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Abstract
In this paper, we discuss the inverse spectral problem for Sturm–Liouville operators with boundary conditions linearly dependent on the spectral parameter and a finite number of interior discontinuities and show that if is given a priori on the interval
for some
, then the potential
on the whole interval
can be uniquely determined either by parts of a finite number of spectra, or by a finite number of subsets of pairs of eigenvalues and the norming constants of the corresponding eigenvalues. We still establish several uniqueness theorems for Sturm–Liouville operators with Robin boundary conditions and interior discontinuous conditions from the above two spectral data.
1 Introduction
Consider the following Sturm–Liouville operator defined by
(1.1)
(1.1) with boundary conditions
(1.2)
(1.2)
(1.3)
(1.3) and interior discontinuities
(1.4)
(1.4) where
,
such that
is the spectral parameter,
is a real-valued function and
.
For (Equation1.1(1.1)
(1.1) )–(Equation1.4
(1.4)
(1.4) ), Yao and Sun [Citation1] showed that all eigenvalues are real and simple and all eigenfunctions of the Sturm–Liouville operator completes in
. Later, Wang [Citation2] discussed two inverse problems for (Equation1.1
(1.1)
(1.1) )–(Equation1.4
(1.4)
(1.4) ) either from partial information on the potential and parts of one spectrum, or from a set of values of eigenfunctions at some interior point and parts of two spectra. More related results for Sturm–Liouville operators with boundary conditions dependent on the spectral parameter were studied by many authors (see [Citation1–Citation14]). In particular, Freiling and Yurko [Citation7] further explored three inverse problems for Sturm–Liouville operators with boundary conditions polynomially dependent on the spectral parameter either from Weyl function, or from discrete spectral data, or from two spectra and provided procedures for reconstructing this differential operator from the above spectral data. Mclaughlin and Polyakov [Citation8] addressed the inverse problem for Sturm–Liouville operators with Dirichlet boundary condition at
and the boundary condition having transcendental functions on the spectral parameter at
and established an interesting uniqueness theorem on the potential
, which is a generalization of Hochstadt–Lieberman theorem.[Citation15] Hald [Citation16] first discussed the half-inverse problem for the Sturm–Liouville operator with one discontinuous condition. More results or generalizations for differential operators with interior discontinuities were studied by many authors (see [Citation1, Citation2, Citation9, Citation10, Citation13, Citation14, Citation16–Citation24]).
Borg [Citation25], or Levinson [Citation26], or Levitan [Citation27], respectively, considered the inverse problem for Equation (Equation1.1(1.1)
(1.1) ) with Robin boundary conditions and showed that the specification of two spectra can uniquely determine the potential
and coefficients
of the boundary conditions. Hochstadt and Lieberman [Citation15] first addressed the half-inverse problem for Equation (Equation1.1
(1.1)
(1.1) ) with Robin boundary conditions and proved that if
is prescribed on
and coefficients
of the boundary condition is given a priori, then the potential
on the interval
can be uniquely determined by one spectrum. Castillo [Citation28], or Suzuki [Citation29] independently showed that the fixed boundary condition at
is necessary for Hochstadt–Lieberman theorem by an example. Hochstadt–Lieberman type theorem for differential operators was established by many authors (see [Citation2, Citation8, Citation10, Citation11, Citation15–Citation17, Citation23, Citation28–Citation36]). Marchenko [Citation37] first adopted an alternative approach to the inverse spectral problem and showed that the Weyl function of Sturm–Liouville operators uniquely determined the potential
and coefficients
of the boundary conditions. Numerous authors discussed the inverse spectral problems for differential operators by the Weyl function of this operator (see [Citation2, Citation5, Citation7, Citation9, Citation11, Citation19–Citation21, Citation30, Citation31, Citation35–Citation41]). Gesztesy and Simon [Citation31] (see [Citation31, Theorem 1.3]) used the Weyl function to study the inverse spectral problem for Equation (Equation1.1
(1.1)
(1.1) ) with Robin boundary conditions and showed that if potential
on the interval
for some
is given a priori, then parts of one spectrum is sufficient to determine the potential
on the whole interval. Suzuki [Citation29] showed that if
is prescribed on
for
, then one spectrum cannot uniquely determine the potential
by a counterexample. Later, using the Weyl function and methods developed in [Citation31], Wei and Xu [Citation36] proved that if
is given a priori on the interval
for some
, then parts of pairs of eigenvalues and the norming constants of the corresponding eigenvalues is sufficient to determine the potential
on the whole interval
. Hence, the following inverse problem is of interest:
Inverse Problem-1: For (Equation1.1(1.1)
(1.1) )–(Equation1.4
(1.4)
(1.4) ), if
is prescribed on
for some
and coefficients
of the boundary condition are given a priori, what extra conditions can ensure the unique determination of the potential
on the interval
?
