Integration-scheme.md

@@ -19,19 +19,17 @@ and $`\varphi_n`$ is shorthand notation for quasiparticle wavefunctions (`qpwfs`
     v_{n,\downarrow}(\bm{r},t)
   \end{pmatrix}
 ```
-Note that all `qpwfs` fluctuate in time with a typical oscillating factor $`\exp(-i E_n t)`$, where $`E_n`$ is a quasi-particle energy. Since we evolve all states with quasi-particle energies from  
-the interval $`[-E_c,E_c]`$, the evolved states exhibit both very slow as well as very rapid oscillations in time.
-In order to improve accuracy of the time integration, in each integration step we subtract from the quasi-particle solution the typical frequency of oscillations, which in practice modifies 
+Note that all `qpwfs` fluctuate in time with a typical oscillating factor $`\exp(-i E_n t)`$, where $`E_n`$ is a quasi-particle energy. Since we evolve all states with quasi-particle energies from the interval $`[-E_c,E_c]`$, the evolved states exhibit both very slow as well as very rapid oscillations in time. In order to improve accuracy of the time integration, in each integration step we subtract from the quasi-particle solution the typical frequency of oscillations, which in practice modifies 
 the equation to the form:  
 ```math
-i \dot{\varphi_n} = (\hat{H}(\{\varphi_n\},t)-\langle H\rangle_n)\varphi_n
+i \dot{\varphi_n}(t) = (\hat{H}(\{\varphi_n\},t)-\langle H\rangle_n(t))\varphi_n(t)
 ```
-where $`\langle H\rangle_n`$ is ``instantaneous'' quasi-particle energy. Although this step introduces a significant numeric cost to the calculations, it greatly improves the accuracy of the time integration. 
+where $`\langle H\rangle_n`$ is _instantaneous_ quasi-particle energy. Although this step introduces a significant numeric cost to the calculations, it greatly improves the accuracy of the time integration. 
 
 # ABM integration algorithm
 We convert time-dependent equations into from:
 ```math
- \dfrac{dy(t)}{dt} = f(y,t)
+ \dot{y}(t) = f(y,t)
 ```
 where:
 ```math