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Xiao-Chuan Xu

Publications and source records attributed to Xiao-Chuan Xu.

11 recordsLinked to original sources
arXiv2025

Direct and inverse spectral problems for the Schrodinger operator with double generalized Regge boundary conditions

In this paper, we study the direct and inverse spectral problems for the Schrodinger operator with two generalized Regge boundary conditions. For the direct problem, we give the properties of the spectrum, including the asymptotic distribution of the eigenvalues. For the inverse problems, we prove several uniqueness theorems, including the cases: even potential, two-spectra, as well as the general partial inverse problem.

math.SP↗
arXiv2024

An inverse problem for the matrix Schrodinger operator on the half-line with a general boundary condition

In this work, we study the inverse spectral problem, using the Weyl matrix as the input data, for the matrix Schrodinger operator on the half-line with the boundary condition being the form of the most general self-adjoint. We prove the uniqueness theorem, and derive the main equation and prove its solvability, which yields a theoretical reconstruction algorithm of the inverse problem.

math.SP↗
arXiv2024

On the local solvability and stability of the partial inverse problems for the non-self-adjoint Sturm-Liouville operators with a discontinuity

In this work, we study the inverse spectral problems for the Sturm-Liouville operators on [0,1] with complex coefficients and a discontinuity at $x=a\in(0,1)$. Assume that the potential on (a,1) and some parameters in the discontinuity and boundary conditions are given. We recover the potential on (0,a) and the other parameters from the eigenvalues. This is the so-called partial inverse problem. The local solvability and stability of the partial inverse problems are obtained for $a\in(0,1)$, in which the error caused by the given partial potential is considered. As a by-product, we also obtain two new uniqueness theorems for the partial inverse problem.

math.SP↗
arXiv2020

An inverse problem to determine the shape of a human vocal tract

The inverse problem of determining the cross-sectional area of a human vocal tract during the utterance of a vowel is considered. The frequency-dependent boundary condition at the lips is expressed in terms of the acoustic impedance of a vibrating piston on an infinite plane baffle. The corresponding pressure at the lips is expressed in terms of the normalized impedance and a key quantity related to the so-called Jost function of an associated Schrödinger equation. Various input data sets are considered to solve the inverse problem, including the pressure at the lips, the absolute pressure at the lips, or the poles of the pressure at the lips. The solution to the inverse problem is obtained with the help of the Gel'fand-Levitan method for the associated Schrödinger equation.

math-ph↗
arXiv2019

Estimates of complex eigenvalues and an inverse spectral problem for the transmission eigenvalue problem

This work deals with the interior transmission eigenvalue problem: $y'' + {k^2}η\left( r \right)y = 0$ with boundary conditions ${y\left( 0 \right) = 0 = y'\left( 1 \right)\frac{\sin k}{k} - y\left( 1 \right)\cos k},$ where the function $η(r)$ is positive. We obtain the asymptotic distribution of non-real transmission eigenvalues under the suitable assumption for the square of the index of refraction $η(r)$. Moreover, we provide a uniqueness theorem for the case $\int_0^1\sqrt{η(r)}dr>1$, by using all transmission eigenvalues (including their multiplicities) along with a partial information of $η(r)$ on the subinterval. The relationship between the proportion of the needed transmission eigenvalues and the length of the subinterval on the given $η(r)$ is also obtained.

math.SP↗
arXiv2019

On the direct and inverse transmission eigenvalue problems for the Schrodinger operator on the half line

For the direct problem, we give the asymptotic distribution of the (real and non-real) transmission eigenvalues for the Schrodinger operator on the half line. For the inverse problem, we prove that the potential can be uniquely determined by all transmission eigenvalues, the parameter in the boundary condition and some non-zero value related to the potential (whereas without the certain constant γ). The reconstruction algorithms are also revealed.

math-ph↗
arXiv2017

Inverse resonance problems for the Schroedinger operator on the real line with mixed given data

In this work, we study inverse resonance problems for the Schrödinger operator on the real line with the potential supported in $[0,1]$. In general, all eigenvalues and resonances can not uniquely determine the potential. (i) It is shown that if the potential is known a priori on $[0,1/2]$, then the unique recovery of the potential on the whole interval from all eigenvalues and resonances is valid. (ii) If the potential is known a priori on $[0,a]$, then for the case $a>1/2$, infinitely many eigenvalues and resonances can be missing for the unique determination of the potential, and for the case $a<1/2$, all eigenvalues and resonances plus a part of so-called sign-set can uniquely determine the potential. (iii) It is also shown that all eigenvalues and resonances, together with a set of logarithmic derivative values of eigenfunctions and wave-functions at $1/2$, can uniquely determine the potential.

math-ph↗
arXiv2017

Determination of matrix potential from scattering matrix

(i) For the matrix Schrödinger operator on the half line, it is shown that if the potential exponentially decreases fast enough then only the scattering matrix uniquely determines the self-adjoint potential and the boundary condition. (ii) For the matrix Schrödinger operator on the full line, it is shown that if the potential exponentially decreases fast enough then the scattering matrix (or equivalently, the transmission coefficient and reflection coefficient) uniquely determine the potential. If the potential vanishes on $(-\infty,0)$ then only the left reflection coefficient uniquely determine the potential.

math-ph↗
arXiv2017

Inverse spectral problems for the Sturm-Liouville operator with discontinuity

In this work, we consider the Sturm-Liouville operator on a finite interval $[0,1]$ with discontinuous conditions at $1/2$. We prove that if the potential is known a priori on a subinterval $[b,1]$ with $b\ge1/2$, then parts of two spectra can uniquely determine the potential and all parameters in discontinuous conditions and boundary conditions. For the case $b<1/2$, parts of either one or two spectra can uniquely determine the potential and a part of parameters.

math.SP↗