Inverse Spectral Problem for Sturm-Liouville Operator with Boundary and Jump Conditions Dependent on the Spectral Parameter ()
1. Introduction
Inverse spectral problems are motivated to recovering operators from the priori known spectral characteristics. These problems often appear in mathematics, physics, mechanics, electronics, and some other sciences and engineering problems, and, hence, are very important to understanding the real world. Significant progress has been made in the inverse problem theory for regular self-adjoint or nonself-adjoint Sturm-Liouville operators [1] [2] [3] [4] .
The inverse problem of Sturm-Liouville operator was initiated by Ambarzumian [5] and Borg [6] , after that, there are various generalizations on the inverse problems of Sturm-Liouville operators. Besides the classical regular Sturm-Liouville operators [2] [3] , in recent years there have been a lot of inverse problems for Sturm-Liouville operators with eigenparameter-dependent boundary conditions and Sturm-Liouville operators with transmission conditions [7] - [10] . Fulton has studied the inverse spectral problem with boundary conditions linearly dependent on the spectral parameter [7] . Binding et al. discussed boundary conditions that depend nonlinearly on the spectral parameter [9] . Hald has studied the discontinuous Sturm-Liouville problem and shown the direct and inverse spectral theory on the Sturm-Liouville problem with internal discontinuous point conditions [10] . The corresponding direct problems of boundary value problems with transmission conditions and/or eigenparameter-dependent boundary conditions, we refer to [20] - [25] and the references therein.
Recent years, the boundary value problems with eigenparameter-dependent transmission conditions have drown scholars’ much attention and have achieved significant progress, including direct and inverse spectral theory and half inverse spectral theory [26] - [35] . In 2005, Akdogan et al. investigated the discontinuous Sturm-Liouville problems, where the spectral parameter not only appears in differential equations, but also in boundary conditions and one of the jump conditions, they got the asymptotic approximation of fundamental solutions and the asymptotic formulae for eigenvalues of such problems [27] . In 2012, Ozkan et al. considered the spectral problems for Sturm-Liouville operator with both boundary and one of the jump conditions linearly dependent on the eigenparameter, and studied the inverse problem of this operator [28] . In 2014, Guo et al. investigated the inverse spectral problem of Sturm-Liouville operator with finite number of jump conditions dependent on the eigenparameter [29] . In 2016, Wei et al. investigated the inverse spectral problem for Dirac operator with boundary and jump conditions dependent on the spectral parameter. Through inducting the generalized normal constants they have proved the uniqueness theorem, then a construction method for solving this inverse problem was given [30] . In 2018 and 2021, Bartels et al. presented Sturm-Liouville problems with transfer condition Herglotz dependent on the eigenparameter, and showed the Hilbert space formulation of the problem and calculated out the eigenvalue and eigenfunction asymptotic formula on this problem [31] [34] . Zhang et al. studied the finite spectrum of Sturm-Liouville problems with both jump conditions dependent on the spectral parameter [35] .
Since then the Sturm-Liouville problems with jump conditions containing the spectral parameter have been widely studied, however, for the problems with both jump conditions containing the spectral parameter attach less attention, which often appear in heat transfer, electronic signal amplifiers and other issues of sciences, hence have high research significance. It’s also a good complement to the study of spectral and inverse spectral problems of boundary value problems of differential equations.
In this paper, we mainly investigate the inverse spectral problem of Sturm-Liouville operator in which the spectral parameter not only appears in the differential equation, but also appears in both of the jump conditions and the boundary conditions. While the spectral parameter appears in equation and boundary conditions and transmission conditions, hence it is much complicated. The studies on such problems play an important role in differential equations and spectral theory. To show the inverse spectral theory of this problem, the operator formulation of this problem is constructed and some spectral properties are given, next the asymptotic behavior of the solutions and eigenvalues is provided, then several uniqueness results for this inverse spectral problem are given. The uniqueness theorem is very important in inverse spectral theory of boundary value problems, and there are many approaches to solve the uniqueness theorem. In this paper, we will use three general methods to solve the uniqueness theorem, which are equivalent to each other, i.e. the Weyl function theory, two spectra and spectral data approach. To analyze this inverse spectral problem, the dependence of eigenvalues of such problems is the theoretical basis of it, and the corresponding results the reader may refer to [36] .
