# som.spectral_stats: Statistical analysis of noisy spectral functions¶

Functions in this module implement the statistical analysis technique for ensembles of spectral functions described in Sections I-II of [GMPPS2017].

som.spectral_stats.spectral_integral()

Evaluate spectral integral

$i_m^{(j)} = \int_{-\infty}^\infty dz \bar K(m, z) A^{(j)}(z)$

for a single energy interval.

Parameters:

Z_m: float, Center of the energy interval. float, Length of the energy interval. som.Configuration, Spectral function $$A^{(j)}(z)$$. str, Name of the resolution function $$\bar K(m, z)$$, one of rectangle, lorentzian, gaussian.

Returns: float, Value of the integral.

Signature(float z_m, float delta_m, configuration c, resolution_function r_func) -> float

Evaluate spectral integral

$i_m^{(j)} = \int_{-\infty}^\infty dz \bar K(m, z) A^{(j)}(z)$

for a single energy interval.

Parameters:

z_m: float, Center of the energy interval. float, Length of the energy interval. som.Configuration, Spectral function $$A^{(j)}(z)$$. str, Name of the resolution function $$\bar K(m, z)$$, one of rectangle, lorentzian, gaussian.

Returns: float, Value of the integral.

Signature(triqs::mesh::refreq mesh, configuration c, resolution_function r_func) -> vector<double>

Evaluate spectral integrals

$i_m^{(j)} = \int_{-\infty}^\infty dz \bar K(m, z) A^{(j)}(z)$

for energy intervals centered around points of a regular energy mesh.

Parameters:

mesh: triqs.gf.meshes.MeshReFreq, Real energy mesh. som.Configuration, Spectral function $$A^{(j)}(z)$$. str, Name of the resolution function $$\bar K(m, z)$$, one of rectangle, lorentzian, gaussian.

Returns: Real 1D NumPy array of values $$i_m^{(j)}$$.

Signature(list[std::pair<double,double>] intervals, configuration c, resolution_function r_func) -> vector<double>

Evaluate spectral integrals

$i_m^{(j)} = \int_{-\infty}^\infty dz \bar K(m, z) A^{(j)}(z)$

for a list of real energy intervals.

Parameters:

intervals: list [float, float], List of pairs (left interval boundary, right interval boundary). som.Configuration, Spectral function $$A^{(j)}(z)$$. str, Name of the resolution function $$\bar K(m, z)$$, one of rectangle, lorentzian, gaussian.

Returns: Real 1D NumPy array of values $$i_m^{(j)}$$.

som.spectral_stats.spectral_avg()

Compute spectral averages over a set of accumulated particular solutions

$i_m = \frac{1}{J} \sum_{j=1}^J i_m^{(j)}$

for energy intervals centered around points of a regular energy mesh.

Parameters:

Cont: Som, Analytic continuation object. int, Index of the diagonal matrix element of the observable used to construct cont. triqs.gf.meshes.MeshReFreq, Real energy mesh. str, Name of the resolution function $$\bar K(m, z)$$, one of rectangle, lorentzian, gaussian.

Returns: Real 1D NumPy array of values $$i_m$$.

Signature(som_core cont, int i, triqs::mesh::refreq mesh, resolution_function r_func) -> vector<double>

Compute spectral averages over a set of accumulated particular solutions

$i_m = \frac{1}{J} \sum_{j=1}^J i_m^{(j)}$

for energy intervals centered around points of a regular energy mesh.

Parameters:

cont: Som, Analytic continuation object. int, Index of the diagonal matrix element of the observable used to construct cont. triqs.gf.meshes.MeshReFreq, Real energy mesh. str, Name of the resolution function $$\bar K(m, z)$$, one of rectangle, lorentzian, gaussian.

Returns: Real 1D NumPy array of values $$i_m$$.

Signature(som_core cont, int i, list[std::pair<double,double>] intervals, resolution_function r_func) -> vector<double>

Compute spectral averages over a set of accumulated particular solutions

$i_m = \frac{1}{J} \sum_{j=1}^J i_m^{(j)}$

for a list of real energy intervals.

Parameters:

cont: Som, Analytic continuation object. int, Index of the diagonal matrix element of the observable used to construct cont. list [float, float], List of pairs (left interval boundary, right interval boundary). str, Name of the resolution function $$\bar K(m, z)$$, one of rectangle, lorentzian, gaussian.

Returns: Real 1D NumPy array of values $$i_m$$.

som.spectral_stats.spectral_disp()

Compute spectral dispersions of a set of accumulated particular solutions

$\sigma^2_m = \frac{1}{J} \sum_{j=1}^J (i_m^{(j)} - i_m)^2$

for energy intervals centered around points of a regular energy mesh.

