本研究是有關微分算子的譜理論。我的主要目的是從不完整的譜集合和節點訊息 去重構微分算子的係數函數。 我們計劃制訂和發展必要的數學方法，以獲取廣 這類問題的廣泛的解決方案，並且也期望獲得一些演算法算。除此之外，我們 也將要將的主要注意與後效微分算子，離散和連續。此外，我們也會進一步研 究具有後續效應二階微分方程和較複雜的高階微分方程並且研究節點反問題和 譜反問題間的關連。 。 主要研究步驟如下： - 將不完整資訊的反問題的分類，尋找對應問題的特徵函數和相關資訊 - 問題可解性的研究和退化情形的研究; - 譜和結點的解析性質研究, 包括漸進行為研究；譜集合和節點資訊一致性的條 件找到一致性條件; - 獲得唯一性定理 - 提供一個可構造出解的方法; - 問題可解性的充要條件研究； - 研究穩定性問題; - 數值計算方法的發展 。 The proposed project is related to the spectral theory of differential operators. The goal of the project is the investigation of nonlinear inverse problems of recovering coefficients of differential operators from incomplete spectral and nodal information. We plan to work out necessary mathematical methods, to obtain solutions for a wide class of incomplete inverse spectral and inverse nodal problems and to develop algorithms for their solutions. We will pay the main attention to differential operators with aftereffect, both discrete and continuous. Moreover, we will study both second-order differential equations and more complicated higher-order differential equations. We plan to investigate connections between inverse nodal and inverse spectral problems. The main stages of the study are: - to classify incomplete inverse problems, to introduce the nodal and spectral characteristics needed for formulations and solutions of the inverse problems; - to obtain information conditions for incomplete inverse problems, to describe degenerate cases of loss of information; - to investigate analytic, asymptotic and structural properties of the nodal and spectral characteristics; to find concordance conditions with given components of operators; - to obtain uniqueness theorems, to indicate nodal and spectral characteristics which give us information on necessary measurements for the solution of the incomplete inverse problems; - to provide constructive procedures for the solutions of incomplete inverse problems for differential equations; - to establish necessary and sufficient conditions for the solvability of this class of nonlinear inverse problems, and to describe the corresponding classes of spectral and nodal characteristics; - to study the stability of the solutions of the incomplete inverse problems; - to develop numerical methods and to conduct numerical experiments.