The purpose of this article is to solve Inverse Problem-1 given by (Equation1.1(1.1)
(1.1) )–(Equation1.4
(1.4)
(1.4) ). At first, we discuss two inverse problems for (Equation1.1
(1.1)
(1.1) )–(Equation1.4
(1.4)
(1.4) ) either from partial information on the potential and parts of a finite number of spectra, or from partial information on the potential and parts of a finite number of subsets of pairs of eigenvalues and the norming constants of the corresponding eigenvalues. Then we establish several uniqueness theorems for Sturm–Liouville operators with Robin boundary conditions and arbitrary finite number of interior discontinuous conditions from the above two spectral data. The techniques used here are based on the Weyl function and methods developed in [Citation7, Citation31, Citation36]. Borg type theorem, or Gesztesy–Simon theorem, or Hochstadt–Lieberman type theorem for (Equation1.1
(1.1)
(1.1) )–(Equation1.4
(1.4)
(1.4) ) is one of Theorem 3.1 under special cases (see below).
This article is organized as follows. In Section 2, we present preliminaries. In Section 3, we discuss the inverse spectral problems for (Equation1.1(1.1)
(1.1) )–(Equation1.4
(1.4)
(1.4) ) from the above two spectral data. In Section 4, we establish several uniqueness theorems for Equation (Equation1.1
(1.1)
(1.1) ) with Robin boundary conditions and interior discontinuous conditions with the above spectral data.
2 Preliminaries
Let and
be solutions of Equation (Equation1.1
(1.1)
(1.1) ) under the initial conditions
(2.1)
(2.1) and satisfying the jump conditions (Equation1.4
(1.4)
(1.4) ), respectively.
Denote . Clearly,
, and
(2.2)
(2.2)
(2.3)
(2.3) The following formula is called as the Green’s formula:
(2.4)
(2.4) where
is called the Wronskian of
and
.
Let(2.5)
(2.5) By calculation, we have
(2.6)
(2.6) Therefore,
does not depend on
and
(2.7)
(2.7) which is called the characteristic function of
.
Let be the set of all eigenvalues of (Equation1.1
(1.1)
(1.1) )–(Equation1.4
(1.4)
(1.4) ). From [Citation1], we see that all zeros
of
are real and simple. Suppose that
and
are eigenfunctions of the corresponding eigenvalue
, then there exists
such that
where
is called the norming constant of the corresponding eigenvalue
. Consequently,
(2.8)
(2.8) and
.
Denote and
. Analogous to [Citation13, Citation14, Citation20], we have the following asymptotic formulae of
and
(2.9)
(2.9)
(2.10)
(2.10) Therefore, for sufficiently large
, we have the asymptotic formula of
(2.11)
(2.11) where
(2.12)
(2.12) where
and
.
Let be zeros of
. Denote
, where
sufficiently small, then there exists a constant
(see [Citation20, Citation40]), such that for sufficiently large
(2.13)
(2.13) Hence,
satisfy
(2.14)
(2.14) and
(2.15)
(2.15) where
.
Let be the solution of Equation (Equation1.1
(1.1)
(1.1) ) under the conditions
,
and the jump conditions (Equation1.4
(1.4)
(1.4) ). Then
(2.16)
(2.16) where
(2.17)
(2.17) which is called the Weyl function of (Equation1.1
(1.1)
(1.1) )–(Equation1.4
(1.4)
(1.4) ).
The following two lemmas are important for proofs of our main results.
Lemma 2.1
([Citation7, Citation14]) Let be the Weyl function of (Equation1.1
(1.1)
(1.1) )–(Equation1.4
(1.4)
(1.4) ) and coefficients
, of the boundary condition be given a priori. Then
uniquely determines the coefficients
, of the boundary condition and
, of the jump conditions as well as
(a.e.) on the interval
.
Lemma 2.2
([Citation31, Proposition B.6]) Let be an entire function such that
(1) |
| ||||
(2) |
|
3 Inverse spectral problems
In this section, we discuss two inverse problems for (Equation1.1(1.1)
(1.1) )–(Equation1.4
(1.4)
(1.4) ) and show that if the potential
is prescribed on the interval
for some
, then the potential
on the whole interval
can be uniquely determined by parts of a finite number of spectra or by parts of a finite number of subsets of pairs of eigenvalues and the norming constants of the corresponding eigenvalues.
