

Definition 12. Let 𝐿 be a differential operator defined by 𝐿𝑓(𝑡)=𝑙𝑓(𝑡),∀𝑡∈𝐼_𝑖∩𝐼,𝑖=0,1,2,…, (22) satisfying the following boundary conditions: for any 𝑖∈ℕ, 𝑓𝑡_𝑖,𝑓^′𝑡_𝑖^𝑇=𝐵𝑡_𝑖𝑓𝑡_𝑖−0,𝑓^′𝑡_𝑖−0^𝑇, (23) where 𝑈_𝑖, 𝑂_𝑖 are the same as in the definition of 𝑇.

Lemma 13 (see [7]). Let 𝐿_𝑀 be the maximal operator of 𝐿, and let 𝐿₀ be the minimal operator. Then the adjoint operator of 𝐿₀ satisfies 𝐿^∗₀=𝐿_𝑀. (24)

Lemma 14 (see [7]). For all 𝑓 in the domain of 𝐿₀, 𝐿𝑓,𝑓≥𝜂𝑓², (25) where 𝜂 is a universal positive constant.

Lemma 15 (see [9]). Let 𝐵 be a densely defined closed operator on a Hilbert space. Denote the range of 𝐵 by ℛ(𝐵) and the nullspace of 𝐵 by 𝒩(𝐵). Then ℛ𝐵^∗⟂=𝒩(𝐵). (26)

Based on all the above properties for operator 𝐿, we obtain the following result.

Lemma 16. Let 𝑔∈𝐿₂([0,∞),ℂ^𝑚). Then there exists 𝑓∈𝐷(𝑇) such that 𝑇𝑓(𝑡)=𝑔,0. (27)

Proof. By Lemma 14, we know that 𝒩(𝐿₀)={0}. From Lemmas 13 and 15, it is easy to show that ℛ(𝐿_𝑀)^⟂=𝒩(𝐿₀). So 𝐿_𝑀 is surjective. Then we have, for all 𝑔∈𝐿₂([0,∞),ℂ^𝑚), ∃𝑓∈𝐷(𝐿_𝑀), such that 𝐿_𝑀𝑓=𝑔. But by the definition of 𝐿, we know that 𝑇𝑓=(𝐿_𝑀𝑓,0), which gives the result.

Lemma 17. Assuming that there exists 𝑑>0, such that 𝑡_𝑖+1−𝑡_𝑖≤𝑑,∀𝑖∈ℕ, (28) then the range of 𝑇 is a subset of ℋ∶=𝐿₂([0,∞),ℂ^𝑚)×𝑙₂^2𝑚.

Proof. Since 𝑓∈𝐿₂([0,∞),ℂ^𝑚) and 𝑙𝑓∈𝐿₂([0,∞),ℂ^𝑚), so 𝑓_2,𝐼0,𝑓_2,𝐼1,…∈𝑙₁,𝑙𝑓_2,𝐼0,𝑙𝑓_2,𝐼1,…∈𝑙₁. (29) There exists 𝐾>0, such that, on each 𝐼_𝑖, 𝑓_∞,𝐼𝑖+𝑓^′_∞,𝐼𝑖≤𝐾𝑓_2,𝐼𝑖+𝑙𝑓_2,𝐼𝑖. (30) Therefore {𝑓(𝑡_𝑖−0)}^∞_𝑖=1, {𝑓(𝑡_𝑖−0)^′}^∞_𝑖=1, {𝑓(𝑡_𝑖)}^∞_𝑖=0, and {𝑓(𝑡^′_𝑖)}^∞_𝑖=0 are all in 𝑙^𝑚₂. It follows that 𝑈_𝑖𝑓𝑡_𝑖−0−𝑓𝑡_𝑖^∞_𝑖=1∈𝑙^𝑚₂,𝑈_𝑖𝑓^′𝑡_𝑖−0+𝑂_𝑖𝑓𝑡_𝑖−0−𝑓^′𝑡_𝑖^∞_𝑖=1∈𝑙^𝑚₂. (31) Thus the range of 𝑇 is contained in 𝐿₂([0,∞),ℂ^𝑚)×𝑙₂^2𝑚.

Next we recall Gronwall’s inequality, which will be used in our proof that 𝑇 is subjective below.

