两区间微分算子自伴域的实参数解刻画及谱的离散性
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摘要
本文上要围绕两区间上微分算子自伴域的刻画及几类微分算子谱的离散性展开研究.多年来带转移条件的Sturm-Liouville司题一直受到很多数学、物理工作者的关注,而具有转移条件的问题也可以理解成两区间上问题的一种特殊情形,即两个区间相邻,重合端点处的左右边界条件构成了转移条件.在这一思想的启发下,1986年,Everitt-Zettl在Hilbert空间的直和框架下研究了两区间上二阶Sturm-Liouville问题的自伴实现理论.然而两个区间上的问题不能简单的看成是各自区间上算子的直和,更有趣的、也是更重要的问题是两个区间之间存在某种程度上的关联.因止Everitt-Zettl在文[9]中研究了两区间理论.由于自伴算子的谱是实的,用实参数解刻画自伴域不仅易于找到显式的解更重要的是会产生与谱相关的信息.
本文在Hilbert空间的直和框架下,利用微分方程的实参数解首先给出一端正则一端奇异的两区间上微分算子自伴域的完全刻画.在直和空间中构造自伴算子的一种简单方式就是取每个空间中自伴算子的直和.如果这样得到的自伴算子就是由两区间上微分方程生成的所有自伴算子,那么我们就没有必要建立“两区间理论”.事实上,正如文[9]中所提到的,有许多自伴算子并不只是每个区间上自伴算子的直和,这些“新的”自伴算子涉及到在这两个区间之间的相互作用.这些相互作用可能‘穿过’正则点,也可能‘穿过’奇异点.其中正则自伴相互作用包含解或其拟导数的跳跃,奇异自伴相互作用包含解的拉格朗日括号的跳跃.
接着我们又给出两端奇异的两区间上最小算子的所有自伴扩张的一个显式刻画.这些扩张产生的“新的”自伴算子不只是每个区间上自伴算子的直和也涉及到两个区间之间的相互作用.这样的相互作用是奇异端点之间的相互作用,在内部奇异点处这些相互作用包含了解的拉格朗日括号的不连续跳跃.该结果同样适用于一个端点是正则的或多个端点是正则的情形.
进一步地,在一个新的带有适当乘数参数的Hilbert空间框架下,我们研究了两个偶数阶实系数微分方程的所有两区间自伴实现理论,给出了一端正则一端奇异的两区间最小算子的所有自伴扩张的描述,这些参数与边界条件相互作用产生的偶数阶自伴性问题使得与其关联的实耦合系数矩阵K更具一般性.
在这个新的带有适当乘数参数的Hilbert空间框架下我们又研究了区间端点都是奇异的情形,给出了两端奇异的两区间上偶数阶实系数微分方程的所有两区间自伴实现的描述,得到的自伴边界条件使得与其关联的实耦合系数矩阵K更具一般性.该结果同样适用于一个端点是正则的或多个端点是正则的情形.
文章还研究了一类四阶正则Sturm-Liouville问题的特征值对问题的依赖性.我们得到特征值不仅连续而且光滑依赖于该问题,同时我们还证明了特征值是所有参数(区间端点、边界条件、方程系数和权函数)的可微函数,并且给出了特征值关于给定参数的微分表达式.特征值与特征函数对参数的连续依赖性除了其理论的重要性,对数值计算而言也是十分重要的.
最后我们研究了几类微分算子谱的离散性,首先研究了一类偶数阶自伴微分算子的谱,当微分算子的系数ak(x)由ex的乘方所控制,该微分算子与具有指数系数的对称微分算子相比较,从而得出其谱是离散的结论；进一步当x→∞时,微分算子的系数ak(x)可能随着ex的乘方增大而增大,我们又给出其谱是离散的充分与必要条件.其次研究了一类具指数系数的对称微分算式生成的自伴微分算子的谱,我们得到该类微分算子的系数满足一定条件时,末项系数按照一定的方式趋于无穷大时,其谱是离散的结论.进一步得到不仅末项系数按照一定的方式趋于无穷大时可以决定此类微分算子谱的离散性,而且,中间项和首项系数按照一定的方式趋于无穷大时也可以决定此类微分算子谱的离散性.最后,我们还研究了一类具指数系数的自伴微分算子本质谱的存在范围.
