Time-dependent basis truncation and convergence

For a fixed orthogonal projector P, replacing the reference Hamiltonian by P H P gives a Hermitian evolution within the retained subspace. That restricted evolution preserves its norm even when the full Hamiltonian couples retained and omitted states. Small final population near a selected cutoff is useful evidence but can miss virtual transitions or earlier excursions. Vary the basis and examine the target observables over the whole time interval rather than relying on normalization alone.

A reconstructed approximate state has a residual equal to i hbar times its derivative minus the reference H acting on it. For a finite Hermitian reference Hamiltonian, variation of constants and unitarity bound state error by initial error plus the integrated residual norm divided by hbar. In a continuum problem this reasoning additionally needs a self-adjoint operator, a common valid domain and a well-defined integrable residual. For a general non-Hermitian reference generator, the estimate needs the corresponding propagation-norm factor.

Let columns of U be orthonormal retained basis vectors that vary with time, and write the reconstructed state as U times a coefficient vector. Substituting that representation into the Schrödinger equation and projecting gives the basis Hamiltonian plus minus i hbar times U-adjoint times its derivative. This connection term accounts for motion of the basis itself. Dropping it mistakes a coordinate rotation in state space for additional physical dynamics.

Differentiating orthonormality shows that U-adjoint times its derivative is anti-Hermitian, so the connection contribution is Hermitian. The resulting projected coefficient dynamics can preserve norm, while residual components orthogonal to the moving subspace still measure missing physics. Nonorthogonal bases require an overlap matrix and related derivative terms. A small coefficient integration tolerance cannot diagnose an incorrectly omitted connection or certify that the moving subspace follows all relevant transitions.

Krylov propagation approximates the action of an exponential in the span of the initial vector and successive Hamiltonian applications. Hermitian Lanczos reduction gives a small tridiagonal problem, with accuracy depending on subspace size, orthogonality and a suitable defect or error estimate. If the generated subspace closes exactly under H, the reduced action is exact for that initial vector. Otherwise norm preservation inside the reduced problem does not bound the omitted full-space dynamics.

Chebyshev propagation instead expands the exponential in polynomials of a shifted, scaled Hermitian Hamiltonian. Verified energy bounds must put its spectrum inside minus one to one; only then does the familiar polynomial norm bound support an omitted-coefficient-tail estimate. An underestimated spectral interval can create large polynomial growth. A single small last coefficient is not automatically a rigorous bound on the whole tail. Driven Hamiltonians also require treatment of their time dependence, separate from polynomial or Krylov approximation of a frozen operator.