Recovering differential pencils with spectral boundary conditions and spectral jump conditions

In 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.

Partial inverse problems for quadratic differential pencils on a graph with a loop

In this paper, partial inverse problems for the quadratic pencil of Sturm–Liouville operators on a graph with a loop are studied. These problems consist in recovering the pencil coefficients on one edge of the graph (a boundary edge or the loop) from spectral characteristics, while the coefficients on the other edges are known a priori. We obtain uniqueness theorems and constructive solutions for partial inverse problems.

Determination of differential pencils with impulse from interior spectral data

In this paper, we are concerned with the inverse spectral problems for differential pencils defined on $[0,\pi ]$ [ 0 , π ] with an interior discontinuity. We prove that two potential functions are determined uniquely by one spectrum and a set of values of eigenfunctions at some interior point $b\in (0,\pi )$ b ∈ ( 0 , π ) in the situation of $b=\pi /2$ b = π / 2 and $b\neq \pi /2$ b ≠ π / 2 . For the latter, we need the knowledge of a part of the second spectrum.

The partial inverse nodal problem for differential pencils on a finite interval

In this paper, the partial inverse nodal problem for differential pencils with real-valued coefficients on a finite interval \([0,1]\) was studied. The authors showed that the coefficients \((q_{0}(x),q_{1}(x),h,H_0)\) of the differential pencil \(L_0\) can be uniquely determined by partial nodal data on the right(or, left) arbitrary subinterval \([a,b]\) of \([0,1].\) Finally, an example was given to verify the validity of the reconstruction algorithm for this inverse nodal problem.

A uniqueness result for differential pencils with discontinuities from interior spectral data

In this work, the interior spectral data is employed to study the inverse problem for a differential pencil with a discontinuity on the half line. By using a set of values of the eigenfunctions at some internal point and eigenvalues, we obtain the functions {q_{0}(x)} and {q_{1}(x)} applied in the diffusion operator.