本文共分八章,第一章绪论,介绍本文所研究问题的背景及本文的主要结果；第二章是文中所涉及相关符号,概念以及性质；第三章研究了一端奇异两区间微分算子自伴域的刻画；第四章研究了两端奇异两区间微分算子自伴域的刻画；第五章研究了含内积倍数的一端奇异两区间微分算子自伴域的刻画；第六章研究了含内积倍数的两端奇异两区间微分算子自伴域的刻画；第七章研究了一类四阶正则Sturm-Liouville问题的特征值；第八章研究了几类自伴微分算子谱的离散性.
In this paper, we study the characterization of self-adjoint domains of two-interval differential operators and the discreteness of spectrum of several ordinary differential operators. Over the years, Sturm-Liouville problems with transmission conditions are concerned by many mathematical and physical researchers. While the problem with trans-mission conditions can be understood as special case of two-interval problem, i.e. with the coincide endpoint of two adjacent intervals whose left and right boundary conditions yield the transmission conditions. Partly motivated by this idea, in1986, Everitt-Zettl devel-oped a theory of self-adjoint realizations of Sturm-Liouville problems on two intervals in the direct sum of Hilbert spaces associated with these intervals. However the two-interval problem which are not merely the sum of self-adjoint operators from each of the separate intervals. It is interesting and important to observe that they involve interactions between the two intervals. Therefore in [9] the authors develop a "two-interval" theory. Since the spectrum of self-adjoint operators is real, the advantage of using real-parameter solutions to characterize self-adjoint domains is not only because it is, in general, easier to find explicit solutions but, more importantly, it yields information about the spectrum.
In this paper, first of all, we give an complete characterization of all self-adjoint domains of differential operators on two intervals in the direct sum of Hilbert spaces in terms of real-parameter solutions. These two intervals with one endpoint of each interval is singular. A simple way of getting self-adjoint operators in a direct sum Hilbert space is to take the direct sum of self-adjoint operators from each of the separate Hilbert spaces. If these were all the self-adjoint operator realization from the two intervals there would be no need for a "two interval" theory. In fact, as noted in [9], there are many self-adjoint operators which are not merely the sum of self-adjoint operators from each of the separate intervals. These "new" self-adjoint operators involve interactions between the two intervals. These interactions may be'through'regular or singular endpoints. The regular self-adjoint interactions can be visualized as jumps of the solutions or their quasi- derivatives and the singular interactions can be described as jumps of Lagrange brackets involving the solutions.
Then we also give an explicit characterization of all self-adjoint extensions of the two-interval minimal operator in terms of real-parameter solutions of the two intervals. These two intervals with all four endpoints are singular. These extensions yield'new' self-adjoint operators which are not merely direct sums of self-adjoint operators from the two intervals but involve interactions between the two intervals. These interactions are the interactions between singular endpoints. At singular interior points these interactions involve jump discontinuities of the Lagrange bracket of solutions. This result reduces to the case when one or two or three or four endpoints are regular.
Furthermore, we study all self-adjoint two-interval realizations of the two equations of even order with real valued coefficients using Hilbert spaces but with the usual inner products replaced by appropriate multiples. we give an characterization of all self-adjoint extensions of the two-interval minimal operator with one endpoint of each interval is singular. The interplay of these multiples with the boundary conditions generates self-adjoint problems of even order with real coupling matrices K which are much more general than the coupling matrices from the'unmodified'theory.
We also study the case when the four endpoints are singular in the modified direct sum Hilbert space with different inner product multiples. we give the characterization of all self-adjoint two-interval realizations of the two equations of even order with real valued coefficients. The self-adjoint boundary conditions obtained by us associated with real coupling matrices K are much more general. This result reduces to the case when one or two or three or four endpoints are regular.
And we also study the eigenvalues of a regular fourth-order Sturm-Liouville problems depends on the problem. We observe that the eigenvalues of the regular fourth-order S-L problems depend not only continuously but smoothly on the problem. An expression for the derivative of the eigenvalues with respect to a given parameter:an endpoint, a boundary condition, a coefficient, or the weight function, are found. Besides its theoretical importance, the continuous dependence of the eigenvalues and the eigenfunctions on the data is fundamental from the numerical point of view.