2. Notation and Basic Properties
Consider the following boundary value problem (denoted by L) consisting of the following Sturm-Liouville equation
(2.1)
together with boundary conditions (BCs)
(2.2)
(2.3)
and jump conditions with spectral parameter
(2.4)
(2.5)
where
is real valued,
,
,
,
,
,
,
. Here
is a spectral parameter.
In order to describe the self-adjointness of the operator corresponding to the problem L, firstly, let us consider the set associated with the functions considered in the present paper as
where
denotes all local absolutely continuous functions on J, then we can introduce an inner product in the Hilbert space
as
(2.6)
for arbitrary
To facilitate the description, the following notation need to be listed. For
, let
then the boundary conditions (2.2), (2.3) and jump conditions (2.4), (2.5) can be written as
In the Hilbert space
we define a linear operator
as
(2.7)
and the domain of the operator
as
Thus, the problem L can be written as the following form
where
.
Then it can be proven that the following theorem about the self-adjointness of the operator
holds.
Theorem 1. [27] The linear operator
is self-adjoint in the Hilbert space
.
Define two fundamental solutions
of Equation (2.1) on whole
satisfying the jump conditions (2.4), (2.5) and the following initial conditions, respectively
Since these solutions
and
satisfy the jump conditions (2.4) and (2.5), the following relations
hold, where
.
For each
, these solutions satisfy the relation
. Then the characteristic function can be introduced as
(2.8)
according to the Liouville’s theorem, the Wronskian
is an entire function in
and the zeros namely
of
coincide with the eigenvalues of the problem L. Substituting
into (2.8) we get
(2.9)
The normal constants
of the problem L can be defined as follows
(2.10)
If the functions
and
are the eigenfunctions of the problem L, then there exists a sequence
such that
(2.11)
Theorem 2. Let
be the zeros of the function
, then
(2.12)
where
, and
are defined by (2.10) and (2.11), respectively.
Proof. Let us write the following equations
(2.13)
(2.14)
Let (2.13), (2.14) multiplied by
and
, respectively, and subtracting them, then the equality
(2.15)
is obtained. Integrating over the interval J
Dividing both sides of the above equality by
, and let
, then we have
Using (2.11)
Thus the equality (2.12) holds.
3. Construction and Asymptotic Approximation of Fundamental Solutions and Eigenvalues
In this section, we will obtain the asymptotic approximation of fundamental solutions and eigenvalues of the problem L.
Lemma 1. Let
. Then the following asymptotics hold.
When
, one has
(3.1)
(3.2)
When
, one has
(3.3)
(3.4)
Where
.
Proof. When
. Let
be the solutions of (2.1) under the conditions
According to [37] , one has
(3.5)
(3.6)
where
.
Suppose
are the solutions of Equation (2.1), and satisfy the jump conditions (2.4), (2.5), and the following initial conditions
Hence as
,
and as
, let
(3.7)
Due to the fact that
meet the jump conditions (2.4), (2.5), and
,
, we can get
Thus by calculation, it can be obtained that
Substituted into Equations (3.7), we have
According to the initial conditions satisfied by
.
For
, as
,
and as
,
Similarly, when
, (3.2) can be obtained.
Lemma 2. The function
has the following asymptotics, for
.
When
, one has
(3.8)
When
, one has
(3.9)
Proof. The proof is the same as lemma 1, hence we omit the details.
Hence, when
, according to (2.9) and (3.8) the characteristic function
as
is
(3.10)
Let
, where
Next, we ready to find the asymptotic formulas for the eigenvalues of the considered problem L.
Theorem 3. Let
be the eigenvalues of the problem L,
, then it has following asymptotics as
(3.11)
Proof. Let
(3.12)
(3.13)
Next we only prove the case of
, and
can be proved in the same way.
