Xin-Jian Xu

• Doctor of Philosophy
• Nanjing University of Science and Technology

The classical Ambarzumyan’s theorem states that if the Neumann eigenvalues of the Sturm-Liouville operator \( - {{{d^2}} \over {d{x^2}}} + q\) with an integrable real-valued potential q on [0, π] are {n2: n ≥ 0}, then q = 0 for almost all x ∈ [0, π]. In this work, the classical Ambarzumyan’s theorem is extended to the Dirac operator on equilateral...

In this work we deal with inverse nodal problems of reconstructing a fourth-order self-adjoint binomial operator d4dx4+q with Dirichlet boundary conditions. We prove that a dense subset of nodal points uniquely determines the potential function q.

In this paper a class of differential operators with retarded arguments on a lasso graph is studied. We derive the asymptotic expressions of its large eigenvalues and obtain a new regularized trace formula for this class of differential operators.

We study inverse spectral problems for radial Schrödinger operators in L2(0, 1). It is well known that for a radial Schrödinger operator, two spectra for the different boundary conditions can uniquely determine the potential. However, if the spectra corresponding to the radial Schrödinger operators with the two potential functions miss a finite num...

We study an inverse eigenvalue problem for the radial Schrödinger operators on the unit interval. This problem consists in the recovery of the potential on a subinterval (0,a), a≤1, from eigenvalues corresponding to the boundary value problems with different boundary conditions. We obtain a sufficient condition for the unique specification of the r...

We consider the inverse problem of the interior transmission eigenvalue problem for spherically stratified media with fixed angular-momentum quantum number \begin{document}$ l $\end{document}. Under the case \begin{document}$ \int^1_0\sqrt{\rho(r)}{\rm d}r\leq1 $\end{document}, we show that a knowledge of all the transmission eigenvalues for \begin...

We consider the inverse spectral problems for Bessel operators on the unit interval subject to a class of discontinuity conditions. In this paper, we prove uniqueness theorems for these inverse spectral problems. That is, we show that either the eigenvalues and the corresponding norming constants or two sets of eigenvalues for different boundary co...

The problem of differential operators with nonlocal boundary condition appears, e.g., in scattering theory, diffusion processes and the other applicable fields. In this paper, we consider a class of differential operators with nonlocal boundary condition, and give its trace formula of Gelfand-Levitan type.

Inverse nodal problem consists in constructing operators from the given zeros of their eigenfunctions. The problem of differential operators with nonlocal boundary condition appears, e.g., in scattering theory, diffusion processes and the other applicable fields. In this paper, we consider a class of differential operators with nonlocal boundary co...

In this work, we consider the interior transmission eigenvalue problem for a spherically stratified medium, which can be formulated as y′′(r) + k2η(r)y(r) = 0 endowed with boundary conditions , where the refractive index η(r) is positive and real. We obtain the distribution of transmission eigenvalues under assumptions that and one of these conditi...