Lemma 18 (Gronwall’s inequality). Let 𝑓(𝑡)>0 be continuous on [𝑎,𝑏] and satisfy 𝑓(𝑡)≤𝛼+^𝑡_𝑎𝛽(𝑠)𝑓(𝑠)𝑑𝑠, (32) where 𝛼>0 and 𝛽>0 is continuous on [𝑎,𝑏]. Then 𝑓(𝑡)≤𝛼exp^𝑡_𝑎𝛽(𝑠)𝑑𝑠, (33) for 𝑡∈[𝑎,𝑏].

Proposition 19. 𝑇∶𝐿₂([0,∞),ℂ^𝑚)→ℋ is surjective.

Proof. Let (𝑔,𝑥,𝑦)∈ℋ. By Lemma 16, there exists 𝑓∈𝐷(𝑇), such that 𝑇𝑓(𝑡)=𝑔,0,0. (34) We just need to find a ℎ∈𝐷(𝑇), such that 𝑇ℎ=(0,𝑥,𝑦). By Definition 4, that is, ℎ^′′(𝑡)=𝐴(𝑡)ℎ(𝑡)𝑡∈𝐼_𝑖𝑥,𝑖=0,1,2,…,_𝑖=𝑈_𝑖ℎ𝑡_𝑖−0−ℎ𝑡_𝑖𝑦,𝑖∈ℕ,_𝑖=𝑈_𝑖ℎ^′𝑡_𝑖−0+𝑂_𝑖ℎ𝑡_𝑖−0−ℎ^′𝑡_𝑖,𝑖=0,1,2,…. (35) For each 𝑖∈ℕ, let 𝐹_𝑖(𝑡) be the fundamental solution matrix of 𝑑ℎ(𝑡)ℎ𝑑𝑡_′(𝑡)=𝐴(𝑡)0ℎ(𝑡)ℎ0𝐼_′(𝑡),𝑡∈𝐼_𝑖; (36) with 𝐹_𝑖(𝑡_𝑖)=𝐼. Since we assumed that |𝐼_𝑖|≤𝑑, for all 𝑖∈ℕ, so there exists a uniform constant 𝐶_𝑑, such that ‖𝐹_𝑖(𝑡)‖_∞≤𝐶_𝑑. By Lemma 11, we choose 𝑄_𝑖=^𝑈𝑖0_{𝑂𝑖𝑈𝑖}, and 𝑋_𝑖⁻¹=𝐹_𝑖(𝑡_𝑖+1−0), then we know that there exist 𝑎,𝑏∈𝑙^𝑚₂, such that 𝑊(𝑎,𝑏)^𝑇=𝑥,𝑦^𝑇. (37) That is, 𝑥_𝑖𝑦_𝑖=𝑄_𝑖𝑎_𝑖𝑏_𝑖−𝑋_𝑖𝑎_𝑖+1𝑏_𝑖+1. (38) Letting ℎ∶[0,∞)→ℂ^𝑚 satisfy ℎ^′∈𝐴𝐶(𝐼_𝑖) and ℎ𝑡_𝑖−0=𝑎_𝑖,ℎ^′𝑡_𝑖−0=𝑏_𝑖, (39) then we have 𝑥_𝑖𝑦_𝑖=𝑄_𝑖ℎ𝑡_𝑖ℎ−0_′𝑡_𝑖−0−𝑋_𝑖ℎ𝑡_𝑖+1ℎ−0_′𝑡_𝑖+1.