Finally, we study the discreteness of spectrum of several ordinary differential opera-tors. First, we study the spectrum of a class of self-adjoint differential operators of even order. When the coefficients ak(x) of the differential operators are restricted by powers of ex, we give a sufficient condition on the coefficients to ensure that the spectrum of the differential operators is discrete; And we formulate necessary and sufficient conditions for the discreteness of the spectrum of the differential operators whose coefficients ak(x) may increase as powers of ex as x→∞. Second, the spectrum of self-adjoint differential operators of a class of symmetric differential expressions with exponential coefficients are considered. The last term coefficient which tends to infinity according to a certain way when the coefficients satisfying certain conditions. A sufficient condition is given which ensure that the spectrum is discrete. We further find that the discreteness of the spectrum of such differential operators not only determined by the last term coefficient which tends to infinity according to a certain way, Moreover, the middle term and the leading coeffi-cient which tends to infinity according to a certain way also can decide the discreteness of the spectrum. Finally, we give the range of essential spectrum of a class of self-adjoint differential operators with exponential coefficients.
This thesis consists of eight chapters. In chapter1, we introduce the background about the problems what we study and the main results of this thesis; Chapter2is related symbols, concepts and properties involved in this thesis; Chapter3study the characterization of self-adjoint domains of differential operators on two intervals with one endpoint is singular of each interval; Chapter4study the characterization of self-adjoint domains of differential operators on two intervals with the endpoints all are singular; Chapter5study the characterization of self-adjoint domains of differential operators on two intervals with inner product multiples and one endpoint is singular of each interval; Chapter6study the characterization of self-adjoint domains of differential operators on two intervals with inner product multiples and the endpoints all are singular; Chapter7study the eigenvalues of a class of regular fourth-order Sturm-Liouville problems; Chapter8study the discreteness of the spectrum of several self-adjoint differential operators.
本文在Hilbert空间的直和框架下,利用微分方程的实参数解首先给出一端正则一端奇异的两区间上微分算子自伴域的完全刻画.在直和空间中构造自伴算子的一种简单方式就是取每个空间中自伴算子的直和.如果这样得到的自伴算子就是由两区间上微分方程生成的所有自伴算子,那么我们就没有必要建立“两区间理论”.事实上,正如文[9]中所提到的,有许多自伴算子并不只是每个区间上自伴算子的直和,这些“新的”自伴算子涉及到在这两个区间之间的相互作用.这些相互作用可能‘穿过’正则点,也可能‘穿过’奇异点.其中正则自伴相互作用包含解或其拟导数的跳跃,奇异自伴相互作用包含解的拉格朗日括号的跳跃.
接着我们又给出两端奇异的两区间上最小算子的所有自伴扩张的一个显式刻画.这些扩张产生的“新的”自伴算子不只是每个区间上自伴算子的直和也涉及到两个区间之间的相互作用.这样的相互作用是奇异端点之间的相互作用,在内部奇异点处这些相互作用包含了解的拉格朗日括号的不连续跳跃.该结果同样适用于一个端点是正则的或多个端点是正则的情形.
进一步地,在一个新的带有适当乘数参数的Hilbert空间框架下,我们研究了两个偶数阶实系数微分方程的所有两区间自伴实现理论,给出了一端正则一端奇异的两区间最小算子的所有自伴扩张的描述,这些参数与边界条件相互作用产生的偶数阶自伴性问题使得与其关联的实耦合系数矩阵K更具一般性.
在这个新的带有适当乘数参数的Hilbert空间框架下我们又研究了区间端点都是奇异的情形,给出了两端奇异的两区间上偶数阶实系数微分方程的所有两区间自伴实现的描述,得到的自伴边界条件使得与其关联的实耦合系数矩阵K更具一般性.该结果同样适用于一个端点是正则的或多个端点是正则的情形.
文章还研究了一类四阶正则Sturm-Liouville问题的特征值对问题的依赖性.我们得到特征值不仅连续而且光滑依赖于该问题,同时我们还证明了特征值是所有参数(区间端点、边界条件、方程系数和权函数)的可微函数,并且给出了特征值关于给定参数的微分表达式.特征值与特征函数对参数的连续依赖性除了其理论的重要性,对数值计算而言也是十分重要的.
