Lung-Hui Chen

The wave travels through the heterogeneity, and the wave-form distorts accordingly. We always compare the dynamics between the before and after wave-forms to retrieve the information inside the heterogeneity. Let us see if a layer of known material places outside the heterogeneity could help to find out more information inside. In this paper we consider the inverse scattering problem in one-dimensional setting. The scattering matrix consists of 2 × 2 entries of meromorphic functions, and the analysis focuses on the zeros and poles of the scattering determinant. We consider the dynamic behavior of scattering entries in transition matrix system. We try to derive the inverse uniqueness from the strength/modulus of scattered wave-fields. Given the strength or the modulus of scattered wave-field over the outside region a priori, it is feasible to deduce the information on perturbed potential inside the region with partial information from the exterior layer. The inverse problem is to find certain information on the internal heterogeneity of certain materials and media with knowledge on certain exterior measurement.

In this paper, we investigate the inverse spectral problem of the Sturm–Liouville operator with two frozen arguments fixed at the points x = a ₁ , a ₂ in (0, π ). We will discuss the non‐uniqueness with certain prior information provided on the potential function , and its support affects the uniqueness or non‐uniqueness results when studying this type of inverse spectral problem. We start the discussion with a certain analysis of the corresponding characteristic function of perturbation , and find that counting the zeros of these entire functions plays a role. This is an ill‐posed inverse spectral problem.

In recent years, there appeared a growing interest in the inverse spectral theory for functional‐differential operators with frozen argument. Such operators are nonlocal and belong to the so‐called loaded differential operators, which frequently appear in mathematics as well as natural sciences and engineering. However, to various classes of nonlocal operators, classical methods of the inverse spectral theory are not applicable. For this reason, it is relevant to develop new methods and approaches for solving inverse problems for operators of this type. We establish a deep connection between the inverse problem for operators with frozen argument and Chebyshev polynomials of the first and the second kinds. Appearing to be of an interest from the point of view of the Chebyshev polynomials theory itself, this connection gives a new perspective method for studying inverse problems for operators with frozen argument. In particular, it allows one to completely describe all non‐degenerate and degenerate cases, that is, when the corresponding inverse problem is uniquely solvable or not, respectively. Moreover, it gives a convenient description of all isospectral potentials in the degenerate case, which is demonstrated by some illustrative examples.

In this paper, we discuss how to partially determine the Fourier transform F ⁢ ( z ) = ∫ - 1 1 f ⁢ ( t ) ⁢ e i ⁢ z ⁢ t ⁢ 𝑑 t , z ∈ ℂ , F(z)=\int_{-1}^{1}f(t)e^{izt}\,dt,\quad z\in\mathbb{C}, given the data | F ⁢ ( z ) | {\lvert F(z)\rvert} or arg ⁡ F ⁢ ( z ) {\arg F(z)} for z ∈ ℝ {z\in\mathbb{R}} . Initially, we assume [ - 1 , 1 ] {[-1,1]} to be the convex hull of the support of the signal f. We start with reviewing the computation of the indicator function and indicator diagram of a finite-typed complex-valued entire function, and then connect to the spectral invariant of F ⁢ ( z ) {F(z)} . Then we focus to derive the unimodular part of the entire function up to certain non-uniqueness. We elaborate on the translation of the signal including the non-uniqueness associates of the Fourier transform. We show that the phase retrieval and magnitude retrieval are conjugate problems in the scattering theory of waves.