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On $(0,L)$, the differential operator $\mathcal{L}={\partial }_x^2+\beta {\partial }_x=e^{-\beta x}{\partial }_x(e^{\beta x}{\partial }_x)$ with homogeneous Neumann boundary conditions is self-adjoint in the weighted inner product with weight $e^{\beta x}$. Its eigenfunctions and nonnegative decay rates areTheir squared norms are $N_0=(e^{\beta L}-1)/\beta$ and $N_k=L[1+{\beta }^2/(4q_k^2)]/2$. The Sturm-Liouville eigenfunction expansion givesThis kernel acts against the measure $e^{\beta y}dy$, not Lebesgue measure alone. It is symmetric in $x,y$; the transition density against Lebesgue measure is $e^{\beta y}{K}_{\beta }(x,y,t)$.

Let $\mu >0$ and let $u$ be a real smooth solution on a bounded regular domain with zero Dirichlet boundary conditions. Green's first identity givesThe two sides have opposite signs unless both vanish, so $u=0$. This energy method also proves uniqueness for prescribed boundary data by applying it to the difference of two solutions. For complex solutions use the complex conjugate in the integral. This argument does not apply with the same sign to the Helmholtz equation, which can have nonzero zero-boundary eigenfunctions.