We agree that together with we consider in this section a boundary value problem
of the same form but with other coefficients. That is, we use
instead of
of the operator
, respectively, such that
for all
.
Let be the spectrum of (Equation1.1
(1.1)
(1.1) )–(Equation1.4
(1.4)
(1.4) ) for
. Since
It is easy to prove
for fixed
or
.
For any , we denote
for all sufficiently large
.
We establish the following uniqueness theorem.
Theorem 3.1
Let be as that defined above,
,
, for each
and coefficients
, of the boundary conditions be given a priori. If
(1) |
| ||||
(2) |
| ||||
(3) | For each |
Let , we have the Borg type theorem except for one eigenvalue.
Theorem 3.2
Let , be as that defined above and coefficients
, of the boundary condition be given a priori. If one of the following conditions hold
(1) | |||||
(2) |
If , Theorem 3.1 leads to the Gesztesy–Simon type theorem for (Equation1.1
(1.1)
(1.1) )–(Equation1.4
(1.4)
(1.4) ). i.e.:
Theorem 3.3
Let be the spectrum of (Equation1.1
(1.1)
(1.1) )–(Equation1.4
(1.4)
(1.4) ) for
,
, and coefficients
, of the boundary condition be given a priori. For some
,
sufficiently small positive number, if
on the interval
and
(3.2)
(3.2) is satisfied for all sufficiently large
, then
In particular, let and
in Theorem 3.3, we have the Hochstadt–Lieberman type theorem for the problem (Equation1.1
(1.1)
(1.1) )–(Equation1.4
(1.4)
(1.4) ). i.e.
Theorem 3.4
Let be prescribed on the interval
and coefficients
, of the boundary condition be given a priori. Then one spectrum is sufficient to determine the potential
on the whole interval
and coefficients
, of the boundary condition and coefficients
,
and
of the jump conditions.
Let ,
,
,
, we get the following uniqueness theorem for the problem (Equation1.1
(1.1)
(1.1) )–(Equation1.4
(1.4)
(1.4) ) for
from two-thirds of three spectra. i.e.
Theorem 3.5
Let ,
, be as that defined above,
and coefficients
, of the boundary condition be given a priori. If
(3.3)
(3.3) then
Next, we present the proof of Theorem 3.1.
Proof of Theorem 3.1
Let , be the solution of Equation (Equation1.1
(1.1)
(1.1) ) for
under the terminal conditions
and
and the interior discontinuities (1.4) for
. By the Green’s formula, we have
(3.4)
(3.4) where
and
(3.5)
(3.5) From
on
together with the terminal conditions
and
, we get
(3.6)
(3.6) In addition, for each
, we obtain
(3.7)
(3.7) This implies
(3.8)
(3.8) Therefore
(3.9)
(3.9) Denote
(3.10)
(3.10) Without loss of generality, let’s assume all eigenvalues
, of (Equation1.1
(1.1)
(1.1) )–(Equation1.4
(1.4)
(1.4) ) for
. By virtue of [Citation1], we get
(3.11)
(3.11) where
is constant.
By virtue of (Equation3.6(3.6)
(3.6) ) and (Equation2.10
(2.10)
(2.10) ) together with Schwarz inequality, this yields
(3.12)
(3.12) where
are constant.
From (Equation2.10(2.10)
(2.10) ), we have
(3.13)
(3.13) Define the functions
, and
by
(3.14)
(3.14) and
(3.15)
(3.15) Then,
is an entire function in
.
By virtue of (Equation3.1(3.1)
(3.1) ), this yields
(3.16)
(3.16) Since
is an entire function in
of order
, there exists a positive constant
such that
(3.17)
(3.17) Therefore,
. For fixed
and
sufficiently large, we have
(3.18)
(3.18) For sufficiently large
, since
then
By the assumption (Equation3.3
(3.3)
(3.3) ) on
of Theorem 3.1, there exist constants
and
such that
By virtue of (Equation3.18
(3.18)
(3.18) ) together with the following relation
we have
(3.19)
(3.19) This implies
(3.20)
(3.20) where
is constant.
