柯西过程的谱性质
Spectral properties of the Cauchy process
摘要：
本文研究了半直线（0，infty）和区间（-1，1）上killed一维Cauchy过程的转移半群的谱性质。这个过程与一维Laplacian A = -sqrt（-d^2/dx ^2）的平方根和Dirichlet外部条件（在区域的补上）有关，并且与半平面中的混合Steklov问题有关。对于半直线，导出了A的广义本征函数psi_lambda的显式表达式，并利用它构造了A的谱表示.给出了半直线上被杀死的柯西过程的跃迁密度（或A在（0，infty）中的热核）和从半直线上第一次离开时间的分布的显式公式。psi_lambda的公式也用于构造区间A的本征函数的近似。对于区间A的特征值λ n，导出了渐近公式λ n = npi/2 - pi/8 + O（1/n），并证明了所有特征值λ n都是简单的.最后，有效的数值方法估计的特征值λ_n的应用，以获得下和上的数值界的前几个特征值的第9个小数点。
英文摘要：
We study the spectral properties of the transition semigroup of the killed one-dimensional Cauchy process on the half-line (0,infty) and the interval (-1,1). This process is related to the square root of one-dimensional Laplacian A = -sqrt(-d^2/dx^2) with a Dirichlet exterior condition (on a complement of a domain), and to a mixed Steklov problem in the half-plane. For the half-line, an explicit formula for generalized eigenfunctions psi_lambda of A is derived, and then used to construct spectral representation of A. Explicit formulas for the transition density of the killed Cauchy process in the half-line (or the heat kernel of A in (0,infty)), and for the distribution of the first exit time from the half-line follow. The formula for psi_lambda is also used to construct approximations to eigenfunctions of A in the interval. For the eigenvalues lambda_n of A in the interval the asymptotic formula lambda_n = n pi/2 - pi/8 + O(1/n) is derived, and all eigenvalues lambda_n are proved to be simple. Finally, efficient numerical methods of estimation of eigenvalues lambda_n are applied to obtain lower and upper numerical bounds for the first few eigenvalues up to 9th decimal point.
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