−0 (40) Let ℎ𝑡_𝑖=𝑈_𝑖ℎ𝑡_𝑖−0−𝑥_𝑖,ℎ^′𝑡_𝑖=𝑈_𝑖ℎ^′𝑡_𝑖−0+𝑂_𝑖ℎ𝑡_𝑖−0−𝑦_𝑖, (41) and then we get 𝑥_𝑖=𝑈_𝑖ℎ𝑡_𝑖−0−ℎ𝑡_𝑖,𝑦_𝑖=𝑈_𝑖ℎ^′𝑡_𝑖−0+𝑂_𝑖ℎ𝑡_𝑖−0−ℎ^′𝑡_𝑖,ℎ𝑡_𝑖+1ℎ−0_′𝑡_𝑖+1−0=𝑋_𝑖⁻¹ℎ𝑡_𝑖ℎ^′𝑡_𝑖=𝐹_𝑖𝑡_𝑖+1ℎ𝑡−0^𝑖ℎ^′𝑡_𝑖. (42) It follows that 𝑇ℎ=(0,𝑥,𝑦).
In the following, it is enough to show that ℎ∈𝐿₂([0,∞),ℂ^𝑚). On each 𝐼_𝑖, we have 𝑑ℎ(𝑡)ℎ𝑑𝑡_′(𝑡)=𝐴(𝑡)0ℎ(𝑡)ℎ0𝐼_′(𝑡). (43) Denote 𝐵(𝑡)=𝐴(𝑡)0(𝑡)=ℎ(𝑡)ℎ0𝐼,𝑋_′(𝑡). (44) Then 𝑋^′(𝑡)=𝐵(𝑡)𝑋(𝑡). (45) By hypothesis of 𝐴(𝑡), we know that ∃𝐾, such that ‖𝐵(𝑡)‖_∞≤𝐾 and |𝐼_𝑖|≤𝑑; thus on each 𝐼_𝑖^𝑡_𝑡𝑖𝑋^′(𝑠)𝑑𝑠=^𝑡_𝑡𝑖𝐵(𝑠)𝑋(𝑠)𝑑𝑠=𝑋(𝑡)−𝑋𝑡_𝑖,‖𝑋(𝑡)‖_∞,𝐼𝑖≤𝑋𝑡_𝑖∞+^𝑡_𝑡𝑖‖𝐵(𝑠)‖_∞‖𝑋(𝑠)‖_∞𝑑𝑠. (46) By Gronwall’s inequality, ‖𝑋(𝑡)‖_∞,𝐼𝑖≤𝑋𝑡_𝑖∞exp^𝑡_𝑡𝑖‖𝐵(𝑠)‖_∞𝑑𝑠≤𝑋𝑡_𝑖∞𝑒^𝑑𝐾. (47) Combining all these facts together, we get ‖𝑋(𝑡)‖_2,𝐼𝑖≤√𝑑𝑚𝑋𝑡_𝑖∞𝑒^𝑑𝐾. (48) Therefore ‖ℎ‖²_[)2,0,∞≤‖𝑋(𝑡)‖²_[)2,0,∞≤^∞_𝑖=0‖𝑋(𝑡)‖²_2,𝐼𝑖≤𝑑𝑚𝑒^∞2𝑑𝐾_𝑖=0𝑋𝑡_𝑖²_∞≤𝑑𝑚𝑒^∞2𝑑𝐾_𝑖=0ℎ𝑡_𝑖²+ℎ^′𝑡_𝑖²<∞. (49) Thus 𝑓+ℎ∈𝐷(𝑇) satisfies 𝑇𝑓+ℎ(𝑡)=𝑔(𝑡),𝑥,𝑦. (50) This completes the proof of Proposition 19.

