Dirichlet heat kernel on an interval ($H_L(x,y,t)$)

For the heat equation with zero Dirichlet boundary conditions on $(0,L)$,It is also obtained by the method of images, taking the difference between Gaussian sources at $y+2jL$ and $-y+2jL$. The eigenfunction expansion is useful at long times; the image expansion is useful at short times.

A transition density with absorbing boundaries counts the joint event of survival up to time $t$ and a specified final position. Its integrated mass is survival probability, not generally one. Dividing by that mass gives the endpoint density conditional on survival, a different object.

For diffusion starting at the centre of $(-L,L)$ with absorbing endpoints, only even-parity spatial eigenfunctions contribute. Integrating their cosine expansion gives the displayed series. The leading late-time term has prefactor $4/\pi$ and decay rate ${\pi }^{2}D/(4L^2)$; its first relative correction is $-\frac{1}{3}{e}^{-2{\pi }^{2}Dt/L^2}$.

For a particle with mass $m$ on $(0,L)$, put $K_0(x;\beta )=\sqrt{m/(2\pi {\hslash }^2\beta )}{e}^{-m{x}^2/(2{\hslash }^2\beta )}$. The method of images gives $K_D(q_f,q_i;\beta )={\sum }_r[K_0(q_f-q_i+2rL;\beta )-K_0(q_f+q_i+2rL;\beta )]$. The diagonal integral isThe reflected intervals tile the real line, giving the subtraction $1/2$. The Poisson summation formula converts this to ${\sum }_{n\ge 1}{e}^{-\beta {\hslash }^2{\pi }^2{n}^2/(2m{L}^2)}$, proving equality between the image and energy eigenstate calculations.