I have an hermitian operator $$ h = -\frac{1}{2} \frac{\partial^2}{\partial x^2} + v(x) $$ acting on square integrable functions that live in some subspace $V$ of $R$.
If I want a computer to calculate the action of this operator on a known function $f$ I would do the following:
- discretize $V$ into the points $x_i$ such that $\Delta x = x_{i+1} -x_i$ is small.
- calculate and store $f(x_i)$ for all discrete points $x_i$.
- calculate the action of $v(x)$ via $$v(x)f(x) \quad\longrightarrow\quad v(x_i)f(x_i) $$
- calculate the action of $-\frac{1}{2} \frac{\partial^2}{\partial x^2}$ via $$ -\frac{1}{2} \left(\frac{\partial^2}{\partial x^2}f\right)(x) \quad\longrightarrow\quad -\frac 1 2 \frac{f(x_{i+2}) - 2f(x_{i+1}) + f(x_{i})}{\Delta x^2} $$
However, how can I calculate the action of the operator $$ \text {exp}({\text i h t}) = \text{exp}\left(\text i \left[ -\frac{1}{2} \frac{\partial^2}{\partial x^2} + v(x) \right] t \right)\;,\quad t\in R $$ on a known function $f$ on a computer?
I know that the standard procedure is to diagonalize $h$ (i.e. find the eigenvalues $\varepsilon_k$ and eigenfunctions $\varphi_k$ of $h$). Then I could write $$ h = \sum_k \varepsilon_k \, \varphi_k \, \varphi^*_k $$ and $$ f(x) = \sum_k c_k \, \varphi_k(x) $$ such that $$ \text{exp}(\text i h t)f(x) = \sum_k c_k \, \text{exp}(\varepsilon_k) \, \varphi_k(x) $$
But I want to avoid the diagonalization since it is expensive.
I also didn't need to diagonalize $\frac{\partial^2}{\partial x^2}$ to let it act on a function!
So is there a clever way to avoid the diagonalization of $h$ in order to let $\text{exp}(\text i h t)$ act on $f$?