具有转移条件的奇异Sturm-Liouville算子
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摘要
本文围绕一类带有转移条件的奇异Sturm-Liouville(S-L)算子展开研究.为方便我们研究此类奇异S-L算子的自共轭性,首先我们对于已有的具有转移条件正则S-L算子自共轭性的相关结果进行了综述,包括带有一个不连续点和多个不连续点的正则S-L算子.在此基础上,我们应用研究奇异S-L算子经典的Weyl圆套的办法,给出了带有转移条件的奇异S-L算子的Weyl圆的方程、圆心、半径以及m(λ)在圆内的充分必要条件,进而得到了Weyl圆套.我们得到圆套Cb在b→∞时的极限是圆或点与我们所研究方程的解在空间H中平方可积的个数的关系,进而给出了带有转移条件奇异S-L算子的极限点型与极限圆型的定义,并给出其亏指数与点、圆型以及方程解的个数的等价关系.为了研究算子的自共轭域,我们给出了与带有转移条件的奇异S-L算子相关联的算子最大算子域和最小算子域的刻画,结合著名的GKN定理,分别在极限圆型与极限点型两种不同的情况下,给出了具有转移条件的奇异S-L算子自共轭域的解析描述.
In this paper,we study a class of singular Sturm-Liouville(S-L) operators with transmission conditions. In order to study the self-adjointness of these singular S-L operators, firstly, we introduced the self-adjointness of regular S-L operators with transmission conditions, including the operators with one discontinuous point or finite discon-tinuous points. On this basis, applying the classical theory of Weyl circle sets Cb, we give the equation, center and radius of the Weyl circle and establish the necessary and sufficient conditions for the function m which in the Weyl circle, and obtained the set of Weyl circle. We give the relationship between the number of square-integrable solutions in H and the limit of Cb which is a circle or a point when b→∞. We give the definition of the limit-point and limit-circle of the singular S-L operators with transmission conditions, and give equivalence relations of the deficiency index, limit-point (limit-circle), and the number of the square-integrable solutions in H. In order to study the self-adjointness of these operators, we give the definition of the maximal and minimal domains associated with transmission conditions. Combining of the well-known GKN theorem, we give analytic description of self-adjoin domain of singular Sturm-Liouville operator with transmission condi-tions in the cases of the limit-circle(limit-point) respectively.
In this paper,we study a class of singular Sturm-Liouville(S-L) operators with transmission conditions. In order to study the self-adjointness of these singular S-L operators, firstly, we introduced the self-adjointness of regular S-L operators with transmission conditions, including the operators with one discontinuous point or finite discon-tinuous points. On this basis, applying the classical theory of Weyl circle sets Cb, we give the equation, center and radius of the Weyl circle and establish the necessary and sufficient conditions for the function m which in the Weyl circle, and obtained the set of Weyl circle. We give the relationship between the number of square-integrable solutions in H and the limit of Cb which is a circle or a point when b→∞. We give the definition of the limit-point and limit-circle of the singular S-L operators with transmission conditions, and give equivalence relations of the deficiency index, limit-point (limit-circle), and the number of the square-integrable solutions in H. In order to study the self-adjointness of these operators, we give the definition of the maximal and minimal domains associated with transmission conditions. Combining of the well-known GKN theorem, we give analytic description of self-adjoin domain of singular Sturm-Liouville operator with transmission condi-tions in the cases of the limit-circle(limit-point) respectively.
引文
[1]Bailey P. B., Sturm-Liouville eigenvalues via a phase function. J.SIAM. Appl. Math.,1966,14:242-249.
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[3]Bailey P. B., Everitt W.N., Zettl A., Computation eigenvalues of singular Sturm-Liouville problems, Results in Mathematics,1999,20:391-423.
[4]Bailey P. B., Everitt W.N., Zettl A., The SLEIGN2 Sturm-Liouville codes. ACM. Trans. Math. Software,2001,21:143-192.
[5]曹之江,常微分算子,上海：上海科技出版社,1987.
[6]郝晓玲,孙炯,Regular Sturm-Liouville operators with Transmission Conditions at finite interior discontinuous points. Journal of Mathematical Sciences:Ad-vances and Applications(已接收)
[7]Likow A. V., Mikhailov YU. A., The theory of heat and mass transfer, Qosen-ergaizdat,1963 (Russian).
