where δ(x-y) is the Dirac delta function.
The heat kernel can be represented in terms of the eigenfunctions and eigenvalues of the Laplace operator. Specifically, if φn is an orthonormal basis of eigenfunctions of Δ with corresponding eigenvalues λn, then the heat kernel can be written as: heat kernels and spectral theory pdf
These properties make the heat kernel a powerful tool for solving the heat equation and other related partial differential equations. where δ(x-y) is the Dirac delta function
K(t,x,y)=∑j=0∞e−λjtϕj(x)ϕj(y)cap K open paren t comma x comma y close paren equals sum from j equals 0 to infinity of e raised to the negative lambda sub j t power phi sub j open paren x close paren phi sub j open paren y close paren ϕjphi sub j Core Concepts Covered Heat Kernel
𝜕u𝜕t=Δupartial u over partial t end-fraction equals delta u Δcap delta is the Laplacian operator. Physically, represents the temperature at point if a unit of heat was concentrated at point Key properties include: , reflecting that heat spreads and cannot be negative.
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This book is a foundational resource for understanding the interplay between the heat equation and the spectral properties of elliptic operators. Core Concepts Covered Heat Kernel