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The Dirichlet difference Laplacian has an orthogonal sine basis with eigenvalues $-{\eta }_j/h^2$, where ${\eta }_j=4{\sin }^2(j\pi /(2M))$ for $j=1,\dots ,M-1$. On a mode the theta method amplification isIf $a\ge 1/2$, the denominator is positive andThus $|G(s)|\le 1$ for every mode and every $\mu >0$. Orthogonality gives a uniform discrete L2 norm contraction, not merely individual eigenvalue estimates.

If $a<0$, a positive $s=-1/a$ makes the implicit matrix singular. If $a=0$, $G=1-s$ becomes arbitrarily large. If $0<a<1/2$, $G(s)$ tends to $-(1-a)/a$, whose magnitude is greater than one. Hence unconditional stability holds exactly for $a\ge 1/2$.

For completeness, $a<1/2$ is contractive under $\mu {\eta }_{\max }\le 2/(1-2a)$; the mesh-independent sufficient restriction is $\mu \le 1/[2(1-2a)]$. This follows from the same squared-modulus difference and keeps the denominator positive even for negative $a$. At $a=1/2$, very high frequencies approach amplification $-1$, so A-stability does not imply damping of rough heat data. At $a=1$ the Backward Euler method damps them strongly; this is the rough-data damping distinction for the theta method.