Guaranteed Lower Eigenvalue Bounds for Spectral Galerkin Methods with Application to Schrödinger Operators

Abstract: Spectral Galerkin methods are renowned for high-precision eigenvalue approximation, yet a rigorous lower bound obtained directly from a spectral discretisation has remained unavailable: the classical Kato and Weinstein--Temple enclosures do apply, but require a~priori information on a neighbouring eigenvalue. This paper resolves the issue by extending the author's projection-based framework for guaranteed lower eigenvalue bounds -- so far realised only through finite element methods -- to conforming spectral Galerkin methods. For trial spaces of exact eigenfunctions the required projection constant is the closed-form optimal value $C_N=λ<em>{M+1}<sup>{-1/2}$, the inverse square root of the first omitted eigenvalue. For $-Δ+V$ with $0\le V\in L<sup>\infty$, a \emph{projection-gap estimate} yields an explicit constant for the standard Galerkin matrix (exact at $V=0$), and a composite discretisation removes the $||V||</em>{L<sup>\infty}$-dependence for large potentials. With Neumann domain truncation these give certified two-sided bounds on $R<sup>d$; for two benchmark potentials on $R<sup>2$ the spectral enclosures match or surpass certified finite element ones at two orders of magnitude fewer degrees of freedom. The same auxiliary-projector mechanism extends to singular potentials with an unbounded $L<sup>\infty$ norm -- in particular to attractive Coulomb singularities in three dimensions, via a localised Hardy inequality -- which we develop in a companion paper.

• The paper introduces a projection-based approach that guarantees certified lower eigenvalue bounds using spectral Galerkin approximations.
• It derives explicit constants and error estimates, outperforming certified FEM methods especially for smooth and confining potentials.
• The method extends to Schrödinger operators with both bounded and singular potentials, enabling rigorous two-sided eigenvalue enclosures.

Guaranteed Lower Eigenvalue Bounds for Spectral Galerkin Methods: An Expert Overview

Abstract and Motivation

This paper addresses the longstanding challenge of producing rigorous, computable lower bounds for eigenvalues of self-adjoint differential operators using spectral Galerkin methods. While spectral methods are known for their rapid convergence in the approximation of eigenvalues—especially for smooth problems—existing frameworks for certified lower bounds have been almost exclusively predicated on finite element techniques. The author generalizes a projection-based lower bound strategy, previously accessible only in the context of finite elements, to a spectral Galerkin context. The approach delivers explicit constants that guarantee lower bounds for eigenvalues approximated in spectral spaces and includes detailed application to Schrödinger operators, including domain truncation and potentials of both bounded and singular types.

Theoretical Framework

Abstract Projection-Based Bound

The foundational result is an abstract lower bound theorem applicable to variational eigenproblems:

$$\lambda_k \ge \frac{\lambda_{k,N}}{1 + C_N^2 \lambda_{k,N}}$$

where $\lambda_{k,N}$ is the $k$-th Galerkin eigenvalue and $C_N$ is a constant determined by the projection error from the (exact or approximate) solution space onto the trial space. This approach neither requires a priori spectral gap estimates nor knowledge of nearby eigenvalues, which distinguishes it from classical techniques such as Temple–Kato or Lehmann–Goerisch methods.

The derivation relies on a projection $\Pi_N$ into the trial space with the Pythagorean property and a norm inequality governed by $C_N$. The central technical goal is to compute or estimate $C_N$ explicitly for spectral Galerkin spaces.

Spectral Constant for Eigenfunction Spaces

When the trial space is spanned by the first $M$ exact eigenfunctions, Parseval's identity yields the optimal projection constant:

$C_N = \lambda_{M+1}^{-1/2}$

This is the sharpest possible bound and is attainable directly, without recourse to mesh-based interpolation or geometrically localized error constants. The result calibrates the framework: for any $v$ orthogonal to the retained eigenbasis,

$\lambda_{k,N}$0

where $\lambda_{k,N}$1 is the orthogonal projection onto the trial space.

This optimal constant is also reflected in explicit one-dimensional examples and validates the projection-based approach as delivering bounds as tight as the data allows.

Schrödinger Operators: Bounded Potentials

For Schrödinger operators $\lambda_{k,N}$2, $\lambda_{k,N}$3, the trial space is not generally built from the true eigenfunctions. Therefore, the exact constant cannot be attained, and the paper introduces a projection-gap mechanism to control the leakage between the Laplacian modal projection and the Ritz projection for $\lambda_{k,N}$4.

The key observation is that the error due to unresolved modes for the Laplacian can be explicitly bounded via the spectral gap ($\lambda_{k,N}$5), while the error due to the potential is governed by a form factor $\lambda_{k,N}$6 that measures the effect of $\lambda_{k,N}$7 relative to the trial space.

