邊界條件依賴譜參數(shù)的非連續(xù)Sturm—Liouville算子的譜問題

閆麗 魏廣生

摘 要：為了豐富Sturm-Liouville（S-L）微分算子的譜理論，研究了閉區(qū)間[0，1]上邊界條件依賴譜參數(shù)的非連續(xù)S-L問題。首先利用該問題在直和空間上的等價刻畫，給出了非連續(xù)S-L問題特征值與連續(xù)S-L問題特征值間的交替關(guān)系，即在非連續(xù)S-L問題的特征值的每個開子區(qū)間內(nèi)都恰有連續(xù)S-L問題的一個特征值，進而由連續(xù)S-L問題的振蕩理論推出非連續(xù)S-L問題的振蕩理論。然后通過Prüfer變換和Hergloz函數(shù)的轉(zhuǎn)換，建立了邊界條件依賴譜參數(shù)的非連續(xù)S-L問題與邊界條件為常值的非連續(xù)S-L問題的轉(zhuǎn)換，得出轉(zhuǎn)換后的特征值與轉(zhuǎn)換前（除去有限個）的特征值相等。最后通過構(gòu)造邊界條件為常值的非連續(xù)S-L問題的特征函數(shù)求得其特征值的漸近式，從而得到了邊界條件依賴譜參數(shù)的非連續(xù)S-L問題的特征值的漸近表達式。新的研究方法可推廣到對間斷點條件依賴譜參數(shù)的S-L問題研究。

關(guān)鍵詞：算子代數(shù)；Sturm-Liouville微分算子；非連續(xù)條件；參數(shù)邊界條件

中圖分類號：O175.1 MSC（2010）主題分類：47A75 文獻標志碼：A

文章編號：1008-1542（2018）04-0321-10doi：10.7535/hbkd.2018yx04005

Abstract：In order to enrich the spectral theory of Sturm-Liouvillel （S-L） differential operators， the discontinuous S-L problem with boundary conditions dependent on spectral parameters on closed interval ＼[0，1＼] is studied. Firstly， by using the equivalent characterization of the problem in the direct sum space， the alternating relation between the eigenvalues of the discontinuous S-L problem and the eigenvalues of the continuous S-L problem is given. That is， there is exactly one eigenvalue of the continuous S-L problem in every open subinterval of the eigenvalues of the discontinuous S-L problem， and then the oscillation theory of the discontinuous S-L problem is derived from the oscillation theory of the continuous S-L problem. Through the transformations of Prüfer and Hergloz function， the transformation between the discontinuous S-L problem with boundary conditions dependent spectral parameters and discontinuous S-L problem with constant boundary conditions is established. The obtained converted eigenvalues are equal to those （excluding the finite eigenvalues） before the conversion. Finally， the asymptotic expressions of eigenvalues of discontinuous S-L problems with boundary conditions dependent on spectral parameters are obtained by constructing the eigenfunctions of discontinuous S-L problems with constant boundary conditions. The new research method can be extended to the study of the S-L problem with boundary conditions dependent spectral parameters.

Keywords：operator algebras； Sturm-Liouville differential operator； discontinuity conditions； eigenparameter-dependent boundary condition

Sturm-Liouville（簡稱S-L）微分算子理論在研究許多數(shù)學(xué)物理問題中有重要的作用，其特征值問題長期以來受到物理學(xué)界和數(shù)學(xué)學(xué)界的關(guān)注。其中，非連續(xù)S-L問題基于許多物理背景和實際應(yīng)用問題，例如：中間有結(jié)點的弦振動問題[1-4]、衍射問題[5-7]、質(zhì)量轉(zhuǎn)移問題[8-10]以及薄的疊層板塊的熱傳導(dǎo)問題[11-13]；再比如地球物理中，地殼底部橫波的反射[14-16]也會導(dǎo)致相應(yīng)的S-L問題不連續(xù)，會產(chǎn)生一個跨越界面的條件，這個條件一般稱之為“界面條件”或“轉(zhuǎn)移條件”，即特征函數(shù)及其導(dǎo)數(shù)產(chǎn)生間斷點。

3 結(jié) 論

基于文獻＼[1＼]中的結(jié)論，針對非連續(xù)且邊界條件含譜參數(shù)的S-L問題（1）—問題（5）的特征值給出了精細估計， 首先利用Hergloz函數(shù)的轉(zhuǎn)換，建立了邊界條件含譜參數(shù)的S-L問題與常值邊界條件S-L問題的轉(zhuǎn)換。然后通過直和空間的等價刻畫， 證明了非連續(xù)S-L問題的特征值與連續(xù)S-L問題的特征值間的交替關(guān)系，并建立了該問題的振蕩理論。最后得到了特征值的漸近表達式。研究結(jié)果為該問題的逆問題提供了理論依據(jù)。

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