4. Exponential Behavior of Solutions: Proof of Theorem 7

In this section, we give the proof of Theorem 7 by considering the exponential behavior of the null space of 𝑇.

For any 𝑓∈ker(𝑇), we denote 𝑘_𝑓(𝑡)=𝑓^′𝑓=𝑓^′⋅𝑓.𝑓⋅𝑓 (51) Note that |𝑓|=(𝑓⋅𝑓), so 𝑓^′=𝑓⋅𝑓^′=𝑓⋅𝑓^′2=𝑓⋅𝑓2𝑓^′⋅𝑓2=𝑓𝑓⋅𝑓^′⋅𝑓.𝑓⋅𝑓 (52)

Lemma 20. Let 𝑓∈Ω_𝜖,𝛿 for interval 𝐼. Then there exists a constant 𝑟>0, and 𝜆=𝜆_𝜖,𝛿>0, such that for any 𝑡>𝑟1𝑡_[0,𝑡]𝑘_𝑓(𝑠)𝑑𝑠>𝜆. (53)

Proof. Assume that there exists {𝑖_𝑘}⊂ℕ, such that lim_𝑛→∞1inf𝑛_𝑘≤𝑛𝑡_𝑖𝑘−𝑡_𝑖𝑘−1≥𝛿, (54) since 𝑓∈ker(𝑇) and 𝑓∈Ω_𝜖,𝛿, so 𝑓^′⋅𝑓𝑡_𝑖𝑘−𝑓^′⋅𝑓𝑡_𝑖𝑘−0=𝑈_𝑖𝑘𝑂_𝑖𝑘𝑓²𝑡_𝑖𝑘−0≥𝜖𝑓²𝑡_𝑖𝑘−0. (55) This implies that 𝑘_𝑓𝑡_𝑖𝑘−𝑘_𝑓𝑡_𝑖𝑘=𝑈−0^𝑖𝑘𝑂_𝑖𝑘𝑓²𝑡_𝑖𝑘−0𝑓⋅𝑓≥𝜖. (56) Thus 1𝑡_[0,𝑡]𝑘_𝑓(𝑠1)𝑑𝑠>𝑡_{𝑡𝑖𝑘<𝑡}𝑘_𝑓𝑡_𝑖𝑘−𝑘_𝑓𝑡_𝑖𝑘>𝜖−0𝑡𝑏_𝑡, (57) where 𝑏_𝑡 is the number of 𝑡_𝑖𝑘, such that 𝑡_𝑖𝑘<𝑡. But we know that lim_𝑡→∞𝑏inf^𝑡𝑡≥𝛿, (58) so there exists 𝑟>0, such that, for any 𝑡>𝑟, 1𝑡_[0,𝑡]𝑘_𝑓(𝑠)𝑑𝑠>𝜆, (59) where 𝜆=𝛿𝜖.

Now we are ready to prove Theorem 7.

Proof. By the definition of 𝑘_𝑓, we know that 𝑘_𝑓(𝑡)=𝑓^′𝑓=ln𝑓^′(𝑡). (60) Thus ln𝑓=^𝑡₀𝑘_𝑓(𝑠)𝑓𝑡)𝑑𝑠+𝑐,(=𝑐𝑒^{∫𝑡0𝑘𝑓(𝑠)𝑑𝑠}=𝑓(0)𝑒^{∫𝑡0𝑘𝑓(𝑠)𝑑𝑠}. (61) (1) Since |𝑓| is nonincreasing, then 𝑘_𝑓(𝑡)≤0; by Lemma 20, we have ^𝑡₀𝑘_𝑓(𝑠)𝑑𝑠=−^𝑡₀𝑘_𝑓(𝑠)𝑑𝑠<−𝜆𝑡, (62) so 𝑓(𝑡)≤𝑒^−𝜆𝑡𝑓(0). (63) (2) Since |𝑓| is nondecreasing, then 𝑘_𝑓(𝑡)≥0; by Lemma 20, we have ^𝑡₀𝑘_𝑓(𝑠)𝑑𝑠=^𝑡₀𝑘_𝑓(𝑠)𝑑𝑠>𝜆𝑡, (64) so 𝑓(𝑡)≥𝑒^𝜆𝑡𝑓(0). (65)

5. Proof of Theorem 8

In this section, we give the proof of Theorem 8. First we obtain the relation between the minimal and maximal operators associated with the weighted operators 𝑀_𝐼 and then characterize the dimension of the kernel of 𝑇.

Definition 21. Let 𝐿 be defined as Definition 12, ⟨𝑡⟩=√𝑡²+1. Define 𝑀_𝐼=⟨𝑡⟩𝐿⟨𝑡⟩,∀𝑡∈𝐼_𝑖∩𝐼,𝑖=0,1,2,…. (66)

Definition 22. Let the maximal operator 𝑀_𝑀,𝐼 associated with 𝑀_𝐼 be the restriction of 𝑀_𝐼 on 𝐷_𝑀, with 𝐷_𝑀=𝑓∈𝐿²(𝐼)∩𝐷(𝐿)∣𝑀_𝐼𝑓∈𝐿²(𝐼),𝑓^′∈𝐴𝐶𝐼_𝑖∩𝐼,∀𝑖=0,1,2,…. (67) Let 𝐷_𝐼 be the set of all 𝑓∈𝐷_𝑀, such that 𝑓∈𝐶^∞(𝐼_𝑖∩𝐼), for all 𝑖∈ℕ and 𝑓 vanishes in a neighborhood of both “end points” of 𝐼 and supp(𝑓) is compact. Denote 𝑀_𝐼 to be 𝑀_𝐼 restricting on 𝐷_𝐼; then we define the minimal operator 𝑀_0,𝐼 to be the smallest closed operator in 𝐿²(𝐼) which extends 𝑀_𝐼 and denote by 𝐷₀ the domain of 𝑀_0,𝐼.

