Issue 
Wuhan Univ. J. Nat. Sci.
Volume 29, Number 3, June 2024



Page(s)  257  262  
DOI  https://doi.org/10.1051/wujns/2024293257  
Published online  03 July 2024 
Mathematics
CLC number: O231.4
The Eigenvalue Properties of a Kind of Singular Differential Equations in 3Dimensional Space
School of Mathematics and Statistics, Central South University, Changsha 410000, Hunan, China
^{†} Corresponding author. Email: ghadirshokor@gmail.com
Received:
10
June
2023
In this paper, we consider the eigenvalue problem of the singular differential equation in a bounded open ball with Dirichlet boundary condition in 3dimensional space, where, , is a given constant. And we have made a detailed characterization of the weak solution space. Furthermore, the existence of the minimum eigenvalue and the fundamental gap are provided.
Key words: singular differential equation / eigenvalue / fundamental gap
Cite this article: GU Mengze, GHADIR Shokor. The Eigenvalue Properties of a Kind of Singular Differential Equations in 3Dimensional Space[J]. Wuhan Univ J of Nat Sci, 2024, 29(3): 257262.
Biography: GU Mengze, male, Ph. D. candidate, research direction: eigenvalue problem of differential operators. Email: 3059268737@gmail.com
© Wuhan University 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
0 Introduction
Consider the following equation:
where is an open and bounded ball with , 0, and is a constant, is the potential function, and .
In the Sobolev space , its norm is:
or the equivalent norm: (by using the Poincaré inequality).
Denote
In quantum mechanics, actually, the eigenvalue problem of operators in the theory of Hilbert space is mentioned by many researchers. In 2013, Tyagi^{[1]} considered a singular eigenvalue problem involving Hardy's potential with Dirichlet boundary condition and gave that the problem possesses a continuous family of eigenvalues. Gesztesy and Zinchenko^{[2]} showed the case of selfadjoint halfline Schrödinger operators on with a potential strongly singular at the endpoint a. Nursultanov^{ [3]} found that asymptotic formulas for the eigenvalues of the SturmLiouville operator on the finite interval, with potentials having a strong negative singular at one endpoint. HaeseHill^{ [4]} studied the spectral properties of its complex regularizations of the form:
where is one of the halfperiods of and is the classical Weierstrass elliptic function. Li and Zhang^{ [5]} studied the numerical approximation of eigenvalue problems of the Schrödinger operator . Homa and Hryniv^{ [6]} proved analogues of the classical Sturm comparison and oscillation theorems for equations on a finite interval with realvalued distributional potentials.
Due to the presence of singular terms such as in the differential equation, the properties of the equation differ from those of traditional differential equations, constituting singular differential equations. The existence of this particular term makes it challenging to directly apply the classical Sobolev space framework and its related properties to the analysis of such equations. To effectively handle this singularity, this paper introduces weighted Sobolev spaces. Within this newly constructed framework, the existence, uniqueness, and regularity of solutions to singular differential equations are successfully established. The establishment of these properties not only deepens our understanding of solutions to singular differential equations but also lays a solid foundation for further research and applications.
Additionally, a comprehensive description of spectral theory for singular differential equations is provided within this new framework. The characteristics of the equation's eigenvalues and eigenfunctions can be more accurately characterized by introducing weighted Sobolev spaces. This theory holds profound mathematical significance and offers new perspectives and methods for solving practical problems.
1 Preliminary
Lemma 1 The space is a Hilbert space.
Proof Let is a Cauchy sequence. Note that , then there exists such that strongly in .
By using the Sobolev Embedding Theorem, we have strongly in for .
By using the HardySobolev inequality^{ [7]}:
holds for ,
where is the completion of in the norm:
We have:
which means that
So, we have , and
Then, there exists such that
strongly in
and
which shows that strongly in . From the assumptions in , then .
Denote
In the sense of distribution, for , is a solution of the following equation
which is equivalent to the following equality:
Definition 1 (i) The bilinear form associated with the elliptic operator defined by (2) is
(ii) We say that is a weak solution of the boundaryvalue problem (3) if
For all , where denotes the inner product in .
Theorem 1 Let be defined as above. Then, there exist constants such that
and
for all .
Proof We readily check that:
for all by CauchySchwarz inequality.
Then we have
by Lemma 1 and Poincaré inequality.
What's more, since a.e., we can obtain that
Corollary 1 Let be a given function, then the elliptic equation , has a unique weak solution.