最后我们研究了几类微分算子谱的离散性,首先研究了一类偶数阶自伴微分算子的谱,当微分算子的系数ak(x)由ex的乘方所控制,该微分算子与具有指数系数的对称微分算子相比较,从而得出其谱是离散的结论；进一步当x→∞时,微分算子的系数ak(x)可能随着ex的乘方增大而增大,我们又给出其谱是离散的充分与必要条件.其次研究了一类具指数系数的对称微分算式生成的自伴微分算子的谱,我们得到该类微分算子的系数满足一定条件时,末项系数按照一定的方式趋于无穷大时,其谱是离散的结论.进一步得到不仅末项系数按照一定的方式趋于无穷大时可以决定此类微分算子谱的离散性,而且,中间项和首项系数按照一定的方式趋于无穷大时也可以决定此类微分算子谱的离散性.最后,我们还研究了一类具指数系数的自伴微分算子本质谱的存在范围.
本文共分八章,第一章绪论,介绍本文所研究问题的背景及本文的主要结果；第二章是文中所涉及相关符号,概念以及性质；第三章研究了一端奇异两区间微分算子自伴域的刻画；第四章研究了两端奇异两区间微分算子自伴域的刻画；第五章研究了含内积倍数的一端奇异两区间微分算子自伴域的刻画；第六章研究了含内积倍数的两端奇异两区间微分算子自伴域的刻画；第七章研究了一类四阶正则Sturm-Liouville问题的特征值；第八章研究了几类自伴微分算子谱的离散性.
In this paper, we study the characterization of self-adjoint domains of two-interval differential operators and the discreteness of spectrum of several ordinary differential operators. Over the years, Sturm-Liouville problems with transmission conditions are concerned by many mathematical and physical researchers. While the problem with trans-mission conditions can be understood as special case of two-interval problem, i.e. with the coincide endpoint of two adjacent intervals whose left and right boundary conditions yield the transmission conditions. Partly motivated by this idea, in1986, Everitt-Zettl devel-oped a theory of self-adjoint realizations of Sturm-Liouville problems on two intervals in the direct sum of Hilbert spaces associated with these intervals. However the two-interval problem which are not merely the sum of self-adjoint operators from each of the separate intervals. It is interesting and important to observe that they involve interactions between the two intervals. Therefore in [9] the authors develop a "two-interval" theory. Since the spectrum of self-adjoint operators is real, the advantage of using real-parameter solutions to characterize self-adjoint domains is not only because it is, in general, easier to find explicit solutions but, more importantly, it yields information about the spectrum.
In this paper, first of all, we give an complete characterization of all self-adjoint domains of differential operators on two intervals in the direct sum of Hilbert spaces in terms of real-parameter solutions. These two intervals with one endpoint of each interval is singular. A simple way of getting self-adjoint operators in a direct sum Hilbert space is to take the direct sum of self-adjoint operators from each of the separate Hilbert spaces. If these were all the self-adjoint operator realization from the two intervals there would be no need for a "two interval" theory. In fact, as noted in [9], there are many self-adjoint operators which are not merely the sum of self-adjoint operators from each of the separate intervals. These "new" self-adjoint operators involve interactions between the two intervals. These interactions may be'through'regular or singular endpoints. The regular self-adjoint interactions can be visualized as jumps of the solutions or their quasi- derivatives and the singular interactions can be described as jumps of Lagrange brackets involving the solutions.
Then we also give an explicit characterization of all self-adjoint extensions of the two-interval minimal operator in terms of real-parameter solutions of the two intervals. These two intervals with all four endpoints are singular. These extensions yield'new' self-adjoint operators which are not merely direct sums of self-adjoint operators from the two intervals but involve interactions between the two intervals. These interactions are the interactions between singular endpoints. At singular interior points these interactions involve jump discontinuities of the Lagrange bracket of solutions. This result reduces to the case when one or two or three or four endpoints are regular.