Lemma 23. If 𝑀_𝐼 is defined as in Definition 21 on 𝐼, then 𝑀_0,𝐼⊆𝑀_𝑀,𝐼, (68) and, for any 𝑓 in the domain of 𝑀_0,𝐼, we have 𝑓(0)=0,𝑓^′(0)=0. If 𝐼 is compact, say 𝐼=[0,𝒯]⊂[0,∞), then for any 𝑓∈𝐷₀, also 𝑓(𝒯)=𝑓^′(𝒯)=0.

Proof. Let {𝑓_𝑛} be a Cauchy sequence of functions from 𝐷_𝐼 converging to 𝑓 with 𝑀_0,𝐼𝑓_𝑛 converging to 𝑀_0,𝐼𝑓 in 𝐿²(𝐼). Then on any compact interval [𝛼,𝛽]⊆𝐼_𝑖,𝑡≥0, 𝑓_𝑛 and 𝑓^′_𝑛 both converge uniformly; therefore 𝑓_𝑛^′′ also converges in each 𝐿²(𝐼_𝑖), so 𝑓_𝑛^′′ converges in 𝐿²(𝐼). It follows that 𝑓^′ is absolutely continuous on each 𝐼_𝑖, and 𝑓^′(0)=0, if 𝐼=[0,𝒯]; then also 𝑓^′(𝒯)=0. Since 𝑓_𝑛^(𝑖) converges to 𝑓^(𝑖) for 𝑖=0,1,2, we see that 𝑀_0,𝐼𝑓=𝑀_𝐼𝑓, 𝑀_𝐼𝑓 is in 𝐿², and so 𝑓∈𝐷_𝑀,𝐼.

Definition 24 (see [10]). A densely defined operator 𝑆 on a Hilbert space is said to be symmetric if [𝑆𝑓,𝑔]=[𝑓,𝑆𝑔] for all 𝑓,𝑔 in the domain of 𝑆.

Lemma 25. 𝑀_0,𝐼 is a symmetric operator, for 𝐼=[0,𝒯] or 𝐼=[0,∞).

Proof. For any 𝑓,𝑔 in the domain of 𝑀_0,𝐼, by Lemma 23, we know that 𝑓(0)=𝑓^′(0)=0 and 𝑔(0)=𝑔^′(0)=0. Without loss of generality, we suppose that 𝑓 and 𝑔 are real vector valued functions: 𝑀_𝐼𝑓,𝑔−𝑓,𝑀_𝐼𝑔=−⟨𝑡⟩⟨𝑡⟩𝑓^′′+𝑡⟩𝐴⟨𝑡⟩+⟨𝑡⟩⟨𝑡⟩𝑔,𝑔⟨𝑓,𝑔𝑓,^′′−𝑓,⟨𝑡⟩𝐴⟨𝑡⟩𝑔=−_𝐼⟨𝑡⟩⟨𝑡⟩𝑓^′′⋅𝑔−𝑓⋅⟨𝑡⟩⟨𝑡⟩𝑔^′′(𝑡)+𝑡⟩𝑑𝑡⟨²𝐴𝑓,𝑔−𝑓,⟨𝑡⟩²⟨𝑡⟩𝑓𝐴𝑔=−^′⋅⟨𝑡⟩𝑔−⟨𝑡⟩𝑓⋅⟨𝑡⟩𝑔^′_𝐼=−⟨𝑡⟩²𝑓^′⋅𝑔−⟨𝑡⟩²𝑓⋅𝑔^′_𝐼. (69)
If 𝐼 is a compact interval [0,𝒯], then by the fact that 𝑓(𝒯)=𝑓^′(𝒯)=𝑔(𝒯)=𝑔^′(𝒯)=0, (70) we get [𝑀_𝐼𝑓,𝑔]=[𝑓,𝑀_𝐼𝑔].
If 𝐼=[0,∞), by hypotheses on the boundary conditions and 𝑈^∗_𝑖𝑈_𝑖=𝐼, we have 𝑓^′⋅𝑔−𝑓⋅𝑔^′𝑡_𝑖=𝑂_𝑖𝑓𝑡_𝑖−0+𝑈_𝑖𝑓^′𝑡_𝑖−0⋅𝑈_𝑖𝑔𝑡_𝑖−0−𝑈_𝑖𝑓𝑡_𝑖⋅𝑂−0_𝑖𝑔𝑡_𝑖−0+𝑈_𝑖𝑔^′𝑡_𝑖=𝑓−0_′⋅𝑔−𝑓⋅𝑔^′𝑡_𝑖−0. (71) Thus we get 𝑀_𝐼𝑓,𝑔−𝑓,𝑀_𝐼𝑔=−^∞_𝑖=1⟨𝑡⟩²𝑓^′⋅𝑔−𝑓⋅𝑔^′𝑡_𝑖−𝑓^′⋅𝑔−𝑓⋅𝑔^′𝑡_𝑖−0=0. (72) That is, [𝑀_𝐼𝑓,𝑔]=[𝑓,𝑀_𝐼𝑔]. So by Definition 24, 𝑀_0,𝐼 is symmetric.

