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In $\mathbb{R}^3$, consider the time-dependent (non-rel) Schrodinger equation with the potential energy $V(\mathbb{x})$. When a small change(e.g., just a small constant $\delta>0$) of V(x) is given, how can I calculate the time-dependent error $\epsilon(t)=\int_{\mathbb{R}^3}|\Psi_t(\mathbb{x})-\Psi_t'(\mathbb{x})|^2$, if the initial conditions are the same ($\Psi_0(\mathbb{x})=\Psi_0'(\mathbb{x})$)? Is there a general method to find $\epsilon(t)$? I have seen some cases where$\epsilon(t) = O(t)$.

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If you add a constant to potential, you will get the same wave function so the error is 0. Do you mean that the variation $V(x)$ at any $x$ position is at most $\delta$? –  hwlau Dec 18 '12 at 0:56
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