Spectrum of a Class of Difference Operators with Indefinite Weights ()
1. Introduction
Spectral theory for Sturm-Liouville boundary value problems has important physical meaning and practical significance. The study of discrete Sturm-Liouville boundary value problems has a very important role in problems when it can be described by discrete and continuous mathematical models. In this paper, we deal with eigenvalue problems for the discrete Sturm-Liouville equations.
In this paper, the following equations and boundary conditions are studied.
,
(1)
. (2)
is the spectrum parameter,
(
is an integer), the real matrix
is positive definite and
. The real function
,
and
are all defined on
, and
,
,
,
,
.
The weight function
on
and changes sign. It’s easy to know
.
In the introduction part, we also mention some other boundary conditions listed below,
, (3)
,
(4)
,
(5)
(6)
,
, (7)
The study of the eigenvalue problem with sign-changing weight has lasted a long time. In 1914, Bôcher [1] studied such problems in the continuous case. Subsequently, many scholars have studied the differential operator problem with indefinite weight function and obtained a series of important results (for example [2] [3]).
However, there are few results on the spectra of discrete second-order linear eigenvalue problems when the weight function
changes its sign on T.
In 2007, Ji and Yang [4] [5] studied the structure of the eigenvalues of problem (1) (3) when the weight function
changes its sign, and they obtained that the number of positive eigenvalues is equal to the number of positive elements in the weight function, and the number of negative eigenvalues is equal to the number of negative elements in the weight function.
In 2008, Ji and Yang [6] discussed the eigenvalues of (1) (4) and (1) (5); by using the matrix theory, they got a very interesting result: the numbers of positive eigenvalues are equal to the numbers of positive elements in the weight function, and the numbers of negative eigenvalues are equal to the numbers of negative elements in the weight function.
In 2013, Ma, Gao and Lu [7] discussed the spectra of the discrete second-order Neumann Eigenvalue problem (1) (6), By the analytical methods, he not only gives the properties of the eigenvalues, but also gives the number of sign changes of the eigenfunction corresponding to the j-th positive/negative eigenvalue.
In 2015, C. Gao, R. Ma [8] studied the problem (1) (4), they find that the problems have T real eigenvalues (including the multiplicity). Furthermore, the numbers of positive eigenvalues are equal to the numbers of positive elements in the weight function, and the numbers of negative eigenvalues are equal to the numbers of negative elements in the weight function.
In 2018, R. Ma, C. Gao, Y. Lu [9] studied the problem (1) (7), by the Sylvester inertia theorem, they obtained the following conclusion:
a) (1) (7) has real and simple eigenvalues, which can be ordered as follows
b) every eigenfunction
corresponding to the eigenvalue
changes its sign exactly
times.
It is the purpose of this paper to establish the discrete analogue of the above conclusion.
In this paper, we will study the problems (1) (2) under coupled boundary conditions, using analytical methods and Sylvester inertia theorem, two conclusions are given. First, the eigenvalues of the questions are real and single, the number of positive eigenvalues is equal to the number of positive elements in the weight function, and the number of negative eigenvalues is equal to the number of negative elements in the weight function. Second, under some conditions, we obtain sign changes of the eigenfunction corresponding to the j-th positive/negative eigenvalue. We discuss different situations and get a series of important conclusions.
2. Theory
2.1. Main Theorem
Theorem 1. The question (1) (2) has T real and simple eigenvalues. And the number of positive eigenvalues is equal to the number of positive elements in the weight function, the number of negative eigenvalues is equal to the number of negative elements in the weight function. And the eigenvalues can be sorted as follows:
.
Theorem 2. In this paper, let
be the eigenvalue of (1) (2), and
be the eigenfunction corresponding to the eigenvalue
, where
. When
, we will give the sign-change times in different cases:
Case 1: If
and
hold, or
hold, where
is the matrix norm of
. Then
1) When
If i is even, the number of sign changes of
is
; if i is odd, the number of sign changes of
is i.
2) When
If i is even, the number of sign changes of
is i; if i is odd, the number of sign changes of
is
.
Case 2: If
and
hold, or
hold. Then
1) When
If i is even, the number of sign changes of
is
; if i is odd, the number of sign changes of
is i.
2) When
If i is even, the number of sign changes of
is i; if i is odd, the number of sign changes of
is
.
2.2. Lemma and Proof of the Theorem
Let
,
,
,
, then problem (1) (2) can be transformed into a matrix equation
,
where
,
.