Lemma 26. For ℎ∈𝐿²(𝐼), the equation 𝑀_𝐼𝑓=ℎ has a solution 𝑓 in the domain of 𝑀_0,𝐼 if and only if ℎ is orthogonal to all solutions of 𝑀_𝑀,𝐼𝑔=0.

Proof. Assume that, for ℎ∈𝐿²(𝐼), the equation 𝑀_𝐼𝑓=ℎ has a solution 𝑓∈𝐷₀. For any 𝑔∈𝐷_𝑀, such that 𝑀_𝐼𝑔=0, we have ℎ,𝑔=𝑀_0,𝐼𝑓,𝑔=𝑓,𝑀_𝐼𝑔=0. (73) Conversely, if ℎ is orthogonal to all solutions of 𝑀_𝑀,𝐼𝑔=0, choose 𝑓 such that 𝑀_𝐼𝑓=ℎ and 𝑓(𝛼)=𝑓^′(𝛼)=0. We need to show that 𝑓(𝛽)=𝑓^′(𝛽)=0. Using the same calculation as in the proof of Lemma 25, we get 𝑀_𝐼𝑓,𝑔−𝑓,𝑀_𝐼𝑔=𝑓𝛽⋅𝑔^′𝛽−𝑓^′𝛽⋅𝑔𝛽. (74) On the other hand, Consider 𝑀_𝐼𝑓,𝑔−𝑓,𝑀_𝑀,𝐼𝑔=𝑀_𝐼𝑓,𝑔=ℎ,𝑔=0. (75) Thus we get that 𝑓(𝛽)⋅𝑔^′(𝛽)−𝑓^′(𝛽)⋅𝑔(𝛽)=0 for any 𝑔 satisfying 𝑀_𝐼𝑔=0. Choose 𝑔∈𝐷_𝑀, s.t. 𝑔(𝛽)≠0, but 𝑔^′(𝛽)=0; then we get 𝑓^′(𝛽)=0; similarly, we can get 𝑓(𝛽)=0.

Corollary 27. 𝐼=[𝛼,𝛽]⊂[0,∞); letting 𝜏 denote the restriction of 𝑀_𝐼 to domain 𝑓∈𝐷_𝑀(𝐼)∶𝑓(𝛼)=𝑓^′(𝛼)=0,𝑓𝛽=𝑓^′𝛽=0, (76) then 𝜏=𝑀_0,𝐼.

Based on all the above results, we can derive the relation between the operators 𝑀_0,𝐼 and 𝑀_𝑀,𝐼.

Proposition 28. If 𝑀_𝐼 is defined as in Definition 21 and 𝐼=[𝛼,𝛽]⊂[0,∞), then 𝑀^∗_0,𝐼=𝑀_𝑀,𝐼. (77)

Proof. Since 𝑀_0,𝐼 is the smallest closed extension of 𝑀_𝐼, it follows that 𝑀^∗_0,𝐼=𝑀^∗_𝐼. (78) As for any 𝑓∈𝐷_𝐼 (then 𝑓∈𝐷₀) and 𝑔∈𝐷_𝑀,𝐼, [𝑀_𝐼𝑓,𝑔]=[𝑓,𝑀_𝑀,𝐼𝑔], then 𝑀_𝑀,𝐼⊆𝑀^∗_𝐼=𝑀^∗_0,𝐼. (79) Let 𝑓 be in the domain of 𝑀^∗_0,𝐼. Since 𝑀_𝑀,𝐼 is surjective for compact 𝐼, so there is a function 𝑔 in 𝐷_𝑀,𝐼 such that 𝑀_𝑀,𝐼𝑔=𝑀^∗_0,𝐼𝑓; thus 𝑀^∗_0,𝐼𝑓−𝑔=0. (80) So if 𝑢 is in the range of 𝑀_0,𝐼 with 𝑀_0,𝐼𝑣=𝑢, we have 𝑓−𝑔,𝑢=𝑓−𝑔,𝑀_0,𝐼𝑣=𝑀^∗_0,𝐼𝑓−𝑔,𝑣=0. (81) Therefore 𝑓−𝑔 is orthogonal to the range of 𝑀_0,𝐼, but, by Lemma 26, 𝑓−𝑔 is in the null space of 𝑀_𝑀,𝐼. Hence 𝑓∈𝐷_𝑀,𝐼; that is, 𝑀^∗_0,𝐼⊂𝑀_𝑀,𝐼.

