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Chuan-Fu Yang

Publications and source records attributed to Chuan-Fu Yang.

At least 19 recordsLinked to original sources
arXiv2025

Weak separability and partial Fermi isospectrality of discrete periodic Schr\"odinger operators

In this paper, we consider the discrete periodic Schr\"odinger operators $\Delta+V$ on $\Z^d$, where $V$ is $\Gamma$-periodic with $\Gamma=q_1 \mathbb{Z}\oplus q_2\mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$ and positive integers $q_j$, $j=1,2,\cdots,d,$ are pairwise coprime. We introduce the notions of generalized partial Fermi isospectrality and weak separability, and prove that two generalized partially Fermi isospectral potentials have the same weak separability. As a direct application, we can prove that two potentials have the same $(d_1,d_2,\cdots,d_r)$-separability by assuming that they are generalized partially Fermi isospectral, instead of the Fermi isospectrality or Floquet isospectrality. Besides, we prove that each couples of components of the generalized Fermi isospectral potentials are Floquet isospectral in some sense.

math.SP↗
arXiv2025

On the inverse transmission eigenvalue problem with a piecewise $W_2^1$ refractive index

In this paper, we consider the inverse spectral problem of determining the spherically symmetric refractive index in a bounded spherical region of radius $b$. Instead of the usual case of the refractive index $\rho\in W^2_2$, by using singular Sturm-Liouville theory, we {first} discuss the case when the refractive index $\rho$ is a piecewise $ W^1_2$ function. We prove that if $\int_0^b \sqrt{\rho(r)} dr<b$, then $\rho$ is uniquely determined by all special transmission eigenvalues; if $\int_0^b \sqrt{\rho(r)} dr=b$, then all special transmission eigenvalues with some additional information can uniquely determine $\rho$. We also consider the mixed spectral problem and obtain that $\rho$ is uniquely determined from partial information of $\rho$ and the ``almost real subspectrum".

math.SP↗
arXiv2025

Borg-type theorem for a class of higher-order differential operators

In this paper, we study an inverse spectral operator for the higher-order differential equation $(-1)^my^{(2m)}+ q y = \lambda y$, where $q \in L^2(0,\pi)$. We prove that if $\|q\|_2$ is sufficiently small, the two spectra corresponding to the both Dirichlet boundary conditions and to the Dirichlet-Neumann ones uniquely determine the potential $q$. The result extends the Borg theorem from the second order to all even higher orders.

math.SP↗
arXiv2024

Borg-type theorem for a class of fourth-order differential operators

In this paper, we study an inverse spectral problem for the fourth-order differential equation $y^{(4)} - (p y')' + q y = \lambda y$ with real-valued coefficients $p$ and $q$ of $L^2(0,1)$. We prove that, for near-constant coefficients, the two spectra corresponding to the Dirichlet and the Dirichlet-Neumann boundary conditions uniquely determine either $p$ or $q$. The result extends the Borg theorem of the second-order case to the fourth-order case.

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↗
arXiv2024

A class of higher order inverse spectral problems

In this paper, we consider the recovery of third-order differential operators from two spectra, as well as fourth-order or fifth-order differential operators from three spectra, where these differential operators are endowed with complex-valued distributional coefficients. For the case of multiple spectra, we first establish the relationship between spectra and the Weyl-Yurko matrix. Secondly, we prove the uniqueness theorem for the solution of the inverse problems. Our approach allows us to obtain results for the general case of complex-valued distributional coefficients.

math.SP↗
arXiv2023

Inverse spectral problem for the Schr\"odinger operator on the square lattice

We consider an inverse spectral problem on a quantum graph associated with the square lattice. Assuming that the potentials on the edges are compactly supported and symmetric, we show that the Dirichlet-to-Neumann map for a boundary value problem on a finite part of the graph uniquely determines the potentials. We obtain a reconstruction procedure, which is based on the reduction of the differential Schr\"odinger operator to a discrete one. As a corollary of the main results, it is proved that the S-matrix for all energies in any given open set in the continuous spectrum uniquely specifies the potentials on the square lattice.

math-ph↗
arXiv2023

Ambarzumyan-type theorem for the Sturm-Liouville operator on the lasso graph

We consider the Sturm-Liouville operator on the lasso graph with a segment and a loop joined at one point, which has arbitrary length. The Ambarzumyan's theorem for the operator is proved, which says that if the eigenvalues of the operator coincide with those of the zero potential, then the potential is zero.

math.SP↗
arXiv2023

Incomplete inverse problem for Dirac operator with constant delay

In this work, we consider Dirac-type operators with a constant delay less than two-fifths of the interval and not less than one-third of the interval. For our considered Dirac-type operators, an incomplete inverse spectral problem is studied. Specifically, when two complex potentials are known a priori on a certain subinterval, reconstruction of the two potentials on the entire interval is studied from complete spectra of two boundary value problems with one common boundary condition. The uniqueness of the solution of the inverse problem is proved. A constructive method is developed for the solution of the inverse problem.

math.SP↗
arXiv2023

Inverse problems for Dirac operators with a constant delay less than half of the interval

In this work, we consider Dirac-type operators with a constant delay less than half of the interval and not less than two-fifths of the interval. For our considered Dirac-type operators, two inverse spectral problems are studied. Specifically, reconstruction of two complex $L_{2}$-potentials is studied from complete spectra of two boundary value problems with one common Dirichlet boundary condition and Neumann boundary condition, respectively. We give answers to the full range of questions usually raised in the inverse spectral theory. That is, we give uniqueness, necessary and sufficient conditions of the solvability, reconstruction algorithm and uniform stability for our considered inverse problems.

math.SP↗
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

Local solvability and stability of inverse problems for Sturm-Liouville operators with a discontinuity

Partial inverse problems are studied for Sturm-Liouville operators with a discontinuity. The main results of the paper are local solvability and stability of the considered inverse problems. Our approach is based on a constructive algorithm for solving the inverse problems. The key role in our method is played by the Riesz-basis property of a special vector-functional system in a Hilbert space. In addition, we obtain a new uniqueness theorem for recovering the potential on a part of the interval, by using a fractional part of the spectrum.

math.SP↗
arXiv2019

Reconstruction and Solvability for Discontinuous Hochstadt-Lieberman Problems

We consider Sturm-Liouville problems with a discontinuity in an interior point, which are motivated by the inverse problems for the torsional modes of the Earth. We assume that the potential on the right half-interval and the coefficient in the right boundary condition are given. Half-inverse problems are studied, that consist in recovering the potential on the left half-interval and the left boundary condition from the eigenvalues. If the discontinuity belongs to the left half-interval, the position and the parameters of the discontinuity also can be reconstructed. In this paper, we provide reconstructing algorithms and prove existence of solutions for the considered inverse problems. Our approach is based on interpolation of entire functions.

math.SP↗
arXiv2017

A partial inverse problem for the Sturm-Liouville operator on the graph with a loop

The Sturm-Liouville operator with singular potentials on the lasso graph is considered. We suppose that the potential is known a priori on the boundary edge, and recover the potential on the loop from a part of the spectrum and some additional data. We prove the uniqueness theorem and provide a constructive algorithm for the solution of this partial inverse problem.

math.SP↗
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↗