【摘要】：Sturm-Liouville問題源于描述固體熱傳導(dǎo)的數(shù)學(xué)模型.近幾年,帶有轉(zhuǎn)移條件的Sturm-Liouville問題在數(shù)學(xué)物理領(lǐng)域已成為重要的研究課題.實際應(yīng)用中往往出現(xiàn)的是區(qū)間內(nèi)帶有不連續(xù)條件的邊界值問題,這類問題與不連續(xù)的物質(zhì)性能有關(guān)系,比如熱量、質(zhì)量的轉(zhuǎn)移,變化的物理轉(zhuǎn)移問題,弦的振動問題,衍射問題等.對應(yīng)區(qū)域的解的結(jié)構(gòu)問題可化為系數(shù)分段連續(xù),在區(qū)間內(nèi)點具有轉(zhuǎn)移條件的二階微分算子的特征值問題.在經(jīng)典Sturm-Liouville理論中,正則或奇異自伴的Sturm-Liouville問題的譜是趨于無窮大的.這個結(jié)果是建立在首項系數(shù)p和權(quán)函數(shù)w都是正的的假設(shè)之下Atkinson在他的書中有陳述：如果Sturm-Liouville問題的系數(shù)滿足r=1/p,q,w∈L(J,C), Sturm-Liouville問題的特征值可能有有限個.2001年, Kong, Wu, Zettl建立了一系列對于任意的正整數(shù)n都恰有n個特征值的Sturm-Liouville問題Kong, Volkmer, Zettl把帶有自伴邊界的有限譜問題用矩陣表示了出來.這一系列結(jié)果表明了Sturm-Liouville司題有有限譜的事實.那么,帶有有限轉(zhuǎn)移條件的Sturm-Liouville問題是否也會有有限譜?本文將會證明結(jié)論是正確的.本文研究帶有有限轉(zhuǎn)移條件的Sturm-Liouville問題的有限譜問題,通過深入的研究得到了一些新的深刻而有趣的成果.本文分兩章.第一章中我們研究了帶有兩個轉(zhuǎn)移條件的Sturm-Liouville司題此處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上勒貝格可積的復(fù)值函數(shù)構(gòu)成的Hilbert空間,邊界條件為其中M2(C)表示復(fù)值2階方陣,以及轉(zhuǎn)移條件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)表示實值2階方陣,得到了帶有兩個轉(zhuǎn)移條件的Sturm-Liouville問題的譜的個數(shù),并建立了帶有兩個轉(zhuǎn)移條件的恰有nl個特征值的Sturm-Liouville問題,同時還證實了這nl個特征值在不自伴的情況下可位于復(fù)平面的任何位置,在自伴的情況下可位于實軸的任何位置.第二章中我們考慮一般情況下的帶有有限轉(zhuǎn)移條件的Sturm-Liouville司題,-(py')'+qy=λwy,這里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上勒貝格可積的復(fù)值函數(shù)構(gòu)成的Hilbert空間,邊界條件其中M2(C)表示復(fù)值2階方陣,轉(zhuǎn)移條件GiY(ci-)+DiY(ci+)=0,Ci,Di∈ M2(R),|Ci|=ρi0,|Di|=θ0,其中M2(R)表示實值2階方陣,得到了帶有n個轉(zhuǎn)移條件的Sturm-Liouville問題譜的個數(shù)的表達(dá)式,并建立了帶有n個轉(zhuǎn)移條件的恰有有限個特征值的Sturm-Liouville問題,同時還證實了這有限個特征值在不自伴的情況下可位于復(fù)平面的任何位置,在自伴的情況下可位于實軸的任何位置.
[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，

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