Non-Bloch Supercell Framework

• Non-Bloch Supercell Framework is a theoretical and practical method for approximating discrete eigenmodes and spectral gaps in quantum and wave physics.
• It employs non-consistent approximations via supercell geometries, ensuring convergence proofs, a priori error estimates, and spectral pollution-free computations.
• Key applications include simulating perturbed periodic Schrödinger operators and localized defect states, achieving exponential convergence for bound state errors.

The Non-Bloch Supercell Framework is a rigorous theoretical and practical method for approximating discrete eigenmodes—particularly isolated eigenvalues located in spectral gaps—of self-adjoint operators in quantum and wave physics. It treats the case where standard discretization techniques (Galerkin, finite element, planewave methods) are applied in a supercell geometry, and the discretized forms used are not mere restrictions of the continuous problem, i.e., the approximations are “non-consistent.” The framework provides convergence proofs, a priori error estimates, and guidelines to guarantee spectral pollution-free computations in, for example, perturbed periodic Schrödinger operators.

1. Abstract Theoretical Framework

The starting point is a variational eigenproblem for a self-adjoint operator, formulated in terms of a continuous bilinear (sesquilinear) form $a(\cdot,\cdot)$ and a scalar product $m(\cdot,\cdot)$ on a Hilbert space $H^1(\mathbb{R}^d)$. Non-consistent approximation entails choosing a finite-dimensional subspace $X_n$ and associated discrete forms $a_n(\cdot,\cdot)$, $m_n(\cdot,\cdot)$, which do not necessarily coincide with the restrictions of the continuous forms; thus, the discrete problem lives in a different “geometry.”

This framework imposes several abstract conditions:

• (A1) Density: Any function in the Hilbert space can be $H^1$-approximated by elements of $X_n$.
• (A2) Boundedness/Coercivity: The discrete forms remain uniformly bounded and coercive, possibly with norm-dependent constants.
• (A3) Discrete Inf–Sup (Stability): For $\mu$ in a compact set away from the continuous spectrum, a lower bound on the discrete form $(a_n-\mu m_n)$ exists, scaling with $\operatorname{dist}(\mu, \sigma(A))$.
• (A4) Consistency: Errors between the continuous and discrete forms are quantified by semi-norms $r_n$ and $s_n$ (e.g., $r_n^a$, $r_n^m$, $s_n^a$, $s_n^m$).

Conditions (B1) and (B2) further ensure absence of spectral pollution (no spurious eigenvalues near isolated $\lambda$).

Main result (Theorem 2.1):

• Each discrete eigenvalue $\lambda$ of the continuous problem is approximated by an eigenvalue of the discrete operator.
• Eigenfunction and eigenvalue errors admit explicit a priori bounds: the eigenvalue error is bounded by a constant times the square of the eigenfunction error (plus consistency terms), reflecting the “doubling” phenomenon typical of Galerkin-type (variational) discretizations.

2. Supercell Modeling of Perturbed Periodic Schrödinger Operators

A central motivating example is

$A = -\Delta + V_\mathrm{per} + W,$

with $V_\mathrm{per}$ a periodic potential and $W$ a localized defect decaying at infinity.

Key features:

• For $W=0$ the spectrum consists of bands (absolutely continuous).
• A localized defect ($W\ne 0$) generates discrete eigenvalues in gaps.

In practice, one simulates the problem by employing a large but finite supercell (size $L$). The space $X_L$ is constructed using planewaves in the supercell, periodically extended, and multiplied by a cut-off $\chi_L$ (unity inside, zero outside a slightly larger domain). The forms $a_L$, $m_L$ are assembled over the supercell (possibly via numerical integration), giving rise to a fundamentally non-consistent method because boundary truncation and integration errors mean the discrete forms differ from those encountered in the infinite periodic medium.

Rigorous verification:

• (A1): Planewave densities, after cut-off, provide density.
• (A2): Uniform coercivity via boundedness on finite supercell.
• (A3): Spectral lower bounds using localization of defect and band theory results.
• (A4): Quantitative error estimates on both truncation (exponential decay in $L$) and integration (controlled by mesh size $M$), accommodating planewave and grid-based integration.