Proposition 29. If 𝑀_𝐼 is defined as in Definition 21 and 𝐼⊂[0,∞), then 𝑀^∗_0,𝐼=𝑀_𝑀,𝐼. (82)

Proof. By Proposition 28, we only need to show 𝑀^∗_0,𝐼⊂𝑀_𝑀,𝐼. If 𝑔 is in the domain of 𝑀^∗_0,𝐼, then, on any compact subinterval [𝛼,𝛽] of 𝐼, we have 𝑀_𝐼𝑓,𝑔=𝑓,𝑀^∗_0,𝐼𝑔,∀𝑓∈𝐷_𝐼. (83) Thus the restriction of 𝑔 to [𝛼,𝛽] is in the domain of 𝑀^∗_0,𝐼. But, by Proposition 28, the restriction of 𝑀^∗_0,𝐼𝑔 to [𝛼,𝛽] must agree with 𝑀_𝑀,𝐼𝑔. Since [𝛼,𝛽] is arbitrary, the proposition is proved.

We can show that 𝑀_0,𝐼 is semibounded in the next lemma.

Lemma 30. If 𝑀_𝐼 is defined as in Definition 21, for all 𝑓∈𝐷₀, 𝑀_𝐼≥1𝑓,𝑓4𝑓². (84)

Proof. By integration by parts we obtain, for 𝑓∈𝐶^∞₀(ℝ⁺), 𝑀_0,𝐼=𝑡⟩𝐿⟨𝑡⟩𝑓𝑡⟩𝑡⟩𝑓𝑓,𝑓⟨,𝑓=−⟨⟨^′′+𝑡⟩𝐴⟨𝑡⟩=⟨𝑡⟩𝑓,𝑓⟨𝑓,𝑓_′,⟨𝑡⟩𝑓^′+⟨𝑡⟩²≥⟨𝑡⟩𝑓𝐴𝑓,𝑓_′,⟨𝑡⟩𝑓^′≥14𝑓², (85) where the last inequality follows from Hardy’s inequality [11].

In order to prove Theorem 8, we need to introduce the Friedrichs extension of 𝑀_0,𝐼.

Definition 31. With 𝑀_𝐼 as in Definition 21, let 𝑄_𝐼(𝑓,𝑔)=[𝑀_𝐼𝑓,𝑔], for all 𝑓,𝑔∈𝐷_𝐼.

It follows from operator theory [9] that the semibounded symmetric operator 𝑀_0,𝐼 has equal deficiency indices, and therefore, by Von Neumann’s theorem, such an operator always has self-adjoint extensions. There is a distinguished extension 𝐻, called the Friedrichs extension [11], which is obtained from the quadratic form associated with 𝑀_0,𝐼.

Proposition 32. 𝑄_𝐼 is a closable quadratic form and its closure 𝑄_𝐼 is the quadratic form of a unique self-adjoint operator 𝐻 defined by 𝐷𝐻_𝐼=𝑓∈𝐿²(𝐼)∶𝑄_𝐼𝑓,𝑔≤𝐶_𝑓𝑔,𝑔,∀𝑔∈𝐷_𝐼,𝑤ℎ𝑒𝑟𝑒𝐶_𝑓,𝐻𝑖𝑠𝑎𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡_𝐼𝑓=𝑀_𝑀,𝐼𝑓,∀𝑓∈𝐷𝐻_𝐼. (86) Furthermore, 𝐻_𝐼≥1𝑓,𝑓4𝑓,𝑓,∀𝑓∈𝐷𝐻_𝐼. (87)

Proof. This is a form of the definition of the Friedrichs extension: for the semibounded symmetric operator 𝑀_0,𝐼 and the closable quadratic form 𝑄_𝐼, the restriction of 𝑀^∗_0,𝐼 to the domain of the closure of 𝑄_𝐼 is in fact the Friedrichs extension. This form is clearly closable. The last inequality follows from the construction of the Friedrichs extension 𝐻_𝐼, together with Lemma 30.