Proof By Theorem 1 and LaxMilgram theorem, we can conclude the corollary.
Corollary 2 The operator is a continuous, coercive, surjective, and linear operator, where is the dual space of .
Proof It is evident that is a linear operator.
Let , by using the LaxMilgram theorem^{[8]}, we can see that there exists a unique element such that
Note that , thus
in the sense of distribution.
Consequently, is a continuous, coercive, surjective, and linear operator.
Lemma 2 Assume , let u be a weak solution to the equation
Then . Moreover,
where .
Proof (1) For all , let be small such that . We take
Select a smooth function satisfying
Since is a weak solution of (5), we have for all . Then
where .
Set
Then, on the one hand, by the Cauchy equality with weight and difference quotient estimate, we have
On the other hand, by difference quotient estimate, we have
Thus, the Cauchy inequality with weight implies
where is depend on .
Consequently,
From the difference quotient, we deduce and thus .
(2) For any , according to (5)
From which, CauchySchwarz inequality and (1), we deduce that:
where we used Poincaré inequality in the equality, consequently, (6) holds.
Remark 1 Under the assumptions of the Lemma 2, (i) It seems that since is singular at ; (ii) From the Sobolev Embedding Theorem, it is clear that .
Proposition 1 Let denotes the spectrum of L. Then and is at most countable. Let , where the eigenvalue is counted according to its multiplicity, then
Finally, there exists an orthonormal basis of , where is an eigenfunction corresponding to , that is
Proof Since is a coercive, surjective, linear operator, set . From the Sobolev Compact Embedding theorem, we deduce that is a bounded, linear, compact operator. Hence , the spectrum of , is at most countable and has as the unique limit point.
For any , denote , , then
Thus,
which shows that is a symmetric operator.
So, we say that is a selfadjoint linear operator. And we thereby obtain the assertion of the Proposition by the Standard Functional Analysis 8^{ [8]}.
Lemma 3 Let be the ith eigenfunction with respect to the eigenfunction of with the potential . Then,
where,
Proof (i) Let and be defined as in Proposition 1. Then
and
We observe that is an orthonormal subset of under the new inner product . In fact, according to Theorem 1, Lemma 1, and the Poincaré inequality, we deduce that
i.e., these two norms and are equivalent on .
For arbitrary with , , it follows that since is an orthonormal basis of . Therefore
We obtain , . Hence .
Finally, we have is an orthonormal subset of under the new inner product .
(ii) Let with . Assume that . We conclude that
according to (i), this proves (7).
(iii) Suppose we have obtained , for any , with , then and . Moreover,
which implies that
By (i), we have , then the result (8) is obtained.
2 Main Results
Theorem 2 There exists , , , such that
and
where is called the fundamental gap with potential .
Proof we will show the proof by the following steps.
Step 1: , exists.
For arbitrary , we have . Let be such that
Then there exists a subsequence of , still denoted by itself, and , such that weakly star in .
Denote be the normalized first eigenfunction of , which means that . From (4), we see
where is the first eigenvalue of with the potential . Thus, there exists a subsequence of , still denoted by itself, and such that
By and Sobolev Embedding theorem, we deduce that
which implies that;
In view of Lemma 1, we have
then there exists a subsequence of , still denoted by itself, and such that
Hence,
which shows that weakly in .
Thus (10) implies that
or we can say in , i.e.
For all ,
Since and (12), we have
Combining (9) (10) (11), we can obtain that
These imply that
Note that , so in the sense of distribution, we have
This proves is an eigenvalue of with potential . Therefore, by the definition of .
Step 2: Choose and such that
For each , there exist , , such that , are the first two normalized eigenfunctions with respect to the eigenvalues and with regard to . By the same argument as in Step 1, by abstracting subsequence, there exist , and such that
Moreover,
and
Consequently, on one side, owing to , we obtain . On the other side, by the same argument as Step 1, we deduce that:
This proves that and are eigenvalues of with potential , in particular .
Obviously, the definition of implies . Suppose , then , i.e., is the only eigenfunction of with potential according to Lemma 1. But the case leads us to , a contradiction. This shows that .
Step 3: Take arbitrary , then
Then there exists a subsequence of , still denoted by itself, and such that
Denote , be the first and second normalized eigenfunctions with respect to , , respectively. By the same argument as in Step 1, by abstracting subsequence, there exists , such that
and weakly in .
By the same argument as in Step 2, we can deduce that
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