Research Article | Open Access
Yurong Li, Jiong Sun, Siqin Yao, "A Class of Generalized Differential Operators with Infinitely Many Discontinuous Points", Abstract and Applied Analysis, vol. 2014, Article ID 947182, 8 pages, 2014. https://doi.org/10.1155/2014/947182
A Class of Generalized Differential Operators with Infinitely Many Discontinuous Points
We define a class of differential operators which have infinitely many discontinuous points, and investigate the kernel of in terms of operator theory. It is shown that the solutions of 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 we study is associated with the linearization of the flow; hence we ask whether solutions to 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 ; 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 . 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  and Zhang and Lian  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.
2. Statement of the Main Results
Throughout this paper, we will let be a partition of , . Denote , for all . Denote as the inner product of vectors . Let be a real interval; denote by the Hilbert space of all measurable functions from to , such that 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 , which satisfy the condition . In this paper, every element of the sequence is a -dimensional column vector; we denote as a set, which satisfy the following condition: ; then () and , for .
Definition 3. Let denote the set of all measurable functions from to , such that is differentiable (in distribution sense) almost everywhere. For any , and the partition of , define the operator: where and is a matrix.
Definition 4. Let be a linear operator defined as with domain where , and denotes the set of absolutely continuous complex valued functions on , and, for each , and the following holds:(1), where is an identity matrix;(2) is a symmetric operator on the ;(3)there exists a universal constant , such that for any unit vector , any , and any ,
Definition 6 ( condition). Let , . be a set of all vectors such that For any , we say that if there exists a subsequence with such that for all , .
Note that any function has a certain nice property that for a subsequence which are not too close to each other.
Theorem 7. Let , and there exist , such that for interval . Then one has the following.(1)If is nonincreasing, then there exists , such that, for all and , (2)If is nondecreasing, then there exists , such that, for all and ,
Next theorem characterizes the dimension of the kernel of .
Theorem 8. There exist exactly linearly independent bounded solutions to the equation .
Example 9. Consider the one-dimensional case; that is, . Let , be a real number, and let be a continuous function of and satisfy the following conditions: , , s.t., ; then the equation 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.
3. Characterization of the Generalized Differential Operator
In this section, we give a characterization on the generalized differential operator .
Definition 10. Define an operator , such that where, for and are nonsingular matrices with , for some , with
Lemma 11. is surjective.
Proof. can be written as Since and are nonsingular matrices, so is an invertible matrix. For any , let . Then we can check that and .
Definition 12. Let be a differential operator defined by satisfying the following boundary conditions: for any , where , are the same as in the definition of .
Lemma 13 (see ). Let be the maximal operator of , and let be the minimal operator. Then the adjoint operator of satisfies
Lemma 14 (see ). For all in the domain of , where is a universal positive constant.
Lemma 15 (see ). Let be a densely defined closed operator on a Hilbert space. Denote the range of by and the nullspace of by . Then
Based on all the above properties for operator , we obtain the following result.
Lemma 16. Let . Then there exists such that
Proof. By Lemma 14, we know that . From Lemmas 13 and 15, it is easy to show that . So is surjective. Then we have, for all , , such that . But by the definition of , we know that , which gives the result.
Lemma 17. Assuming that there exists , such that then the range of is a subset of .
Proof. Since and , so There exists , such that, on each , Therefore , , , and are all in . It follows that Thus the range of is contained in .
Next we recall Gronwall’s inequality, which will be used in our proof that is subjective below.
Lemma 18 (Gronwall’s inequality). Let be continuous on and satisfy where and is continuous on . Then for .
Proposition 19. is surjective.
Proof. Let . By Lemma 16, there exists , such that
We just need to find a , such that . By Definition 4, that is,
For each , let be the fundamental solution matrix of
with . Since we assumed that , for all , so there exists a uniform constant , such that . By Lemma 11, we choose , and , then we know that there exist , such that
Letting satisfy and
then we have
and then we get
It follows that .
In the following, it is enough to show that . On each , we have Denote Then By hypothesis of , we know that , such that and ; thus on each By Gronwall’s inequality, Combining all these facts together, we get Therefore Thus satisfies 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 .
For any , we denote Note that , so
Lemma 20. Let for interval . Then there exists a constant , and , such that for any
Proof. Assume that there exists , such that since and , so This implies that Thus where is the number of , such that . But we know that so there exists , such that, for any , where .
