AbstractIn this work, we discuss the inverse problem for second order differential pencils with boundary and jump conditions dependent on the spectral parameter. We establish the following uniqueness theorems: $(i)$
(
i
)
the potentials $q_{k}(x)$
q
k
(
x
)
and boundary conditions of such a problem can be uniquely established by some information on eigenfunctions at some internal point $b\in (\frac{\pi }{2},\pi )$
b
∈
(
π
2
,
π
)
and parts of two spectra; $(ii)$
(
i
i
)
if one boundary condition and the potentials $q_{k}(x)$
q
k
(
x
)
are prescribed on the interval $[\pi /2(1-\alpha ),\pi ]$
[
π
/
2
(
1
−
α
)
,
π
]
for some $\alpha \in (0, 1)$
α
∈
(
0
,
1
)
, then parts of spectra $S\subseteq \sigma (L)$
S
⊆
σ
(
L
)
are enough to determine the potentials $q_{k}(x)$
q
k
(
x
)
on the whole interval $[0, \pi ]$
[
0
,
π
]
and another boundary condition.
A half-inverse problem for the singular diffusion operator with jump conditions
