Regularization-schemes-of-the-pairing-field.md

@@ -56,4 +56,12 @@ where $`K=2.442 75`$ is a numerical constant. In this formula, we assume that al
 # Custom renormalization scheme
 Static codes allow for defining your own renormalization scheme. You need to provide the formula in `void modify_potentials(...)` function. See [here](Strict 2D or 1D mode) for example. 
 
-# Regularization scheme and the energy conservation in td calculations
+# Regularization scheme and the energy conservation in td calculations
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+In publication [arXiv:1606.02225](https://arxiv.org/abs/1606.02225) it was pointed that TDBdG like equations, formally conserve energy only if all quasiparticle states are evolved, see discussion of Eqs.(25)-(26). This situation corresponds to the cubic cutoff. If the space is truncated eg.  by introducing a spherical cutoff at some initial time then in general energy maybe not conserved.  
+
+In practical applications, we observe that the energy when applied the spherical regularization scheme is conserved only with some accuracy, which is not related to the integrator accuracy. Below we provide an example of (3d calculation), where for the time interval $`te_F<170`$ we apply an external time-dependent potential (we pump energy into the system), and for $`te_F>170`$ the system evolves without any external perturbation. 
+![wiki1](uploads/8ef4aba0e9c8051ef9f3361adc4b2e76/wiki1.png)
+![wiki2](uploads/3c18f9d061adcea98d658125b498a1ad/wiki2.png)
+It is clearly visible, that for evolution with the cubic cutoff the energy is conserved up to high accuracy, while for spherical cutoff the quality of the energy conservation is significantly lower.  
+
+In conclusion, we find that typically for trajectories of length $`te_F\approx1000`$ the spherical cutoff provides reasonable accuracy, while for generation of long trajectories $`te_F\gg 1000`$ it is recommended to use the cubic cutoff. 
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