Now we are ready to prove Theorem 7.
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 and then characterize the dimension of the kernel of .
Definition 21. Let be defined as Definition 12, . Define
Definition 22. Let the maximal operator associated with be the restriction of on , with Let be the set of all , such that , for all and vanishes in a neighborhood of both “end points” of and is compact. Denote to be restricting on ; then we define the minimal operator to be the smallest closed operator in which extends and denote by the domain of .
Lemma 23. If is defined as in Definition 21 on , then and, for any in the domain of , we have . If is compact, say , then for any , also .
Proof. Let be a Cauchy sequence of functions from converging to with converging to in . Then on any compact interval , and both converge uniformly; therefore also converges in each , so converges in . It follows that is absolutely continuous on each , and , if ; then also . Since converges to for , we see that , is in , and so .
Definition 24 (see ). A densely defined operator on a Hilbert space is said to be symmetric if for all in the domain of .
Lemma 25. is a symmetric operator, for or .
Proof. For any in the domain of , by Lemma 23, we know that and . Without loss of generality, we suppose that and are real vector valued functions:
If is a compact interval , then by the fact that we get .
If , by hypotheses on the boundary conditions and , we have Thus we get That is, . So by Definition 24, is symmetric.
Lemma 26. For , the equation has a solution in the domain of if and only if is orthogonal to all solutions of .
Proof. Assume that, for , the equation has a solution . For any , such that , we have Conversely, if is orthogonal to all solutions of , choose such that and . We need to show that . Using the same calculation as in the proof of Lemma 25, we get On the other hand, Consider Thus we get that for any satisfying . Choose , s.t. , but ; then we get ; similarly, we can get .
Corollary 27. ; letting denote the restriction of to domain then .
Based on all the above results, we can derive the relation between the operators and .
Proposition 28. If is defined as in Definition 21 and , then
Proof. Since is the smallest closed extension of , it follows that As for any (then ) and , , then Let be in the domain of . Since is surjective for compact , so there is a function in such that ; thus So if is in the range of with , we have Therefore is orthogonal to the range of , but, by Lemma 26, is in the null space of . Hence ; that is, .
Proposition 29. If is defined as in Definition 21 and , then
Proof. By Proposition 28, we only need to show . If is in the domain of , then, on any compact subinterval of , we have Thus the restriction of to is in the domain of . But, by Proposition 28, the restriction of to must agree with . Since is arbitrary, the proposition is proved.
We can show that is semibounded in the next lemma.
Lemma 30. If is defined as in Definition 21, for all ,
Proof. By integration by parts we obtain, for , where the last inequality follows from Hardy’s inequality .
In order to prove Theorem 8, we need to introduce the Friedrichs extension of .
Definition 31. With as in Definition 21, let , for all .
It follows from operator theory  that the semibounded symmetric operator 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 , which is obtained from the quadratic form associated with .
Proposition 32. is a closable quadratic form and its closure is the quadratic form of a unique self-adjoint operator defined by Furthermore,
Proof. This is a form of the definition of the Friedrichs extension: for the semibounded symmetric operator and the closable quadratic form , the restriction of 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 and .
Proposition 33. is nonincreasing if and only if is a nonincreasing solution to .
Proof. Let ; if is nonincreasing, then . So by east calculation, we have which implies the result.
Now we are ready to prove Theorem 8.
Proof. It is sufficient to consider real solutions. Let be any real solution of ; for any , let , . Using the fact that is positive semidefinite for each , we have
On the other hand, for any ,
This implies that is nondecreasing on . Choosing , such that , then , from (52); is nondecreasing. Then, by Theorem 7, we know that those solutions have exponentially increasing amplitude; that is, they are unbounded solutions. Clearly . The set is dimensional, which means that there are at most linearly independent bounded solutions for .
Now we prove , using the properties of Friedrichs extension of . Let be the Friedrichs extension of ; then, by Proposition 32, is self-adjoint and positive definite. Also for any in , . We choose compactly supported piecewise smooth functions on , such that the space spanned by has dimension . Then are linearly independent mod . It is clear that for ; this forces the Fredholm index to increase by . Since is already surjective, this produces an dimensional nonincreasing solution space to , so ; by Proposition 33, we know that there are at least linearly independent bounded solutions for . This completes the proof.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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).
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