We can know
and
(
) all are positive definite matrixes. In fact, for every
, where
, we have
If
, then
, so
is a positive definite matrix. Similarly, all matrixes
(
) are positive definite matrixes.
Let
denote the j-th principal subdeterminant of
deleting the first row and the first column, suppose that
,
, and
; (8)
; (9)
;
, (10)
. (11)
As we know, finding the eigenvalues of (1) (2) is equivalent to finding the zeros of
.
For
, let
be the number of the elements in
for some
, and
be the number of the elements in
for some
.
Lemma 1. For
, we have
,
.
Proof. For
, we have
,
then
when
, we have
,
Similarly, we can get that
(
).
Lemma 2. For
, the roots of
are real. Moreover,
has
positive roots and
negative roots.
Lemma 3. Two continuous polynomials
and
have no common zeros, for
The proof of lemma 2 and Lemma 3 are similar to the proof of lemma 2 and Lemma 3 of [9] respectively.
Lemma 4. Suppose that
is a root of
, then we have
, for
.
Proof. For
, since
, by lemma 3, we have
, combining this with (8)-(10), we infer that
.
when
, since
, by lemma 3, we have
, combining this with (11), we infer that
.
The proof is completed.
Lemma 5. For
, the roots of
is simple. The positive roots of
and
separate one another and the negative roots of
and
separate one another.
Proof. 1) When
,
From (9), we have
, if
, then
,
,
and
. If
, then
,
,
and
.
From (10),
, therefore
has two different roots:
,
.
If
, then
,
,
, by calculation, we know that
, let
and
, we have
.
If
, then
,
,
, we know that
, let
and
, we have
.
If
, then
,
,
, we know that
, let
and
, we have
.
If
, then
,
,
, we know that
, let
and
, we have
.
Thus
, the result holds.
2) When
, the proof is similar to Lemma 5 of [9].
From the above five lemmas, we can get Theorem 1.
Lemma 6. Let
be the sign-change times of
, then for
, we have that
,
,
where
means that
from left hand side of C, and
means that
from right hand side of C.
Proof. This proof is similar to the lemma 6 of [9].
Note. From the proof process, we can see that
is a nondecreasing function.
Lemma 7. Let
be the eigenfunction of (1) (2). When
, we have that
,
, (12)
. (13)
Proof. This proof is similar to the lemma 7 of [9].
Finally, we prove theorem 2.
Proof. We only prove case 1. The proof of case 2 is similar.
From (12), we know that the sign of
is equal to the sign of
, for
, so the sign-change times of
is equal to the number of sign changes of
.
From lemma 5, we have
. Combining this with lemma 6 and note, we infer that
. (14)
i) If
and
hold,
Since
, by the theory of matrices, we get that
, therefore,
, that is
.
From (11), we know
. (15)
then by the (15) with
, we have
, (16)
which implies that the sign-change times of
is
, so the number of sign changes of
is
.
From lemma 5, we have
, for
.
From (2), we have
, (17)
. (18)
1) When
From (13), we have
, (19)
combining (17) with (19), we have
, (20)
due to
, by (18) and (20), then
.
So we can get the sign change.
2) When
From (13), we have
, (21)
combining (17) with (21), we have
, (22)
due to
, by (18) and (22), then
.
So we can get the sign change.
ii) If
hold, we infer that
.
combining this with (15), then (16) holds, which implies that the sign-change times of
is
, so we have that the number of sign changes of
is
.
The rest proof of the section is as same as the corresponding section in part (i).
3. Conclusion
In this paper, we study a class of second order difference operators with indefinite weights and coupled boundary conditions. Firstly, we transform the problem into matrix form. Then, by using the inertia theorem and analytic method, we get some important conclusions about eigenvalues and eigenfunctions (see Theorem 1 and Theorem 2). Our work has great significance to the perfection of the theory of the discrete Sturm-Liouville problems.
Funding
This work was supported by the National Natural Science Foundation of China (11661059) and the Natural Science Foundation of the Inner Mongolia Autonomous Region of China (2017MS(LH)0103).
Nomenclature
= forward difference operator
= spectrum parameter
= weight function·
,
(
) = the number of elements in
(
)
= matrix
R = real number field
= vector
= determinant of
= the j-th principal submatrix of
deleting the first row and the first column
= the j-th principal submatrix of
deleting the first row and the first column
= j-th principal subdeterminant of
deleting the first row and the first column
= j-th
root of
= matrix norm of
= spectral radius of
= eigenfunction corresponding to the eigenvalue