2
|
R. Zhang et al., Inverse spectral problem
and the jump conditions
π
π
{
y( + 0) = βy( − 0),
ꢂ
ꢂ
2
2
(1.4)
ꢀ
ꢂ
ꢂ
π
2
π
2
π
2
耠
−1
耠
y ( + 0) = β y ( − 0) + ay( − 0).
ꢁ
2
Here λ is the spectral parameter, M(x) is a real-valued function in L (0, π), h, H, β, a are real, and β > 0.
Problem (1.1)–(1.4), denoted by L = L(M(x), ρ(x), h, H, β, a), is called a boundary value problem for the
π
Sturm–Liouville equation with the discontinuity conditions at
.
2
The boundary value problems with a discontinuous point inside the interval frequently appear in math-
ematics, physics, geophysics, and other aspects of natural sciences (see [1, 2, 6, 10, 14]). Generally, such
problems are related to discontinuous material characters of a intermediary. This kind of problem has been
studied by many authors (see, e.g., [3, 5, 7, 23, 27]).
In general, for reconstructing the potential on the whole interval and all parameters about the Sturm–
Liouville operator, it is necessary to specify two spectra of the problem with different boundary conditions
(
see, e.g., [16, 17, 26]). Hochstadt and Lieberman (see [8]) showed that if the potential M(x) is known a priori
π
π
on the half-interval ( , π), then a single spectrum is sufficient to determine M(x) on the half-interval (0, ).
2
2
This is the so-called half-inverse problem which has been generalized into many cases (see [9, 11, 18, 20–
2
2, 24, 25] and the references therein).
Nabiev and Amirov (see [14]) studied the boundary value problem L = L(M(x), ρ(x), h, H, 1, 0), where
β = 1 and a = 0, and gave some integral representations for the solutions of equation (1.1). In 2008,
Shieh and Yurko (see [19]) gave the uniqueness theorem of the half-inverse problem for the problem
L = L(M(x), 1, h, H, β, a), where ρ(x) ≡ 1, β, a and H are assumed to be known a priori. In [28], Yurko
studied the problem L = L(M(x), ρ(x), h, H, β, a), and proved that the potential M(x) and the coefficients in
the boundary conditions and the jump conditions can be uniquely determined from the Weyl-type function
or from two spectra.
In 2001, Mochizuki and Trooshin (see [12]) studied the problem L(M(x), 1, h, H, 0, 1), where ρ(x) ≡ 1,
a = 1 and β = 0, and proved that a set of values of the logarithmic derivative of eigenfunctions at some
an internal point and spectrum can uniquely determine the potential M(x) on (0, π). They used the same
method for reconstructing the potential for Dirac operator (see [13]). Yang (see [25]) considered the problem
L(M(x), 1, h, H, β, a), where ρ(x) ≡ 1, and showed that the potential M(x) can uniquely be determined by
π
a set of values of eigenfunctions at some an internal point and one spectrum. For the Dirac operator, the
2
similar problems were studied in [6, 24].
In [15], Ozkan, Keskin and Cakmak considered the problem L(M(x), ρ(x), 0, 0, β, 0), where h, H and a are
π
assumed to be zero. They showed that if the potential M(x) is prescribed on (0, ) (see Figure 1), then only
2
one spectrum is sufficient to determine M(x) on the interval (0, π) and ρ(x), β. The assumptions proposed
in [15] to reconstruct the potential are overdetermined. In fact, it is enough to assume that the potential M(x)
(
1+α)π
is given on a smaller interval (0,
) (see Figure 2).
4
In this paper, we consider the problem L = L(M(x), ρ(x), h, H, β, a) and prove that if the potential M(x)
(
1+α)π
on (0,
) (see Figure 2) and h are given, then only a single spectrum is sufficient to determine M(x) on
4
(
0, π), ρ(x), H, β and a. We also consider the case that the potential M(x) is given on the right “half-interval”
(
see Figure 3), and prove a uniqueness theorem. Also it is shown that potential M(x) on (0, π), ρ(x), β, a, h
Figure 1: The case in [15].
Figure 2: Case (i) in this paper.
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