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Lung-Hui Chen

Publications and source records attributed to Lung-Hui Chen.

At least 19 recordsLinked to original sources
arXiv2026

Topology in One Point Interaction Problem on Extended Non-Local Star Graphs and its Eigenvalues

The author studies the inverse spectral problem of Sturm-Liouville operator on a star-like metric graph. At the vertex of this star-like graph, there are attached $m$ edges that imposed with non-local Sturm-Liouville operator satisfying some suitable non-local boundary conditions. At the vertex, we consider one point interaction condition to model a metric graph that fixed on the end of edges of the graph. This models the vibration or flux that changes over time that monitored at the vertex which serves as certain control/regulation center. The author shows that the system is solvable under very necessary conditions. It is crucial to recover the topology of the network/metric graph which the topology is given. To begin the analysis, one constructs the special solution fixed on one end of edges while maintaining continuous at the vertex. This models a string that is vibrating vertically at the vertex according to certain frequencies. The non-local characteristic function plays a role, and then, one tries to find a non-trivial non-local eigenvalue.

math-ph↗
arXiv2025

Partial Information for Inverse Spectral Uniqueness in Vibration System with Multiple Frozen Arguments

In this paper, we investigate the inverse spectral problem of the Sturm-Liouville operator with many frozen arguments fixed at the points $\{a_{1}, a_{2},\ldots,a_{N}\}$ in $(0,π)$. We start with counting the zeros or the eigenvalues of characteristic function, and then discuss how certain information provided a priori on the point set $\{a_{1}, a_{2},\ldots,a_{N}\}$ would affect the uniqueness or non-uniqueness of this vibration system with many frozen points. The knowledge at the frozen or regulator points are practical in many on-site problems. Parallelly, certain irrational independence assumption assures the inverse spectral uniqueness as well.

math.SP↗
arXiv2025

Plancherel-Pólya's Type of Instability in Vibration System with Multiple Frozen Arguments

We discuss the problem of the inverse spectral problem of Sturm-Liouville operator with multiple frozen arguments at $\{a_{1}, a_{2},\ldots,a_{N}\}$ in $(0,π)$. One may consider the characteristic functions as perturbation of sine or of cosine functions depending on the boundary problem prescribed. However, such perturbation is represented in the form of Fourier transform of certain function which may or may not bring in Riesz basis theory and classical perturbation theory in functional analysis. We shall demonstrate the spectral perturbation in Plancherel-Pólya's type of inequality and connect to perturbation of related potential functions in $L^{2}$-functional norm.

math-ph↗
arXiv2025

Uniqueness of Inverse Spectral Problem of Non-Local Sturm-Liouville Operators on Star Graph

In this paper, we explore the inverse spectral problem of Sturm-Liouville operator on a star-like graph. To this fixed star-like graph centered at the origin as its vertex, we attach $m$ edges. On each edge, we impose the Sturm-Liouville operator with certain non-local potential functions with some suitable non-local boundary value conditions. At the vertex, we consider a frozen argument type of condition at zero to model a network that fixed on the end of each edge on the graph. The vibration and flow changes are monitored at that vertex which serves as certain control center. There is an inverse uniqueness subject to the suitable non-local boundary condition. We show that the system is solvable. Additionally, we give a Weyl's type of spectral asymptotics.

math-ph↗
arXiv2025

Stability of Inverse Resonance Problem on the Half Line

We consider the inverse resonance problem in one-dimensional scattering theory. The scattering matrix consists of $2\times 2$ entries of meromorphic functions, which are quotients of certain Fourier transform. The resonances are expressed as the zeros of Fourier transform of wave field. For compactly-supported perturbation, we are able to quantitatively estimate the zeros and poles of each meromorphic entry. The size of potential support is connected to the zero distribution of scattered wave field. We derive the inverse stability on scattering source based on certain knowledge on the perturbation theory of resonances. When the resonances are distributed regularly, there is certain natural stability through the value distribution theory.

math.SP↗
arXiv2025

Recovering the Topology in One Point Interaction Problem on Extended Non-Local Star Graphs

The author studies the inverse spectral problem of Sturm-Liouville operator on a star-like graph. To this star-like graph centered at the origin as its vertex, there are attached $m$ edges that imposed the Sturm-Liouville operator with certain non-local potential functions with some suitable local boundary value conditions. At the vertex, we consider one point interaction condition at vertex to model a network that fixed on the end of the edges on the graph. The vibration and flow changes are monitored at that vertex which serves as certain control/regulation center. The author shows that the system is solvable under very necessary conditions. It is crucial to recover the topology of the network. In this paper, author constructs the special solution edge by edge and point to point.

