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documentation:language_reference:objects:responsefunction:start [2024/10/05 17:29] – Sina Shokridocumentation:language_reference:objects:responsefunction:start [2024/10/07 10:00] (current) – Sina Shokri
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 A more compact way to store this spectrum, as a function of $\omega$ (and $\gamma$), is by just using two arrays for the values of $ \{R_k\} $ (residues) and $ \{\omega_k\} $ (poles). This is precicely the purpose of the object //Response Function//. In other words, response functions in Quanty are functions that, given a complex number as input ($\omega + i \gamma/2$) return a complex number (single valued functions). Additionally, the output could be a matrix (matrix functions) when the response function is defined using array of matrices, instead of array of numbers. Response functions fullfil the Kramers Kronig relations and "causality". A more compact way to store this spectrum, as a function of $\omega$ (and $\gamma$), is by just using two arrays for the values of $ \{R_k\} $ (residues) and $ \{\omega_k\} $ (poles). This is precicely the purpose of the object //Response Function//. In other words, response functions in Quanty are functions that, given a complex number as input ($\omega + i \gamma/2$) return a complex number (single valued functions). Additionally, the output could be a matrix (matrix functions) when the response function is defined using array of matrices, instead of array of numbers. Response functions fullfil the Kramers Kronig relations and "causality".
  
-The response functions can also be defined by matrices such that the function is given by $ A_0 + B_0^* \frac{ 1 }{\omega - H + i \gamma / 2} B_0^{T} $, where $A_0$, $B_0$ and $H$ are matrices. $H$ can have different forms whereby only a few elements/blocks are non-zero. In Quanty, there are 4 different forms (types) for the matrix $H$:+The response functions can also be defined by matrices such that the function is given by  
 +$$ A_0 + B_0^* \frac{ 1 }{\omega - H + i \gamma / 2} \Bigg|_{[0,0]} B_0^{T} $$  
 +where $A_0$, $B_0$ and $H$ are matrices. Any unitary transformation that leaves the [0,0] element of $H$ unchanged results in the same spectrum. Hence $H$ can have different forms, whereby only a few elements/blocks are non-zero. In Quanty, there are 4 different forms (types) for the matrix $H$:
   * list of poles (//ListOfPoles//)   * list of poles (//ListOfPoles//)
   * tri-diagonal (//Tri//)   * tri-diagonal (//Tri//)

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