On the Local Solvability and Stability for the Inverse Spectral Problem of the Generalized Dirichlet–Regge Problem

For the generalized Dirichlet–Regge problem with complex coefficients,we prove the local solvability and stability for the inverse spectral problem,which indicates an improved result of the previous work([Journal of Geometry and Physics,159,103936(2021)]).

Inverse problems for radial Schrodinger operators with the missing part of eigenvalues

We study inverse spectral problems for radial Schrodinger operators in L^(2)(0,1).It is well known that for a radial Schrodinger operator,two spectra for the different boundary conditions can uniquely determine the potential.However,if the spectra corresponding to the radial Schrodinger operators with the two potential functions miss a finite number of eigenvalues,what is the relationship between the two potential functions?Inspired by Hochstadt(1973)'s work,which handled the Sturm-Liouville operator with the potential q∈L^(1)(0,1),we give a corresponding result for radial Schrodinger operators with a larger class of potentials than L^(1)(0,1).When q∈L^(1)(0,1),we also consider the case where the spectra corresponding to the radial Schrodinger operators with the two potential functions miss an infinite number of eigenvalues and the eigenvalues are close in some sense.

The inverse problem for differential pencils on a star-shaped graph with mixed spectral data

The partial inverse problem for differential pencils on a star-shaped graph is studied from mixed spectral data.More precisely,we show that if the potentials on all edges on the star-shaped graph but one are known a priori then the unknown potential on the remaining edge can be uniquely determined by partial information on the potential and a part of eigenvalues.

Reconstruction of the Sturm-Liouville operator with discontinuities from a particular set of eigenvalues

Sturm-Liouville operators on a finite interval with discontinuities are considered. We give a uniqueness theorem for determining the potential and the parameters in boundary and under discontinuous conditions from a particular set of eigenvalues, and provide corresponding reconstruction algorithm, which can be applicable to McLaughlin-Rundell＇s uniqueness theorem （see J. Math. Phys. 28, 1987）.

The Dbar-dressing method for the(2+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

In this work,the(2+1)-dimensional Date–Jimbo–Kashiwara–Miwa(DJKM)equation is studied by means of the ■-dressing method.A new ■ problem has been constructed by analyzing the characteristic function and the Green’s function of its Lax representation.Based on solving the ■ equation and choosing the proper spectral transformation,the solution of the DJKM equation is constructed.Furthermore,the more general solution of the DJKM equation can be also obtained by ensuring the evolution of the time spectral data.

Ambarzumyan's Theorem for the Dirac Operator on Equilateral Tree Graphs

The classical Ambarzumyan’s theorem states that if the Neumann eigenvalues of the Sturm-Liouville operator-d^(2)/dx^(2)+q with an integrable real-valued potential q on[0,π] are {n^(2):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 tree graphs.We prove that if the spectrum of the Dirac operator on graphs coincides with the unperturbed case,then the potential is identically zero.

Ambarzumyan Theorems for Dirac Operators

We consider the inverse eigenvalue problems for stationary Dirac systems with differentiable selfadjoint matrix potential.The theorem of Ambarzumyan for a Sturm-Liouville problem is extended to Dirac operators,which are subject to separation boundary conditions or periodic(semi-periodic)boundary conditions,and some analogs of Ambarzumyan's theorem are obtained.The proof is based on the existence and extremal properties of the smallest eigenvalue of corresponding vectorial Sturm-Liouville operators,which are the second power of Dirac operators.

INVERSION OF TRACE FORMULAS FOR A STURM-LIOUVILLE OPERATOR

This paper revisits the classical problem“Can we hear the density of a string?”,which can be formulated as an inverse spectral problem for a Sturm-Liouville operator.Instead of inverting the map from density to spectral data directly,we propose a novel method to reconstruct the density based on inverting a sequence of trace formulas which bridge the density and its spectral data clearly in terms of a series of nonlinear integral equations.Numerical experiments are presented to verify the validity and effectiveness of the proposed numerical algorithm.The impact of different parameters involved in the algorithm is also discussed.