Next proposition tells us the relation between ker(𝑇) and ker(𝑀_𝐼).

Proposition 33. 𝑓∈ker(𝑇) is nonincreasing if and only if 𝑔=𝑓/⟨𝑡⟩ is a nonincreasing solution to 𝑀_𝐼𝑔=0.

Proof. Let 𝑓∈ker(𝑇); if |𝑓| is nonincreasing, then 𝑓^′⋅𝑓<0. So by east calculation, we have 𝑔⋅𝑔^′(𝑡)=2𝑔^′(𝑡)(𝑡)𝑓⋅𝑔=2^′⋅𝑓1+𝑡²−𝑓²𝑡1+𝑡²²<0, (88) which implies the result.

Now we are ready to prove Theorem 8.

Proof. It is sufficient to consider real solutions. Let 𝑓 be any real solution of 𝑇𝑓=0; for any 𝑖∈ℕ, let 𝑡∈𝐼_𝑖, (𝑓⋅𝑓)^′(𝑡)=2𝑓^′(𝑡)⋅𝑓(𝑡). Using the fact that 𝐴(𝑡) is positive semidefinite for each 𝑡≥0, we have 𝑓⋅𝑓^′′(𝑡)=2𝑓^′2(𝑡)+2𝐴(𝑡)𝑓(𝑡)⋅𝑓(𝑡)≥0. (89) On the other hand, for any 𝑖∈ℕ, 𝑓⋅𝑓^′𝑡_𝑖−𝑓⋅𝑓^′𝑡_𝑖−0=𝑈^∗_𝑖𝑂_𝑖𝑓,𝑓𝑡_𝑖−0≥0. (90) This implies that (𝑓⋅𝑓)^′ is nondecreasing on [0,∞]. Choosing 𝑓, such that 𝑓(0)=0,𝑓^′(0)≠0, then (𝑓⋅𝑓)^′(𝑡)>(𝑓⋅𝑓)^′(0)=0, from (52); |𝑓| is nondecreasing. Then, by Theorem 7, we know that those solutions have exponentially increasing amplitude; that is, they are unbounded solutions. Clearly dim(ker(𝑇))≤2𝑚. The set {𝑓∈ker(𝑇)∣𝑓(0)=0,𝑓^′(0)≠0} is 𝑚 dimensional, which means that there are at most 𝑚 linearly independent bounded solutions for 𝑇𝑓=0.
Now we prove dim(ker(𝑀_𝑀,𝐼))≥𝑚, using the properties of Friedrichs extension of 𝑀_0,𝐼. Let 𝐻_𝐼 be the Friedrichs extension of 𝑀_0,𝐼; then, by Proposition 32, 𝐻_𝐼 is self-adjoint and positive definite. Also for any 𝑓 in 𝐷(𝐻_𝐼), 𝑓(0)=0. We choose 𝑚 compactly supported piecewise smooth functions 𝑓₁,𝑓₂,…,𝑓_𝑚 on 𝐼, such that the space spanned by {𝑓_𝑘(0)∣𝑘=1,…,𝑚} has dimension 𝑚. Then {𝑓_𝑘∣𝑘=1,…,𝑚} are linearly independent mod 𝐷(𝐻_𝐼). It is clear that 𝑓_𝑘∈𝐷_𝑀 for 𝑘=1,…,𝑚; this forces the Fredholm index to increase by 𝑚. Since 𝐻_𝐼 is already surjective, this produces an 𝑚 dimensional nonincreasing solution space to 𝑀_𝐼𝑓=0, so dim(ker(𝑀_𝑀,𝐼))≥𝑚; by Proposition 33, we know that there are at least 𝑚 linearly independent bounded solutions for 𝑇𝑓=0. This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referee for her/his suggestions which have improved the presentation of the paper. The work of the authors is supported by the National Nature Science Foundation of China (Grant no. 11161030) and the third author is supported by the Program of Higher Level Talents of Inner Mongolia University (SPH-IMU).