By virtue of the assumption of Theorem 3.1 together with (Equation3.20(3.20)
(3.20) ), we get
(3.21)
(3.21) By virtue of (Equation3.12
(3.12)
(3.12) ), (Equation3.15
(3.15)
(3.15) ) and (Equation3.21
(3.21)
(3.21) ), for
sufficiently large, we obtain
(3.22)
(3.22) From Lemma 2.2 together with (Equation3.22
(3.22)
(3.22) ), we have
Hence,
(3.23)
(3.23) By virtue of (Equation3.6
(3.6)
(3.6) ) and (Equation3.23
(3.23)
(3.23) ), this yields
This implies
Consequently,
(3.24)
(3.24) By virtue of Lemma 2.1 together with (Equation3.24
(3.24)
(3.24) ), we obtain
This completes the proof of Theorem 3.1.
Instead of partial information on the potential and parts of a finite number of spectra, one can use partial information on the potential and a finite number of subsets of pairs of eigenvalues and the norming constants of the corresponding eigenvalues to establish the following uniqueness theorem for (Equation1.1(1.1)
(1.1) )–(Equation1.4
(1.4)
(1.4) ). The techniques used here are based on the methods in [Citation20, Citation36].
Theorem 3.6
Let be as that defined above,
,
, for
, and coefficients
, of the boundary conditions be given a priori. If
(1) |
| ||||
(2) |
| ||||
(3) |
| ||||
(4) | For each |
Clearly, Theorem 3.6 leads to the following Corollary 3.7, which is a generalization of Theorem 4.1 in [Citation36].
Corollary 3.7
Let be the spectrum of (Equation1.1
(1.1)
(1.1) )–(Equation1.4
(1.4)
(1.4) ) for
,
and coefficients
of the boundary condition be given a priori. Denote the subset
. If
(1) |
| ||||
(2) | |||||
(3) | The subset |
Let and
, we have the following corollary.
Corollary 3.8
Let be the spectrum of (Equation1.1
(1.1)
(1.1) )–(Equation1.4
(1.4)
(1.4) ) and coefficients
, of the boundary condition be given a priori. If
is prescribed on the interval
, then the even spectral data
, or the odd spectral data
is sufficient to determine the potential
on the whole interval
and coefficients
, of the boundary condition and coefficients
and
, of the jump conditions.
In the rest parts of this section, we prove Theorem 3.6.
Proof of Theorem 3.6
We use the same symbols as these in Theorem 3.1. Let be the solution of Equation (Equation1.1
(1.1)
(1.1) ) for
.
Denote(3.27)
(3.27) By virtue of (Equation3.27
(3.27)
(3.27) ), for each
, we have
(3.28)
(3.28) Without loss of generality, we assume that
for all
. This implies
(3.29)
(3.29) From (Equation3.28
(3.28)
(3.28) ) and (Equation3.29
(3.29)
(3.29) ), we get
(3.30)
(3.30) By virtue of the assumption
on
of Theorem 3.6 together with the Green’s formula, we obtain
(3.31)
(3.31) Next, we will prove that
(3.32)
(3.32) First, we show that
, holds.
Let be the solution of Equation (Equation1.1
(1.1)
(1.1) ) for
with the initial conditions
and
. Consequently,
(3.33)
(3.33) Define the entire functions
and
by
(3.34)
(3.34) By virtue of (Equation2.10
(2.10)
(2.10) ), for sufficiently large
, this yields
(3.35)
(3.35) Similar to (Equation3.21
(3.21)
(3.21) ), by calculation, we have
(3.36)
(3.36) for each
, where
is a positive constant.
From (Equation3.36(3.36)
(3.36) ) together with the assumption of Theorem 3.6, we get
(3.37)
(3.37) By virtue of (Equation3.34
(3.34)
(3.34) ), (Equation3.35
(3.35)
(3.35) ) and (Equation3.37
(3.37)
(3.37) ), this yields
(3.38)
(3.38) By Lemma 2.2 together with (Equation3.38
(3.38)
(3.38) ), we obtain
(3.39)
(3.39) Therefore,
(3.40)
(3.40) Second, we prove
.
Let be the solution of Equation (Equation1.1
(1.1)
(1.1) ) for
with the initial conditions
and
. Therefore,
(3.41)
(3.41) Define the entire function
by
(3.42)
(3.42) Since
, we get
(3.43)
(3.43) From (Equation2.10
(2.10)
(2.10) ), (Equation3.12
(3.12)
(3.12) ) and (Equation3.43
(3.43)
(3.43) ), we obtain the following asymptotic formula
(3.44)
(3.44) This implies
(3.45)
(3.45) for sufficiently large
.
