= Spectral shift in an adjoint weak formulation
{title2=$B^*[u,v]-\lambda(u,v)=(f,v)$}
The weak form of $(L^*-\lambda)u=f$ includes the same <scalar> shift as the <partial differential equation>[differential equation]. The unshifted identity $B^*[u,v]=(f,v)$ instead represents $L^*u=f$ and cannot imply the shifted equation for nonzero $\lambda$ unless $u=0$. On a smooth <bounded> <domain> choose any nonzero compactly supported smooth $u$ and take $L=-\Delta+1$, $f=Lu$ for a direct counterexample. For constant drift coefficients, <integration by parts> reverses the drift sign in $L^*$ without changing this spectral term.
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