Denote
, where
, and
are square roots of
, then from [3] we know that for any
, there exists a constant
, such that
thus for sufficiently large
, when
and
, it has
(3.14)
It’s easy to know
. Clearly,
for
, according to Rouchè’s theorem, it is clear that the number of zeros of
inside
coincides with the number of zeros of
. Applying Rouchè’s theorem again to the circle
, for sufficiently large n, in each
, there exits a unique zero of
, namely
. Because of
is sufficiently small, when
, we have
(3.15)
Let
. Substituting (3.15) into (3.10), we can obtain that
By the well-known formula
, the above equation can be changed to the following formula
When
, then
is true.
Therefore, (3.11) can be rolled out.
4. Inverse Problems
In this section, we mainly consider the reconstruction of the problem L, from the Weyl function, from the spectral data
, and from two spectra
.
Denote
(4.1)
where
are not 0 at the same time. Let
be the solution of (2.1), satisfying the following initial conditions and jump conditions (2.4) and (2.5)
Because of
, we have
or
(4.2)
Denote
(4.3)
Thus
is the solution of (2.1) that satisfies the conditions
,
and the jump conditions (2.4) and (2.5), where
is defined in (2.8).
The functions
and
are called the Weyl solution and the Weyl function for the boundary value problem L.
Next, the uniqueness theorem for problem L will be given by the Weyl function. For studying the inverse problem we agree that together with L consider a boundary value problem
of the same form but with different coefficients
.
Theorem 4. If
, then
, i.e.
a.e. J, and
,
,
,
,
,
,
.
Proof. Let us define the matrix
by the formula
then we can calculate that
(4.4)
and
(4.5)
According to (4.2) and (4.4), the following equations can be obtained
(4.6)
Denote
,
, where
is sufficiently small number,
and
are square roots of the eigenvalues of the problems L and
, respectively. By virtue of (3.1), (3.8) and (3.10), for sufficiently large
, there exists a constant
such that
(4.7)
Thus, if
, then for each fixed
, the functions
and
are entire in
. Combined with (4.7), and according to Liouville’s theorem, we can get
(4.8)
Substituting (4.8) into (4.5), then for each
and
we have
(4.9)
Due to
and
, ones have
.
On the other hand, the asymptotic expressions
(4.10)
can be easily verified. Here
Without loss of generality, assume
. From (4.9), (4.10) we get
for
. When
, one has
(4.11)
By letting
, in (4.11) we contradict
. Thus
and
. Hence
,
.
Finally, if
holds, then we can conclude
, a.e. J and
,
,
,
,
,
,
. So consequently,
.
Lemma 3. [29] For the function
defined in (4.1), the following expression can be established
(4.12)
Theorem 5. If
and
, then
a.e. J, and
,
,
,
,
,
,
, i.e.
.
Proof. From lemma 3, if
and
, then
. According to Theorem 4, this theorem can be proved.
Lastly, through the two spectra
, let us prove the uniqueness theorem. Let
be the spectra of the problem
consisting of the Equation (2.1) with condition
(where
are not 0 at the same time) and conditions (2.3), (2.4) and (2.5). It is obvious that
are the zeros of
, where
is the characteristic function of the problem
.
Theorem 6. If
, then
a.e. J, and
,
,
,
,
,
,
,
.
Proof. Since the functions
and
are entire of order
, we can write by Hadamards factorization theorem (methods popularized by the literature [3] )
Thus
and
are uniquely determined up to a multiplicative constant by their zeros (the case when
requires minor modifications). Therefore, one has
,
, i.e.
when
,
. Consequently
, according to Theorem 4, the proof is completed.
5. Conclusion
In the present work, the inverse spectral problem of Sturm-Liouville operator with boundary conditions and jump conditions dependent on the spectral parameter is investigated. Such problems are connected with fields such as mechanical engineering, and acoustic wave propagation problems, etc. Here the uniqueness theorems of this problem are given by using Weyl function theory, two spectra and spectral data approaches. However, we only discuss the uniqueness theorem of the problem, the reconstruction formulae and stability of this problem have not been considered, we plan to consider these problems in future studies.
Acknowledgements
This work is supported by National Natural Science Foundation of China (Grant No. 12261066), Natural Science Foundation of Inner Mongolia (Grant Nos. 2021MS01020 and 2023LHMS01015).
NOTES
*Corresponding author.