From (Equation3.37(3.37)
(3.37) ), (Equation3.42
(3.42)
(3.42) ) and (Equation3.45
(3.45)
(3.45) ), we have
(3.46)
(3.46) By Lemma 2.2 together with (Equation3.46
(3.46)
(3.46) ), we get
(3.47)
(3.47) Therefore,
(3.48)
(3.48) From (Equation3.40
(3.40)
(3.40) ) and (Equation3.48
(3.48)
(3.48) ), we get
(3.49)
(3.49) In virtue of
on
and (Equation3.49
(3.49)
(3.49) ), this yields
(3.50)
(3.50) Consequently,
(3.51)
(3.51) From Lemma 2.1 together with (Equation3.51
(3.51)
(3.51) ), we obtain
By now, the proof of Theorem 3.6 is completed.
4 Uniqueness theorems
In this section, we present several uniqueness theorems for Sturm–Liouville operators with Robin boundary conditions and interior discontinuities from partial information on the potential and parts of a finite number of spectra or from partial information on the potential and a finite number of subsets of pairs of eigenvalues and the norming constants of the corresponding eigenvalues.
Denote the Sturm–Liouville operators ,
, of the form
(4.1)
(4.1) with boundary conditions
(4.2)
(4.2)
(4.3)
(4.3) and interior discontinuities (Equation1.4
(1.4)
(1.4) ), where
,
, and
are real-valued functions and
.
Let , be the spectrum of (Equation4.1
(4.1)
(4.1) )–(Equation4.3
(4.3)
(4.3) ) and (Equation1.4
(1.4)
(1.4) ) for
. Since
, it is easy to prove that
for fixed
or
.
We obtain the following uniqueness theorem for the discontinuities Sturm–Liouville operator.
Theorem 4.1
Let be as that defined above,
,
, for each
and coefficients
, of the boundary condition be given a priori. If
(1) |
| ||||
(2) |
| ||||
(3) | For each |
Proof
Applying the same arguments as that in the proof of Theorem 3.1, we can prove Theorem 4.1 and omit the proof of Theorem 4.1 here.
Remark 1
(1) | Let | ||||
(2) | If | ||||
(3) | Let | ||||
(4) | Let |
Remark 2
If for each
, then
,
, for each
in Theorem 4.1 is an infinite set. Using the condition ‘coefficient
, of the boundary condition is given a priori’ instead of ‘coefficients
, for each
of the boundary conditions are given a priori’ in Theorem 4.1 and other conditions remain unchanged, we can also prove
for each
Therefore, Theorem 4.1 still holds.
Using partial information on the potential and a finite number of subsets of pairs of eigenvalues and the norming constants of the corresponding eigenvalues as the spectral data, we obtain the following uniqueness theorem for (Equation4.1(4.1)
(4.1) )–(Equation4.3
(4.3)
(4.3) ) and (Equation1.4
(1.4)
(1.4) ).
Theorem 4.2
Let be as that defined above,
,
, for each
and coefficients
, of the boundary conditions be given a priori. Suppose that the following conditions hold:
(1) |
| ||||
(2) |
| ||||
(3) | |||||
(4) | For each |
Proof
The proof is analogous to that of Theorem 3.6 in Section 3.
Remark 3
Clearly, if , Theorem 4.2 leads to Theorem 4.1 in [Citation36].
Let ,
, we then have the following corollary.
Corollary 4.3
Let be the spectrum of (Equation4.1
(4.1)
(4.1) )–(Equation4.3
(4.3)
(4.3) ) and (Equation1.4
(1.4)
(1.4) ) for
and coefficient
of the boundary condition be given a priori. Suppose that the following conditions hold.
(1) |
| ||||
(2) | The even spectral data |
Finally, we present an example for Theorem 4.1 for the case , which is given in data [Citation14, Citation20]. Denote
, where
is the eigenfunction of the eigenvalue
. It is well known that the spectral data
is equivalent to the spectral
in the inverse spectral problem. Therefore, we reconstruct the potential
and coefficients by the spectral data
. Since some symbols are undefined in this paper, the reader might check out these in [Citation14, Citation20].
Example
[Citation14, Citation20] Take such that
,
,
and
. Let
be the spectral data of
. Clearly
Let
and
, and
be an arbitrary positive number. Denote
. By virtue of (73) and (77) in [Citation14, Citation20], this yields
Therefore
where
. Using (78)–(80) in [Citation14, Citation20], we calculate
Acknowledgements
The author would like to express his gratitude to anonymous referees, editors and Professor D. Lesnic, Department of Applied Mathematics, University of Leeds, UK., for careful examination and valuable suggestions.
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