Open Access
Wuhan Univ. J. Nat. Sci.
Volume 29, Number 3, June 2024
Page(s) 257 - 262
Published online 03 July 2024

© Wuhan University 2024

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (, 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).


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 self-adjoint half-line Schrödinger operators on with a potential strongly singular at the endpoint a. Nursultanov [3] found that asymptotic formulas for the eigenvalues of the Sturm-Liouville operator on the finite interval, with potentials having a strong negative singular at one endpoint. Haese-Hill [4] studied the spectral properties of its complex regularizations of the form:

where is one of the half-periods 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 real-valued 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 Hardy-Sobolev 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


which shows that strongly in . From the assumptions in , then .



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 boundary-value problem (3) if

For all , where denotes the inner product in .

Theorem 1   Let be defined as above. Then, there exist constants such that



for all .

Proof   We readily check that:

for all by Cauchy-Schwarz 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 Lax-Milgram 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 Lax-Milgram 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 .


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 .


From the difference quotient, we deduce and thus .

(2) For any , according to (5)

From which, Cauchy-Schwarz 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


which shows that is a symmetric operator.

So, we say that is a self-adjoint linear operator. And we thereby obtain the assertion of the Proposition by the Standard Functional Analysis 8 [8].

Lemma 3   Let be the i-th eigenfunction with respect to the eigenfunction of with the potential . Then,




Proof   (i) Let and be defined as in Proposition 1. Then


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


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


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



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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