[8]刘景麟,常微分算子谱论,科学出版社,2009.
[9]Mukhtarov 0. Sh., Yakubov S., Problems of differential equations with trans-mission conditions, Applicable Analysis.2002,81:1033-1064.
[10]Naimark M. A., Linear differential operators, English Transl. Ungar, New York, 1968.
[11]孙炯,王忠,线性算子的谱分析,科学出版社,2005.
[12]Tikhonov A.N., Samarskii A.A., Equations of mathematical physics, Oxofrd and New York, Pergamon,1963.
[13]Titchmarsh E. C., Eigenfunction expansions associated with second order dif-ferential equations, Oxford, Part Ⅰ,1946; Part Ⅱ,1962.
[14]Titeux I., Yakubov Ya., Completeneess of root functions for thermal conduction in a strip with piecewise continuous coefficients, Math. Models Methods Appl. Sc.,1997,7(7):1035-1050.
[15]Voitovich N. N., Katsenelbaum B.Z.,Sivov A.N.,Generalized method of eigenvi-bration in the theory of diffraction,Nauka,Moskov,1997(Russian).
[16]王爱平,关于Weidmann猜想及具有转移条件微分算子的研究(博士学位论文),内蒙古大学,2006.
[17]王桂霞,Sturm-Liouville问题的谱分析与数值计算(博士学位论文),内蒙古大学,2008.
[18]Weyl H., Uber gewohnliche differentialgleichungen mit singularitaten und die zugehorigen entwicklungen willkurlicher funktionen, Math. Ann.,1910,68:220-269.
[2]Bailey P. B., Gordon M.K., Shampine L. F., Automatic solution of the Sturm-Liouville problems, ACM. TOMS.1978,4:193-208.
[3]Bailey P. B., Everitt W.N., Zettl A., Computation eigenvalues of singular Sturm-Liouville problems, Results in Mathematics,1999,20:391-423.
[4]Bailey P. B., Everitt W.N., Zettl A., The SLEIGN2 Sturm-Liouville codes. ACM. Trans. Math. Software,2001,21:143-192.
[5]曹之江,常微分算子,上海：上海科技出版社,1987.
[6]郝晓玲,孙炯,Regular Sturm-Liouville operators with Transmission Conditions at finite interior discontinuous points. Journal of Mathematical Sciences:Ad-vances and Applications(已接收)
[7]Likow A. V., Mikhailov YU. A., The theory of heat and mass transfer, Qosen-ergaizdat,1963 (Russian).
[8]刘景麟,常微分算子谱论,科学出版社,2009.
[9]Mukhtarov 0. Sh., Yakubov S., Problems of differential equations with trans-mission conditions, Applicable Analysis.2002,81:1033-1064.
[10]Naimark M. A., Linear differential operators, English Transl. Ungar, New York, 1968.
[11]孙炯,王忠,线性算子的谱分析,科学出版社,2005.
[12]Tikhonov A.N., Samarskii A.A., Equations of mathematical physics, Oxofrd and New York, Pergamon,1963.
[13]Titchmarsh E. C., Eigenfunction expansions associated with second order dif-ferential equations, Oxford, Part Ⅰ,1946; Part Ⅱ,1962.
[14]Titeux I., Yakubov Ya., Completeneess of root functions for thermal conduction in a strip with piecewise continuous coefficients, Math. Models Methods Appl. Sc.,1997,7(7):1035-1050.
[15]Voitovich N. N., Katsenelbaum B.Z.,Sivov A.N.,Generalized method of eigenvi-bration in the theory of diffraction,Nauka,Moskov,1997(Russian).
[16]王爱平,关于Weidmann猜想及具有转移条件微分算子的研究(博士学位论文),内蒙古大学,2006.
[17]王桂霞,Sturm-Liouville问题的谱分析与数值计算(博士学位论文),内蒙古大学,2008.
[18]Weyl H., Uber gewohnliche differentialgleichungen mit singularitaten und die zugehorigen entwicklungen willkurlicher funktionen, Math. Ann.,1910,68:220-269.