Parameters:

Cont: Som, Analytic continuation object. int, Index of the diagonal matrix element of the observable used to construct cont. triqs.gf.meshes.MeshReFreq, Real energy mesh. Real 1D NumPy array of precomputed averages $$i_m$$. str, Name of the resolution function $$\bar K(m, z)$$, one of rectangle, lorentzian, gaussian.

Returns: Real 1D NumPy array of values $$\sigma_m$$.

Signature(som_core cont, int i, triqs::mesh::refreq mesh, vector<double> avg, resolution_function r_func) -> vector<double>

Compute spectral dispersions of a set of accumulated particular solutions

$\sigma^2_m = \frac{1}{J} \sum_{j=1}^J (i_m^{(j)} - i_m)^2$

for energy intervals centered around points of a regular energy mesh.

Parameters:

cont: Som, Analytic continuation object. int, Index of the diagonal matrix element of the observable used to construct cont. triqs.gf.meshes.MeshReFreq, Real energy mesh. Real 1D NumPy array of precomputed averages $$i_m$$. str, Name of the resolution function $$\bar K(m, z)$$, one of rectangle, lorentzian, gaussian.

Returns: Real 1D NumPy array of values $$\sigma_m$$.

Signature(som_core cont, int i, list[std::pair<double,double>] intervals, vector<double> avg, resolution_function r_func) -> vector<double>

Compute spectral dispersions of a set of accumulated particular solutions

$\sigma^2_m = \frac{1}{J} \sum_{j=1}^J (i_m^{(j)} - i_m)^2$

for a list of real energy intervals.

Parameters:

cont: Som, Analytic continuation object. int, Index of the diagonal matrix element of the observable used to construct cont. list [float, float], List of pairs (left interval boundary, right interval boundary). Real 1D NumPy array of precomputed averages $$i_m$$. str, Name of the resolution function $$\bar K(m, z)$$, one of rectangle, lorentzian, gaussian.

Returns: Real 1D NumPy array of values $$\sigma_m$$.

som.spectral_stats.spectral_corr()

Compute spectral two-point correlators of a set of accumulated particular solutions

$\sigma_{mm'} = \frac{1}{J} \sum_{j=1}^J (i_m^{(j)} - i_m) (i_{m'}^{(j)} - i_{m'})$

for energy intervals centered around points of a regular energy mesh.

Parameters:

Cont: Som, Analytic continuation object. int, Index of the diagonal matrix element of the observable used to construct cont. triqs.gf.meshes.MeshReFreq, Real energy mesh. Real 1D NumPy array of precomputed averages $$i_m$$. str, Name of the resolution function $$\bar K(m, z)$$, one of rectangle, lorentzian, gaussian.

Returns: Real 2D NumPy array of values $$\sigma_{mm'}$$.

Signature(som_core cont, int i, triqs::mesh::refreq mesh, vector<double> avg, resolution_function r_func) -> matrix<double>

Compute spectral two-point correlators of a set of accumulated particular solutions

$\sigma_{mm'} = \frac{1}{J} \sum_{j=1}^J (i_m^{(j)} - i_m) (i_{m'}^{(j)} - i_{m'})$

for energy intervals centered around points of a regular energy mesh.

Parameters:

cont: Som, Analytic continuation object. int, Index of the diagonal matrix element of the observable used to construct cont. triqs.gf.meshes.MeshReFreq, Real energy mesh. Real 1D NumPy array of precomputed averages $$i_m$$. str, Name of the resolution function $$\bar K(m, z)$$, one of rectangle, lorentzian, gaussian.

Returns: Real 2D NumPy array of values $$\sigma_{mm'}$$.

Signature(som_core cont, int i, list[std::pair<double,double>] intervals, vector<double> avg, resolution_function r_func) -> matrix<double>

Compute spectral two-point correlators of a set of accumulated particular solutions

$\sigma_{mm'} = \frac{1}{J} \sum_{j=1}^J (i_m^{(j)} - i_m) (i_{m'}^{(j)} - i_{m'})$

for a list of real energy intervals.

Parameters:

cont: Som, Analytic continuation object. int, Index of the diagonal matrix element of the observable used to construct cont. list [float, float], List of pairs (left interval boundary, right interval boundary). Real 1D NumPy array of precomputed averages $$i_m$$. str, Name of the resolution function $$\bar K(m, z)$$, one of rectangle, lorentzian, gaussian.

Returns: Real 2D NumPy array of values $$\sigma_{mm'}$$.