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(jià)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(gè)開子區(qū)間內(nèi)都恰有連續(xù)S-L問題的一個(gè)特征值，進(jìn)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(gè)）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(dá)式。新的研究方法可推廣到對間斷點(diǎn)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(xiàn)標(biāo)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(shí)際應(yīng)用問題，例如：中間有結(jié)點(diǎn)的弦振動(dòng)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(gè)跨越界面的條件，這個(gè)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(diǎn)。

3 結(jié) 論

基于文獻(xiàn)＼[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(xì)估計(jì)， 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(jià)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(dá)式。研究結(jié)果為該問題的逆問題提供了理論依據(jù)。

參考文獻(xiàn)/References：

[1] BENEDEK A I， PANZONE R. On Sturm-Liouville problems with the square root of the eigenvalue parameter conditions contained in the boundary conditions[J]. Notas Algebra Analysis， 1981， 10： 1-62.

[2] BINDING P A， HRYNIV R， LANGER H，et al. Elliptic eigenvalue problems with eigenparameter dependent boundary conditions[J]. J Differential Equations， 2001， 174： 30-54.

[3] DIJKSMA A. Eigenfunction expansions for a class of J-selfadjoint ordinary differential operators with boundary conditions containing the eigenvalue parameter[J]. Proc Roy Soc Edinburgh Sect A， 1980， 86： 1-27.

[4] EBERHARD W， FREILING G， SCHNEIDER A. Eigenfunction expansion for a regular fourth order eigenvalue problem with eigenvalue parameter in the boundary condition[J]. Int J Math Math Sci， 1992， 15： 809-811.

[5] FULTON C T. Singular eigenvalue problems with eigenvalue-parameter contained in the boundary conditions[J]. Proc Roy Soc Edinburgh Sect A， 1980， A 87： 1-34.

[6] HINTON D B， SHAW J K. Differential operators with spectral parameter incompletely in the boundary conditions[J]. Funkcial Ekvac， 1990， 33： 363-385.

[7] RUSSAKOVSKII E M. The matrix Sturm-Liouville problem with spectral parameter in the boundary conditions[J]. Algebraic and operator aspects， Trans Moscow Math Soc， 1996， 57： 159-184.

[8] HINTON D B. Eigenfunction expansions for a singular eigenvalue problem with eigenparameter in the boundary conditions[J]. SIAM J Math Anal， 1981， 12： 572-584.

[9] KOZHEVNIKOV A， YAKUBOV S. On operators generated by elliptic boundary problems with a spectral parameter in boundary conditions[J]. Integral Equations Operator Theory， 1995， 23： 205-231.

[10]RUSSAKOVSKII E M. Operator treatment of boundary problems with spectral parameters entering via ploynomials in the boundary conditions[J]. Funct Anal Appl， 1975， 9： 358-359.

[11]SHKALIKOV A A. Boundary problems for ordinary differential equations with parameter in the boundary conditions[J]. J Soviet Math， 1986， 33： 1311-1342.

[12]TRETTER C. On lambda-nonlinear boundary eigenvalue problems[J]. Mathematics Research Akademie， 1993， 71：1208-1216.

[13]ZAYED E M E， IBRAHIM S F. An expansion theorem for an eigenvalue problem on an arbitrary conditions[J]. Acta Math Sin （Engl Ser）， 1995， 11： 399-407.

[14]WEI G， XU H K. Inverse spectral problem with partial information given on the potential and norming constants [J]. Trans Amer Math Soc， 2012， 364： 3265-3288.

[15]PIVOVARCHIK V N. An inverse Sturm-Liouville problem by three spectra[J]. Integral Equ Oper Theory， 1999， 34： 234-243.

[16]GESZTESY F， SIMON B. Inverse spectral analysis with partial information on the potential. II. The case of discrete sprctrum [J]. Trans Amer Math Soc， 2000， 35： 2765-2787.

[17]江衛(wèi)華，郭彥平，王斌.二階微分方程組邊值問題2個(gè)正解的存在性[J].河北科技大學(xué)學(xué)報(bào)，2006，27（1）：15-17.

JIANG Weihua， GUO Yanping， WANG Bin. Existence of two positive solutions to boundary value problems of second order systems[J]. Journal of Hebei University of Science and Technology， 2006，27（1）：15-17.

[18]郭彥平，苗素榮，禹長龍.無窮區(qū)間上二階三點(diǎn)差分方程邊值問題正解的存在性[J].河北科技大學(xué)學(xué)報(bào)，2016，37（6）：556-561.

GUO Yanping， MIAO Surong， YU Changlong. Existence of positive solutions to boundary value problem of second-order three-point difference equations on infinite intervals[J]. Journal of Hebei University of Science and Technology， 2016，37（6）：556-561.

[19]WEI G， XU H K. Inverse spectral problem for a string equation with partial information [J]. Inverse Problems， 2010， 26： 115004.

[20]GESZTESY F， SIMON B. On the determination of a potential from three spectra[J]. Amer Math Soc Transl， 1997， 2： 18-26.

[21]GESZTESY F， SIMON B. Inverse spectral analysis with partial information on the potential， Ⅰ. The case of an a.c. component in the sprctrum[J]. Helv Phys Acta， 1997， 70： 66-71.

[22] BINDING P A， BROWNE P J， WATSON B A. The Sturm-Liouville problem with the discontinuity conditions at an interior point and boundary conditions depending on the eigenparameter[J]. Proc Edinb Math Soc， 2002， 45： 631-645.

[23] 傅守忠， 王忠， 魏廣生. Sturm-Liouville問題及其逆問題[M]. 北京： 科學(xué)出版社， 2015.