带有限个转移条件的Sturm-Liouville问题的有限谱问题
发布时间：2018-07-17 04:18
【摘要】：Sturm-Liouville问题源于描述固体热传导的数学模型.近几年,带有转移条件的Sturm-Liouville问题在数学物理领域已成为重要的研究课题.实际应用中往往出现的是区间内带有不连续条件的边界值问题,这类问题与不连续的物质性能有关系,比如热量、质量的转移,变化的物理转移问题,弦的振动问题,衍射问题等.对应区域的解的结构问题可化为系数分段连续,在区间内点具有转移条件的二阶微分算子的特征值问题.在经典Sturm-Liouville理论中,正则或奇异自伴的Sturm-Liouville问题的谱是趋于无穷大的.这个结果是建立在首项系数p和权函数w都是正的的假设之下Atkinson在他的书中有陈述：如果Sturm-Liouville问题的系数满足r=1/p,q,w∈L(J,C), Sturm-Liouville问题的特征值可能有有限个.2001年, Kong, Wu, Zettl建立了一系列对于任意的正整数n都恰有n个特征值的Sturm-Liouville问题Kong, Volkmer, Zettl把带有自伴边界的有限谱问题用矩阵表示了出来.这一系列结果表明了Sturm-Liouville司题有有限谱的事实.那么,带有有限转移条件的Sturm-Liouville问题是否也会有有限谱?本文将会证明结论是正确的.本文研究带有有限转移条件的Sturm-Liouville问题的有限谱问题,通过深入的研究得到了一些新的深刻而有趣的成果.本文分两章.第一章中我们研究了带有两个转移条件的Sturm-Liouville司题此处y = y(t),t∈J = (a,c1)∪(c1,c2)∪(c2,b), -∞ab +∞, r =1/p,q,w∈ L(J,C), L(J,C)表示在J上勒贝格可积的复值函数构成的Hilbert空间,边界条件为其中M2(C)表示复值2阶方阵,以及转移条件C1Y(C1-) +D1Y(c1+) = 0,C1,D1∈M2(R),|C1| = ρ1 0,|D| = θ1 0,C2F(c2-) + D2Y(c2+) = 0,C2,D2∈M2(R), |C2| =ρ2 0,|D2| = θ2 0,其中M2(R)表示实值2阶方阵,得到了带有两个转移条件的Sturm-Liouville问题的谱的个数,并建立了带有两个转移条件的恰有nl个特征值的Sturm-Liouville问题,同时还证实了这nl个特征值在不自伴的情况下可位于复平面的任何位置,在自伴的情况下可位于实轴的任何位置.第二章中我们考虑一般情况下的带有有限转移条件的Sturm-Liouville司题,-(py')'+qy=λwy,这里y=y(t),t∈J=(a,c1)∪(c1,c2)∪…∪(ci-1,ci)∪(ci,b),-∞ab +∞,1≤i≤n,n∈N+,r=1/p,q,w∈L(J,C),L(J,C)表示在J上勒贝格可积的复值函数构成的Hilbert空间,边界条件其中M2(C)表示复值2阶方阵,转移条件GiY(ci-)+DiY(ci+)=0,Ci,Di∈ M2(R),|Ci|=ρi0,|Di|=θ0,其中M2(R)表示实值2阶方阵,得到了带有n个转移条件的Sturm-Liouville问题谱的个数的表达式,并建立了带有n个转移条件的恰有有限个特征值的Sturm-Liouville问题,同时还证实了这有限个特征值在不自伴的情况下可位于复平面的任何位置,在自伴的情况下可位于实轴的任何位置.
[Abstract]:The Sturm-Liouville problem originates from the mathematical model of solid heat conduction. In recent years, the Sturm-Liouville problem with transition conditions has become an important research topic in the field of mathematics and physics. In practical applications, boundary value problems with discontinuous conditions are often found, which are related to discontinuous material properties, such as heat, mass transfer, changing physical transfer, string vibration. Diffraction problems, etc. The structural problems of solutions of corresponding regions can be transformed into eigenvalue problems of second-order differential operators with piecewise continuity of coefficients and transition conditions at points within an interval. In the classical Sturm-Liouville theory, the spectrum of regular or singular self-adjoint Sturm-Liouville problems tends to infinity. This result is based on the assumption that both the first coefficient p and the weight function w are positive. Atkinson has stated in his book that if the coefficients of the Sturm-Liouville problem satisfy rn 1 / pqqnw 鈭，
本文编号：2128977
[Abstract]:The Sturm-Liouville problem originates from the mathematical model of solid heat conduction. In recent years, the Sturm-Liouville problem with transition conditions has become an important research topic in the field of mathematics and physics. In practical applications, boundary value problems with discontinuous conditions are often found, which are related to discontinuous material properties, such as heat, mass transfer, changing physical transfer, string vibration. Diffraction problems, etc. The structural problems of solutions of corresponding regions can be transformed into eigenvalue problems of second-order differential operators with piecewise continuity of coefficients and transition conditions at points within an interval. In the classical Sturm-Liouville theory, the spectrum of regular or singular self-adjoint Sturm-Liouville problems tends to infinity. This result is based on the assumption that both the first coefficient p and the weight function w are positive. Atkinson has stated in his book that if the coefficients of the Sturm-Liouville problem satisfy rn 1 / pqqnw 鈭，
本文编号：2128977
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