math-ph↗
arXiv2021

Sturm-Liouville-type operators with frozen argument and Chebyshev polynomials

The paper deals with Sturm-Liouville-type operators with frozen argument of the form $\ell y:=-y''(x)+q(x)y(a),$ $y^{(α)}(0)=y^{(β)}(1)=0,$ where $α,β\in\{0,1\}$ and $a\in[0,1]$ is an arbitrary fixed rational number. Such nonlocal operators belong to the so-called loaded differential operators, which often appear in mathematical physics. We focus on the inverse problem of recovering the potential $q(x)$ from the spectrum of the operator $\ell.$ Our goal is two-fold. Firstly, we establish a deep connection between the so-called main equation of this inverse problem and Chebyshev polynomials of the first and the second kinds. This connection gives a new perspective method for solving the inverse problem. In particular, it allows one to completely describe all non-degenerate and degenerate cases, i.e. when the solution of the inverse problem is unique or not, respectively. Secondly, we give a complete and convenient description of iso-spectral potentials in the space of complex-valued integrable functions.

math.SP↗
arXiv2020

Wave Support Theorem and Inverse Resonant Uniqueness on the Line

In the paper, we experimentally study the inverse problem with the resonant scattering determinant. We analyze the structure of characteristics of perturbed linear waves. Assuming there is the common part of potential perturbation propagating along the same strips, we estimate the common part of the perturbed wave, and its Fourier transform. We deduce the partial inverse uniqueness from the Nevanlinna type of representation theorem.

math.SP↗
arXiv2020

Certain Inverse Resonance Uniqueness on the Line with Super-Exponentially Decaying Potential

In the paper, we study the inverse problem with the resonant data of fast decaying potential $V$. We review Froese' construction of the Born's approximation and Neumann series to analyze the growth of scattering determinant. Assuming all the the resonances are given, we deduce the certain inverse uniqueness on $V$ from the Nevanlinna type of representation theorem.

math.AP↗
arXiv2016

An Inverse Uniqueness of a Phaseless Scattering Problem by Zero-Crossings

We discuss the inverse uniqueness problem in phaseless scattering by counting the zeros of its modulus of the scattering amplitude. The phase linearization of scattered wave field disturbs the originally uniform distribution of the zero set. There is a connection between the perturbation of the index of refraction and the zero distribution of the modulus. We conclude the inverse uniqueness of the phaseless problem from the point of view of interior transmission problem.

math.SP↗
arXiv2016

A Proof of Schiffer's Conjecture in Starlike Domain by Far-Field Patterns

We formulate the Schiffer's conjecture in spectral geometry in the context of scattering theory. The problem is equivalent to finding a non-trivial solution in an interior transmission problem. We compare the back-scattering data of the perturbation along all incident angles. The uniqueness of the inverse scattering problem along each incident direction proves the Schiffer's conjecture.

math.AP↗
arXiv2016

Some Inverse Spectral Results in Exterior Transmission Problem

We consider an inverse spectral theory in a domain with the cavity that is bounded by a penetrable inhomogeneous medium. An ODE system is constructed piecewise through the solutions inside and outside the cavity. The ODE system is connected to the PDE system via the analytic continuation. For each scattered angle, we describe its eigenvalue density in the complex plane, and prove an inverse uniqueness on the inhomogeneity by the measurements in the far-fields.

math.AP↗
arXiv2016

A Near-Field Basis in Radially Symmetric Interior Transmission Problem

The spectrum of interior transmission problem is the zero set of certain entire functional determinant. It is classic that we deploy the series of exponential polynomials to approximate the distribution of the roots of the entire functions of exponential type. We construct an exponential system in the form of $\{e^{ik_jr}\}$ according to the set of interior transmission eigenvalues $\{k_{j}\}$. The eigenvalues are the zeros of a sine-type function. In particular, they are intersection points of two asymptotically periodic entire functions. The intersection set is asymptotically sine-like near the real axis, so we may manage to construct a basis according to the class of spectral objects. Due to the result of Paley-Wiener theorem, the zero set generates a natural duality in the form of Fourier transform associated with exponential polynomials. Whenever there is a sufficient quantity of transmission eigenvalues, we are given a series of exponential polynomials to saturate the functional density, which completes a $L^{2}$-Riesz basis in a suitable ball.

math.AP↗
arXiv2015

A Fixed Energy Fixed Angle Inverse Uniqueness in Interior Transmission Problem

We transform an inverse scattering problem to be an interior transmission problem. We find an inverse uniqueness on the scatterer with a knowledge of a fixed interior transmission eigenvalue. By examining the solution in a series of spherical harmonics at far fields, we can decide the perturbation uniquely for the radially symmetric perturbations.

math-ph↗
arXiv2015

An Inverse Uniqueness in Interior Transmission Problem and Its Eigenvalue Tunneling in Penetrable Simple Domains

We study an inverse uniqueness with a knowledge of spectral data in the interior transmission problem defined by an index of refraction in a simple domain. We expand the solution in such a domain into a series of one dimensional problems. For each one dimensional problem, we apply a value distribution theory in complex analysis to describe the eigenvalues of the system. By the orthogonality of the one dimensional system, we consider the uniqueness on the perturbation along each given incident angle.

math.AP↗
arXiv2013

An Uniqueness Result on Spherically Stratified Media in Constant Absorbing Background with Interior Transmission Eigenvalues

Given a set of transmission eigenvalues, its density function inversely determines the form of the indicator function. This is one application of the Cartwright's theory in inverse problems. We use the indicator function inversely to determine the form of the functional determinant d(z). Such an asymptotic expansion is uniquely determined while considered in constant absorbing medium.

math.AP↗