A Class of Generalized Differential Operators with Infinitely Many Discontinuous Points

and Applied Analysis 3 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, ∃C 1 > 0, s.t., ‖a(t)‖ < C 1 ; then the equation Tf = 0 becomes f 󸀠󸀠 (t) = a (t) f, f (t i − 0) − f (t i ) = 0, f 󸀠 (t i − 0) + O i f (t 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 operatorW : D(W) ⊂ l 2 → l 2m 2 , such that W(a) = a, ∀a ∈ l 2m 2 , a i = Q i a i − X i a 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 ∈ l 2m 2 | W (a) ∈ l 2m 2 } . (20) Lemma 11. W is surjective. Proof. W can be written as W =( Q 1 −X 1 0 0 ⋅ ⋅ ⋅


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  we study is associated with the linearization of the flow; hence we ask whether solutions to  = 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  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 ; then, in the last two sections, we give the proof of the two main theorems.

Statement of the Main Results
Throughout this paper, we will let with inner product Definition 1.Let  be the differential expression on an interval  defined by where () is a positive semidefinite symmetric real  ×  matrix for each  ∈  + and a continuous function of .
Definition 2. Let   be a set of all sequences of complex numbers {  } ∞ =1 , which satisfy the condition ∑ ∞ =1 |  |  < ∞.In this paper, every element of the sequence is a dimensional column vector; we denote    as a set, which satisfy the following condition: such that  is differentiable (in distribution sense) almost everywhere.For any  ∈ ([0, ∞), C  ), and the partition P of [0, ∞), define the operator: where be a linear operator defined as () =  () ,  ∈   ,  = 0, 1, 2, . . ., with domain where l = ((  ), −)Λ, and (  ) denotes the set of absolutely continuous complex valued functions on   , and, for each  ∈ N, and the following holds: (1)  *    = , where  is an identity  ×  matrix; (2)  *    is a symmetric operator on the R  ; (3) there exists a universal constant  1 > 0, such that for any unit vector  ∈ R  , any  > 0, and any  ∈ N, ) . ( Definition 6 (Ω , condition).Let  > 0,  > 0. Ω , be a set of all vectors V ∈ R  such that For any  : [0, ∞) → R  , we say that  ∈ Ω Next theorem characterizes the dimension of the kernel of .Theorem 8.There exist exactly  linearly independent bounded solutions to the equation  = 0. Example 9. Consider the one-dimensional case; that is,  = 1.Let   = 1,   be a real number, and let () = () be a continuous function of  and satisfy the following conditions: () ≥ 0, ∃ 1 > 0, s.t., ‖()‖ <  1 ; then the equation  = 0 becomes 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.

Characterization of the Generalized Differential Operator 𝑇
In this section, we give a characterization on the generalized differential operator .
Proof. can be written as Since   and   are nonsingular matrices, so  is an invertible matrix.For any â ∈  2 2 , let  =  −1 â.Then we can check that () = â and  ∈  2  2 .In order to study the operator , we first introduce an operator  which was studied in [7,9] and review some properties of .
Definition 12. Let  be a differential operator defined by satisfying the following boundary conditions: for any  ∈ N, where   ,   are the same as in the definition of .
Lemma 13 (see [7]).Let   be the maximal operator of , and let  0 be the minimal operator.Then the adjoint operator of  0 satisfies Lemma 14 (see [7]).For all  in the domain of  0 , where  is a universal positive constant.
Lemma 15 (see [9]).Let  be a densely defined closed operator on a Hilbert space.Denote the range of  by R() and the nullspace of  by N().Then Based on all the above properties for operator , we obtain the following result.
Lemma 17.Assuming that there exists  > 0, such that then the range of  is a subset of Therefore Next we recall Gronwall's inequality, which will be used in our proof that  is subjective below.
In the following, it is enough to show that ℎ ∈  2 ([0, ∞), C  ).On each   , we have Denote Then Combining all these facts together, we get Therefore Thus  + ℎ ∈ () satisfies

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 .

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   and then characterize the dimension of the kernel of .
Definition 21.Let  be defined as Definition 12, ⟨⟩ = √  2 + 1. Define Definition 22.Let the maximal operator  , associated with   be the restriction of   on   , with Let D be the set of all  ∈   , such that  ∈  ∞ (  ∩ ), for all  ∈ N and  vanishes in a neighborhood of both "end points" of  and supp() is compact.Denote M to be   restricting on D ; then we define the minimal operator  0, to be the smallest closed operator in  2 () which extends M and denote by  0 the domain of  0, .
Proof.Let {  } be a Cauchy sequence of functions from D converging to  with  0,   converging to  0,  in  2 ().
Definition 24 (see [10]).A densely defined operator  on a Hilbert space is said to be symmetric if [, ] = [, ] for all ,  in the domain of .
Proof.For any ,  in the domain of  0, , by Lemma 23, we know that (0) =   (0) = 0 and (0) =   (0) = 0. Without loss of generality, we suppose that  and  are real vector valued functions: Let  be in the domain of  * 0, .Since  , is surjective for compact , so there is a function  in  , such that  ,  =  * 0, ; thus So if  is in the range of  0, with  0, V = , we have Therefore  −  is orthogonal to the range of where the last inequality follows from Hardy's inequality [11].
In order to prove Theorem 8, we need to introduce the Friedrichs extension of  0, .It follows from operator theory [9] that the semibounded symmetric operator  0, has equal deficiency indices, and therefore, by Von Neumann's theorem, such an operator always has self-adjoint extensions.There is a distinguished extension , called the Friedrichs extension [11], which is obtained from the quadratic form associated with  0, .Proposition 32.  is a closable quadratic form and its closure Q is the quadratic form of a unique self-adjoint operator  defined by  (  ) = { ∈  Proof.This is a form of the definition of the Friedrichs extension: for the semibounded symmetric operator  0, and the closable quadratic form   , the restriction of  * 0, to the domain of the closure of   is in fact the Friedrichs extension.This form is clearly closable.The last inequality follows from the construction of the Friedrichs extension   , together with Lemma 30.
Next proposition tells us the relation between ker() and ker(  ).