关于微分运算特征值的渐近分布

On the Asymtotic Distribution of Eigenvalues of Linear Differential Operators.

本文讨论了微分方程, 在下列边界条件下的特征值分布问题。 当v固定时,系数α_(vj)不全是零,β_(vj)也不全是零。 方程式(1)中P_2(x),P_3(x),…P_n(x)在[0,1]连续,得到下列结果:当n为奇数时则其特征值的分布为式中ω_μ为x^n+1=0的—个根,a_0/b_0为一常数,(m_1-m_2)为固定的整数,k为任意充分大的整数。 当n为偶数时则特征值分布有下列两种情况可能出现。式中(?),ω_(μ+1)表示x^n+1=0,的根,m_4,m_1表示固定整数,a_0/b_0为一常数,k为充分大的整数。

in his book 'Linear differential operators', has discussed the asymtotlcdistribution of eigenvalues of linear differential operators of the so called normal boundary con-ditions and the broken down boundary conditions in the sense ofLet(?) the linear differential eguation which satisfies the following boundary conditions:in which y_0^((j)), y_1^((j))express y^((j))(0), y^((j))(1). By normal boundary conditions we mean that By broken down conditons we mean thatBoth (3)and (4)are narrow conditions on the coefficients, In this paper, a broader condition hasbeen discussed, the only restriction upon the coefficients in the formula (2) being that when v isfixed, not all α_(vj) egual zero. and also not all β_(vj) egual zero. This is called non-degenerated con-dition and includes the normal condition. Under this condition the following result has beencstablished.Theorem. Let the differential equation beof which the coefficients p2(x), p3(x),…pn(x)are continuous on the interval[0, 1], and satisfythe following boundary conditions:(for fixed V not all α_(vj) equal zero, also not all β_(vj) equal zero). When n is an odd number, theeigenvalues of(5)can be expressed in the following form:where a_0, b_0 are constants depending upon the coefficients α_(vj), β_(vj), and m1, m2 are positiveintegers ≤n(n-1), k is any sufficiently large integers and ω_μ is a root of x^n+1=0,When n is an even number, the following two cases may occur:where a_0, b_0, k have the same meaning as above, and m_1, m_4 are positive integers≤n(n-1).