Lanczos–Krylov spectra and dynamics

Starting from a unit vector $q_1$, the m-dimensional Krylov space spans $q_1,Hq_1,\ldots,H^{m-1}q_1$. Those powers usually become nearly parallel, so Lanczos constructs orthonormal directions instead. At step j, subtract components along the current and previous vectors, then normalize the remainder. For Hermitian H, exact arithmetic makes the remaining earlier overlaps vanish automatically. The diagonal coefficient is a Rayleigh expectation, while the nonnegative off-diagonal coefficient is a norm.

A zero remainder means the generated space is invariant under H, so recurrence should terminate rather than divide by zero. A small floating-point remainder can instead amplify roundoff when normalized. Loss of orthogonality can produce repeated approximations to an already found eigenvalue, sometimes called ghosts. Reorthogonalization and independent residual checks help distinguish such numerical repetitions from a genuine degenerate eigenspace. One starting vector samples only its projection into a degenerate subspace and cannot reveal every independent vector there.

Collect the basis vectors as columns of Q and form $T_m=Q^\dagger HQ$. Its eigenvalues are Ritz values, and lifting a normalized small eigenvector y gives the approximation Qy. In exact orthogonality, the final Lanczos coefficient and the last component of y determine the full-space residual norm. In finite precision, applying H again to the recovered vector is the safer verification. A lowest Ritz value remains variationally above the true lowest energy in the searched invariant sector.

A spectral function uses an operator-excited starting vector, whose norm carries the total spectral weight. Renormalizing that vector for the recurrence must not erase its prefactor from the final correlation function. Finite lattices have discrete spectral poles. Adding an imaginary broadening parameter makes a smooth plot but does not create a physical lifetime. Converge Krylov dimension first, then examine system size and artificial broadening to decide which features are supported by the calculation.

For a state initially equal to $q_1$, projected real-time propagation evaluates a small matrix exponential and maps its coefficients back to the original space. This preserves norm in an orthonormal basis because the projected Hamiltonian is Hermitian. Norm preservation is necessary but insufficient: a tiny Krylov space can evolve unitarily within itself while missing important physical amplitudes outside it. Increasing the subspace dimension or reducing the time interval tests that error.

For long calculations, rebuild the Krylov space after short propagation intervals and control accumulated error, rather than relying on one fixed space indefinitely. Real time uses an imaginary exponential and rotates phases. Imaginary time uses a decaying exponential and, after normalization, filters higher-energy contributions if the initial state overlaps the desired ground sector. These evolutions have different numerical and physical roles. A geometric vector display can illustrate orthogonality but cannot generate a Krylov basis, a spectral function, or either many-body exponential.