3. A Priori Error Estimates and Convergence Rates

The framework expresses error estimates in terms of:

• Projection error: $\|\left(1 - \Pi_{X_n}\right)\psi\|_{H^1}$
• Consistency errors: seminorms ($r_n^a$, $r_n^m$, $s_n^a$, $s_n^m$)
• Numerical integration error: linked to discretization parameters (e.g., mesh size $M$)

In planewave discretization (e.g., $Y_{L,N}$ = Fourier modes with $|k|$ bounded), the approximation error for $\psi\in H^r$ locally behaves as

$$\|u - \Pi_{L,N} u\|_{H^{r-1}} \leq C \left(\frac{L}{N}\right)^{r-1} \|u\|_{H^r}.$$

Bound state localization further gives exponential decay of the truncation error with $L$, per Simon’s estimates.

Overall, eigenfunction norm error scales as

$$\|\psi - R_n \psi\|_{H^1} \leq C\left[e^{-\delta L} + \left(\frac{L}{N}\right)^{r-1} + \text{integration error}\right],$$

where $R_n$ is a (possibly non-orthogonal) spectral projector onto the discrete eigenspace. The corresponding eigenvalue error is then

$$|\lambda_n - \lambda| \leq C \left( \|\psi - \Pi_{X_n} \psi\|^2_{H^1} + \text{consistency errors} \right ),$$

yielding a quadratic convergence rate with respect to the eigenfunction error (in cases where $s_n$-type consistency errors vanish).

4. Numerical Illustrations

Numerical experiments are presented for 1D periodic Schrödinger operators plus defects:

• Discrete spectra from the supercell approach precisely converge to the true (infinite-domain) spectrum.
• Eigenvalue errors decay nearly exponentially with $L$ due to bound state localization.
• The predicted “doubling” of convergence rate (eigenvalue error $\sim$ square of eigenvector error) is confirmed.
• Numerical integration errors (as in grid-based integration schemes) fit the theoretical scaling—often $(L/M)^{r-2}$ for mesh size $M$.

5. Key Analytical Formulas

Table: Main error estimates

Quantity	Formula / Bound	Description
Eigenfunction error	$\|\psi - R_n \psi\|_{H^1} \leq C[\text{sum}]$	sum of projection, truncation, consistency, integration errors
Eigenvalue error	$|\lambda_n - \lambda| \leq C[\text{err.}^2]$	quadratic dependence on eigenvector error, in absence of $s_n$ terms
Planewave approximation	$\|u - \Pi_{L,N} u\|_{H^{r-1}} \leq C (L/N)^{r-1}$	Jackson-type inequality

All such bounds involve explicit constants (dependent on localization rate, smoothness order $r$, mesh size, etc.).

6. Implications, Trade-offs, and Deployment Guidance

• The Non-Bloch Supercell Framework certifies “spectral pollution-free” computation—no spurious eigenvalues near gap states of interest—provided the assumptions (density, stability, consistency, no pollution) hold.
• Optimal convergence is achieved for localized defect states: exponential decay with $L$ and algebraic convergence with $N$ or integration mesh $M$.
• The framework is robust against non-consistency—including domain truncation and numerical integration errors.
• For practical deployment, the supercell size should be chosen so that exponential truncation errors are smaller than algebraic discretization or integration errors. Planewave basis density ($N$) and numerical grid ($M$) must be scaled in accordance with required accuracy and localization properties of the bound states.
• For systems with deep spectral gaps and exponentially localized states, the supercell method achieves rapid convergence; for shallow gaps, however, larger supercells and finer discretization may be required.
• The method underpins modern defect simulations in condensed matter and photonic crystals, where reliable and efficient resolution of gap states is critical.

7. Summary

The Non-Bloch Supercell Framework rigorously unifies the analysis of non-consistent approximations for self-adjoint operator eigenproblems, with detailed error estimates, absence of spectral pollution, and optimal convergence rates. Explicit a priori bounds for eigenfunctions and eigenvalues, verified numerically, demonstrate its utility for supercell simulations of localized defect states in periodic media. The framework provides systematic prescriptions for basis choice, integration strategies, and supercell scaling, ensuring accurate and efficient approximation of spectral properties in a range of physical systems, including planewave expansions and finite element schemes.