薛丁格方程解在黎曼幾何上的分析

A note on the parabolic kernel of the Schrodinger operator

巫玟錚(Wen-Cheng Wu)

指導教授 ： 宋瓊珠

國立清華大學/理學院/數學系/碩士(2011年)

https://doi.org/10.6843/NTHU.2011.00604

摘要

In this note, we study parabolic equations of the type (△- λ^{2}q(x,t)- d/dt)u(x,t)=0 on an n-dimensional Riemannian manifold M. By studying the parabolic equation of this type, we discuss the gradient estimates and the Harnack inequality for positive solutions. In some case, we utilize the Harnack inequality to estimate the upper bound and the lower bound for positive solutions. Applications of these estimates for the equation are also discussed. Finally, we study the asymptotic behaviour for the fundamental solution of the operator △- λ^{2}q(x,t)- d/dt as λ--> ∞.

關鍵字

薛丁格方程 ； 熱方程

並列摘要

無資料

並列關鍵字

Gradient estimate ； Harnack inequality ； asympototic ； Schrodinger ； parabolic ； heat kernel ； Green's function ； eigenvalue

參考文獻

[A] ARONSON, D. G., Uniqueness of positive weak solutions of second order parabolic equations. Ann. Polon. Math., (1965), 285-303

[Az] AZENCOTT, R., Behavior of diffusion semi-groups at infinity. Bull. Soc. Math. France, 102 (1974), 193-240

[C-G] CHEEGER, J. & GROMOLL, D., The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differential Geom., 6 (1971), 119-128.

[C-G-T] CHEEGER, J., GROMOV, M. & TAYLOR, M., Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differential Geom., 17 (1983), 15-33.

[C-Y] CHENG, S. Y. & YAU, S. T., Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math., 28 (1975), 333-354.