[Paper Review] Spectral Pollution and How to Avoid It (With Applications to Dirac and Periodic Schrödinger Operators)

Mathieu Lewin, Éric Séré|Dec 11, 2008

Spectral Theory in Mathematical Physics23 references24 citations

TL;DR

This paper provides a rigorous framework to identify and eliminate spectral pollution in Galerkin approximations of self-adjoint operators with spectral gaps, such as periodic Schrödinger and Dirac operators. By exploiting a decomposition of the Hilbert space via a fixed projector P, the authors derive exact conditions under which spurious eigenvalues do not appear, proving that certain bases—like Wannier functions for periodic systems or free-Dirac spectral projectors—completely avoid pollution.

ABSTRACT

This paper, devoted to the study of spectral pollution, contains both abstract results and applications to some self-adjoint operators with a gap in their essential spectrum occuring in Quantum Mechanics. First we consider Galerkin basis which respect the decomposition of the ambient Hilbert space into a direct sum $H=PH\oplus(1-P)H$, given by a fixed orthogonal projector $P$, and we localize the polluted spectrum exactly. This is followed by applications to periodic Schrödinger operators (pollution is absent in a Wannier-type basis), and to Dirac operator (several natural decompositions are considered). In the second part, we add the constraint that within the Galerkin basis there is a certain relation between vectors in $PH$ and vectors in $(1-P)H$. Abstract results are proved and applied to several practical methods like the famous "kinetic balance" of relativistic Quantum Mechanics.

Motivation & Objective

• Address the persistent issue of spectral pollution in numerical approximations of self-adjoint operators with gaps in their essential spectrum.
• Identify conditions on approximation bases that prevent spurious eigenvalues from appearing in spectral gaps.
• Provide a theoretical foundation for understanding why certain numerical methods in quantum mechanics (e.g., kinetic balance) succeed or fail in avoiding pollution.
• Apply the abstract framework to concrete physical systems: periodic Schrödinger operators and Dirac Hamiltonians with various decompositions.
• Establish a rigorous link between the choice of basis and the absence of pollution, particularly in relativistic quantum systems.

Proposed method

• Introduce a fixed orthogonal projector P on the Hilbert space H, decomposing it as H = P H ⊕ (1−P) H.
• Define P-spurious eigenvalues as limits of Ritz values from Galerkin approximations using subspaces respecting the P-decomposition.
• Derive exact characterization of the polluted spectrum using spectral theory and strong convergence of projectors.
• Establish a simple criterion: if the spectral gap of A avoids the range of certain operators related to P, no pollution occurs.
• Apply the framework to Dirac operators using different decompositions: upper/lower spinors, dual basis, and free-Dirac spectral projectors.
• Analyze balanced bases where relations between P H and (1−P) H components are enforced, such as in kinetic balance methods.

Experimental results

Research questions

• RQ1 Under what conditions on the approximation basis can spectral pollution be completely avoided in a gap of the essential spectrum?
• RQ2 Why do certain numerical methods in relativistic quantum mechanics—like kinetic balance—fail or succeed in reducing spurious eigenvalues?
• RQ3 How does the choice of basis (e.g., Wannier functions, spinor decomposition) affect the presence of spectral pollution in periodic and Dirac operators?
• RQ4 Can the polluted spectrum be exactly localized when the Galerkin basis respects a given Hilbert space decomposition?
• RQ5 What is the role of the spectral projector of the free Dirac operator in eliminating pollution?

Key findings

• Pollution is completely avoided in the spectral gap of a self-adjoint operator if the associated projector P satisfies a specific spectral condition related to the gap.
• For periodic Schrödinger operators, using a Wannier-type basis associated with the unperturbed Hamiltonian eliminates spectral pollution.
• Decomposition into upper and lower spinors for the Dirac operator always leads to pollution, as shown by explicit construction of spurious eigenvalues.
• Using the spectral projector of the free Dirac operator as the decomposition projector results in a completely pollution-free approximation scheme.
• The dual basis and kinetic balance methods do not in general eliminate pollution, and their effectiveness depends on the size of the perturbation and the parameter ε.
• For the dual kinetic balance method, the polluted spectrum is explicitly characterized as the union of two intervals depending on ε and the bounds of the potential V.