Kronecker Sum Separable Spectra
Build a separable four-state operator by adding factor spectra.
Free theory articleBuild a separable four-state operator by adding factor spectra.
Concept & interpretation
The Kronecker sum is an operation that constructs a large composite operator by linearly combining smaller, independent factor matrices. It is commonly used to build the Hamiltonian of a system composed of non-interacting subsystems, or to represent multidimensional discrete Laplacians. A remarkable property of the Kronecker sum is that the spectrum (the eigenvalues) of the resulting operator is simply the pairwise sum of the spectra of the original factors.
Experiment
Open the Kronecker Sum Separable Spectra module to build the four-state operator interactively.
- Select A diagonal (a), A coupling (c), and B coupling (d) using the discrete selects in the sidebar.
- Observe the 4-node coupling diagram update to reflect the transition weights and node self-energies.
- Verify the sorted eigenvalues reported in the Results panel match the mathematical pairwise sums a+2 ±c ±d.
- Observe that varying a coupling only shifts the corresponding factor's contribution, while the other factor's relative spectrum remains unaffected.