Negative-energy direction gives exponential growth in a self-adjoint wave equation ($\Vert q(t)\Vert \ge \Vert q_0\Vert e^{\alpha t}$)

Let $\ddot{q}=Fq$ on a Hilbert space, with time-independent self-adjoint operator $F$, and conserved quadratic energy $E=\Vert \dot{q}{\Vert }^2/2-⟨q,Fq⟩/2$. If $⟨q_0,F{q}_0⟩>0$, set $\dot{q}(0)=\alpha q_0$ with ${\alpha }^2=⟨q_0,F{q}_0⟩/\Vert q_0{\Vert }^2$ and $\alpha >0$. This gives $E=0$. For $N=\Vert q{\Vert }^2$, $N^{\prime }=2\mathrm{Re}⟨q,\dot{q}⟩$ and $N^{\prime \prime }=4\Vert \dot{q}{\Vert }^2$, soby the Cauchy-Schwarz inequality. Since $(\ln N{)}^{\prime }(0)=2\alpha$, $N(t)\ge N(0)e^{2\alpha t}$. Positivity of $N$ follows from $N^{\prime }(0)>0$ and $N^{\prime \prime }\ge 0$. Thus a negative direction of the restoring energy $W=-⟨q,Fq⟩/2$ yields an exponentially growing solution without a normal-mode expansion or a variational spectral theorem. If $W$ is positive definite, conserved $E$ instead controls the energy norm; a bound in the original Hilbert space norm additionally follows from a coercive lower bound on $W$.