## Test note

$$
\sum_i^N K(x_i)M^i
$$

Inline math?: $\sum_i^N K(x_i) M^i$

# Large Mass Harmonic Oscillator on Eucledian time

Info:
* Eucledian time points: 3
* N = 80
* Matrix size of N^3
* Action: $S = \frac12 \sum_i \Delta_\tau \sigma x_i^2 $
* $\sigma = 1$
* Runtime ~10 minutes
* Fokker-Planck equation
* Should be positive eigenvalues of the equation:
* $F_{\textrm{FP}} P(\textbf{x};\tau) = -\lambda P(\textbf{x};\tau)$

## Eigenvalues:

| k | $||Ax-kx||/||kx||$ |
| :-------------: | :-------------: |
| -1.843422+0.001579i | 0.998479 |
| -2.616751-0.107733i | 0.965833 |
| -2.729703+0.136284i | 1.01945 |
| -3.640527+0.073839i | 1.03712 |
| -3.781522-1.430333i | 0.762849 |
| -3.894783+1.512824i | 0.745834 |
| -5.352922-2.580163i | 0.538589 |
| -5.508457+2.678970i | 0.437366 |
| -7.712095+3.674526i | 0.42868 |
| -10.555826-4.278824i | 0.434935 |
| -10.950730+4.342183i | 0.438883 |
-19.321643-0.247089i | 0.817175 |
| -45.441350+0.332406i | 0.5352 |
| -80.623264+0.052920i | 0.376649 |
| -110.389972-0.384503i | 0.273537 |
| -146.793279+0.052263i | 0.168638 |
| -173.062205+0.048124i | 0.0891449 |
| -186.049725+0.026397i | 0.0361323 |
---------------------- --------------------

## Ground state
Under is the ground state for the 3 eucledian time points
| $x_0$ | $x_1$ | $x_2$ |
|:-------------:|:-------------:|:-----:|
| |  |  |

## First exited states
Under is a plot of the first 3 exited states in euledian time $x_0 = 0$.
| $1$ | $2$ | $3$ |
| ------------- |:-------------:|:-----:|
| |  |  |