Furthermore, we study all self-adjoint two-interval realizations of the two equations of even order with real valued coefficients using Hilbert spaces but with the usual inner products replaced by appropriate multiples. we give an characterization of all self-adjoint extensions of the two-interval minimal operator with one endpoint of each interval is singular. The interplay of these multiples with the boundary conditions generates self-adjoint problems of even order with real coupling matrices K which are much more general than the coupling matrices from the'unmodified'theory.
We also study the case when the four endpoints are singular in the modified direct sum Hilbert space with different inner product multiples. we give the characterization of all self-adjoint two-interval realizations of the two equations of even order with real valued coefficients. The self-adjoint boundary conditions obtained by us associated with real coupling matrices K are much more general. This result reduces to the case when one or two or three or four endpoints are regular.
And we also study the eigenvalues of a regular fourth-order Sturm-Liouville problems depends on the problem. We observe that the eigenvalues of the regular fourth-order S-L problems depend not only continuously but smoothly on the problem. An expression for the derivative of the eigenvalues with respect to a given parameter:an endpoint, a boundary condition, a coefficient, or the weight function, are found. Besides its theoretical importance, the continuous dependence of the eigenvalues and the eigenfunctions on the data is fundamental from the numerical point of view.
Finally, we study the discreteness of spectrum of several ordinary differential opera-tors. First, we study the spectrum of a class of self-adjoint differential operators of even order. When the coefficients ak(x) of the differential operators are restricted by powers of ex, we give a sufficient condition on the coefficients to ensure that the spectrum of the differential operators is discrete; And we formulate necessary and sufficient conditions for the discreteness of the spectrum of the differential operators whose coefficients ak(x) may increase as powers of ex as x→∞. Second, the spectrum of self-adjoint differential operators of a class of symmetric differential expressions with exponential coefficients are considered. The last term coefficient which tends to infinity according to a certain way when the coefficients satisfying certain conditions. A sufficient condition is given which ensure that the spectrum is discrete. We further find that the discreteness of the spectrum of such differential operators not only determined by the last term coefficient which tends to infinity according to a certain way, Moreover, the middle term and the leading coeffi-cient which tends to infinity according to a certain way also can decide the discreteness of the spectrum. Finally, we give the range of essential spectrum of a class of self-adjoint differential operators with exponential coefficients.
This thesis consists of eight chapters. In chapter1, we introduce the background about the problems what we study and the main results of this thesis; Chapter2is related symbols, concepts and properties involved in this thesis; Chapter3study the characterization of self-adjoint domains of differential operators on two intervals with one endpoint is singular of each interval; Chapter4study the characterization of self-adjoint domains of differential operators on two intervals with the endpoints all are singular; Chapter5study the characterization of self-adjoint domains of differential operators on two intervals with inner product multiples and one endpoint is singular of each interval; Chapter6study the characterization of self-adjoint domains of differential operators on two intervals with inner product multiples and the endpoints all are singular; Chapter7study the eigenvalues of a class of regular fourth-order Sturm-Liouville problems; Chapter8study the discreteness of the spectrum of several self-adjoint differential operators.
引文
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[70]王忠.一类自伴微分算子谱的离散性,数学学报,2001,44(1)：95-102.
[71]王忠.高阶对称微分算子谱是离散的一个充分必要条件,系统科学与数学,2000,20(2)：224-227.
[72]王忠.复系数2n阶微分算子的谱,数学学报,2000,43(5)：787-796.
[73]郭古宽,孙炯.一类具指数系数的对称微分算子的谱及亏指数,系统科学与数学,2003,23(2)：182-189.
[74]杨传富,黄振友,杨孝平.偶阶非对称微分算子离散谱准则,数学进展,2007,36(2)：160-168.
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[82]Hao X.L.Sun J.and Zettl A.,Real-parameter square-integrable solutions and the spectrum of differential operators,J.Math.Anal.Appl.,2011,376:696-712.
[83]Makin A.s.,On the spectrum of the Sturm-Liouville operator with degenerate boundary conditions, Dokl.Akad.Nauk,2011,437(2),154-157(Russian).
[84]王忠,孙炯Euler微分算子谱是离散的充分必要条件,系统科学与数学,2001,21(4)：497-506.
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[88]曹之江,孙炯,Edmunds D.E.,On self-adjointness of the product of two 2-nd order differential operators,Acta.Math.Sincia,English Series,1999,15(3):375-386.
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