For $\lambda_{k,N}$8 bounded and $\lambda_{k,N}$9, the constants take the form:

$k$0

with $k$1 and $k$2 the first omitted Laplacian eigenvalue. This yields explicit, certified lower bounds from a single standard Galerkin computation.

However, as $k$3 becomes large (e.g., in truncated domains with confining potentials), this bound deteriorates since $k$4 scales with $k$5. The author introduces a tail-refined form factor $k$6 that sharply reduces this pessimism by reflecting only the part of $k$7 leaving the trial space, computable as a largest eigenvalue problem on known matrices.

Composite Discretization: Removing $k$8-Dependence

To fully decouple the lower bound quality from $k$9, the paper constructs a composite discretization based on:

• A bandlimited under-approximation of $C_N$0,
• A weighted projection of reduced degree,
• An algebraic enforcement (via the slaving identity) that ensures the product of the trial functions and the under-approximated $C_N$1 stays within the trial space.

This structure allows the use of the spectral gap $C_N$2 directly, with no inflation due to the potential magnitude:

$C_N$3

In practice, one solves an augmented eigenproblem involving low-rank updates to the original matrices, maintaining rigorous lower bounds and vastly improved efficiency for confining/large potentials.

Domain Truncation and Two-Sided Enclosures

A principal application is to certified eigenvalue enclosures for Schrödinger operators on unbounded domains (e.g., $C_N$4). By combining:

• Neumann truncation lower bound (using the monotonicity under Neumann restriction),
• The composite spectral Galerkin lower bound, and
• The Dirichlet Rayleigh–Ritz upper bound,

the framework provides fully computable two-sided enclosures for the spectrum, with the truncation error controlled by classical exponential decay estimates.

Extension to Singular Potentials

The projection-gap mechanism generalizes beyond bounded $C_N$5. For singular (e.g., Coulomb) potentials with unbounded $C_N$6 norm, Hardy-type inequalities and auxiliary shifting render the form factor finite, enabling rigorously certified lower bounds for quantum chemistry applications. This is reserved for a companion manuscript.

Numerical Results

Numerical experiments, including high-DOF benchmarks in one and two dimensions, demonstrate:

• Spectral Galerkin lower bounds match or outperform certified finite element (CECR FEM) bounds at orders of magnitude fewer DOFs (e.g., $C_N$7, $C_N$8 DOFs, outperforms $C_N$9 DOFs in FEM).
• Confirmed rates of convergence ($\Pi_N$0) and explicit demonstration of the saturation effect near the truncation index.
• The tail-refined form factor significantly improves the lower bound compared to crude $\Pi_N$1-based constants, especially for high-frequency modes and smooth potentials.
• The composite method is essential for confining potentials, while the auxiliary-projector approach suffices for moderate $\Pi_N$2.

Analysis on disk domains (via Fourier–Bessel bases) confirms the transferability of the core machinery beyond rectangular geometry, with domain-specific adaptivity in the realization of the method.

Implications and Outlook

Practical

This work removes the longstanding barrier to producing certified lower bounds in spectral Galerkin discretizations, making high-precision, rigorous eigenvalue computation for quantum and PDE models practical for a significantly broader class of problems. Particularly for smooth and confining potentials, spectral methods offer up to two orders of magnitude more efficiency in degrees of freedom than existing certified finite element approaches.

Theoretical

The methodology demonstrates that:

• Rigorous projection-based bounds can be constructed without dependence on a priori spectral separation,
• Closed-form constants analogous to finite element interpolation errors can be identified for global spectral bases,
• The unconditional nature of the bounds allows them to seed more delicate (possibly exponential-rate) lower bound techniques, such as Lehmann–Goerisch post-processing.

There remains a fundamental information-theoretic limit to single-constant frameworks: the so-called index saturation effect limits lower bound sharpness near the highest-resolved mode. Transferring spectral convergence of upper bounds to lower bounds remains an open question.

Conclusion

The paper establishes a general, efficient, and rigorously computable framework for certified lower eigenvalue bounds using spectral Galerkin methods, with explicit projection error constants, application to both bounded and singular potentials (including quantum chemistry models), and full computability in terms of standard spectral matrices. The method is both theoretically transparent and practically superior to finite element based bounds in smooth, confined settings, and lays a critical foundation for further improvement of rigorous error control in spectral numerical analysis for PDE eigenproblems.

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Reference:

"Guaranteed Lower Eigenvalue Bounds for Spectral Galerkin Methods with Application to Schrödinger Operators" (2607.04247)