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Matthew J. Colbrook

Publications and source records attributed to Matthew J. Colbrook.

At least 19 recordsLinked to original sources
arXiv2026

Sharp Computational Bounds for Spectral Types of Schr\"odinger Operators

We prove sharp bounds for determining the spectral-type decomposition of Schr\"odinger operators in the spirit of Smale's program on the foundations of computation. For explicit one-dimensional self-adjoint Schr\"odinger operators $H=-{\mathrm d^2}/{\mathrm dx^2}+V$ on $L^2(\mathbb R)$, where $V\in C^\infty(\mathbb R;\mathbb R)$ is given by a finite description of all derivatives and derivative bounds, the pure point and absolutely continuous spectral sets cannot, in general, be recovered by any single limiting procedure. The singular continuous spectral set is strictly harder: it cannot, in general, be recovered by two nested limiting procedures. Analytic constructions of dichotomies realize the lower bounds: Gordon-type repetitions for pure point spectrum, high barriers for absolutely continuous spectrum, and an inverse spectral construction for singular continuous spectrum based on Riesz products, moment-killing perturbations, and a computational Gelfand--Levitan scheme. The finite-description framework also implies corresponding limitations on what can be certified in fixed formal systems (e.g., when used in computer-assisted proofs). Conversely, using wavelet-based certified computation, we prove matching upper bounds for broad classes of self-adjoint differential operators on $\mathbb R^d$ with coefficients of locally bounded variation and quantitative local variation control: two limits suffice for the pure point and absolutely continuous parts, and three for the singular continuous part. This provides a sharp hierarchy for spectral types.

math.NA↗
arXiv2026

A counterexample to Kenig's conjecture for the Laplace double-layer operator

Layer potentials provide a classical approach to boundary value problems for Laplace's equation on Lipschitz domains. Kenig's 1994 spectral-radius conjecture for the double-layer operator would ensure operator-norm convergence of the associated Neumann series on mean-zero $L^2$ densities when the boundary is connected. We disprove this conjecture by constructing a bounded simply connected planar Lipschitz domain whose double-layer operator on arclength $L^2$ has essential spectral radius strictly greater than $1/2$. More precisely, for every $t>1/2$ sufficiently close to $1/2$, we obtain such a domain with $\pm i t$ in its Fredholm essential spectrum. The construction starts from smooth graphs whose shapes repeat under translation. In the limit of separated scales, refinement makes solutions of adjoint resolvent equations grow with fixed forcing. The graph slopes remain uniformly bounded. A computer-assisted certificate proves this growth through an inequality for Hermitian $2\times2$ matrices. Its strict margin at $- i/2$ persists at nearby spectral parameters. Normalisation and a Floquet transform then give compactly supported densities with small residuals on the full graphs. We insert rescaled segments of successive graphs into one bounded boundary, where these densities form a weakly null sequence of approximate eigenvectors. The same spectral conclusion holds on a single periodic Lipschitz graph. The certificate combines continuous estimates, exact rational arithmetic and rigorous interval enclosures.

math.AP↗
arXiv2026

The Spectral Skeleton of Chaos: Koopman Wave Packets on Poincar\'e Sections

A Poincar\'e section replaces a flow by a return map, but for a chaotic system this map is usually known only from sampled crossings. We show that coarse transport can be read directly from Koopman spectral data, without fitting the map. Measure-preserving EDMD retains the isometric structure; riggedDMD then approximates spectral measures and constructs finite regularized wave packets. Packet phase supplies a finite-resolution transport coordinate; low modulus marks a singular skeleton where the phase becomes ill-conditioned. We demonstrate the idea on the R\"ossler system, a 32-mode Kuramoto--Sivashinsky Galerkin system, and the forced Duffing oscillator. The packets yield coarse symbolic models on sections ranging from an almost one-dimensional curve to a visibly thick set. Their graphs organize observed low-period orbits and guide targeted searches for others. In Duffing Regime~II, a seven-region rule accounts for $91\%$--$94\%$ of filtered one-step transitions, while failures in the lowest retained modulus decile occur at $5.08$--$5.20$ times the overall rate. The packets are not Koopman eigenfunctions, nor are the regions exact Markov partitions. Together these computations show how spectral information beyond isolated eigenpairs can expose chaotic transport directly from trajectories.

nlin.CD↗
arXiv2026

A Complete Resolution of Forsythe's Conjecture for Restarted Conjugate Gradients

Forsythe's conjecture, published in 1968, asserts that for each restart length $s$, every exact-arithmetic restarted conjugate-gradient iteration on a real symmetric positive definite problem either terminates or has normalised residuals that converge separately along the even and odd restart subsequences. Apart from the classical steepest-descent case, this asymptotic question remained unresolved in full generality for nearly six decades. We give a complete classification by restart length in the original finite-dimensional setting and identify a sharp threshold. For $s=2$ and $s=3$, every problem either terminates or has convergent even and odd residual directions. For every $s\ge4$, there is a diagonal positive definite counterexample of dimension $s+4$ which never terminates and whose even residual directions do not converge. Together with Akaike's theorem for $s=1$, this shows that the conjectured universal conclusion is true precisely for $s\in\{1,2,3\}$ and false for every $s\ge4$. The positive results follow from a degree-independent double-orthogonality identity and an analysis of the low-degree fixed-point sets. At restart length four, rational interval arithmetic and Sturm sequences certify a transverse Hopf point of the leading vector field of the rescaled squared-weight map. Analytic periodic-orbit and shadowing arguments yield the counterexample at restart length four, and degree elevation extends the construction to every larger restart length. The classification for all $s\ge2$ is also formally verified in Lean.

math.NA↗
arXiv2026

A sharp isoperimetric inequality for the Neumann--Poincar\'e operator in every dimension

Let $\Omega\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,\alpha}$, where $0<\alpha<1$. For the adjoint Neumann--Poincar\'e operator $K^*_{\partial\Omega}$, normalised so that its distinguished eigenvalue is $1/2$, let $\lambda_j^+(\Omega)$ denote the upper min--max values on the mean-zero energy space. We prove $\sum_{j=1}^{d}\lambda_j^+(\Omega)\ge \frac{d-2}{2}.$ It follows that $\lambda_1^+(\Omega)\ge \frac{d-2}{2d},$ with equality if and only if $\Omega$ is a ball. In dimension three this proves the $1/6$-conjecture of Miyanishi and Suzuki. The proof uses the coordinate boundary-charge densities induced by uniform applied fields. Their energy Gram matrix is the perfect-conductor polarization tensor $M_\infty$. Positivity of a $2d\times2d$ Gram matrix yields the endpoint Hashin--Shtrikman inequality $|\Omega|\mathrm{Tr}(M_\infty^{-1})\le1,$ and bounds the trace of the compression of $K^*_{\partial\Omega}$ to the applied-field space. Equality in the inverse-trace inequality makes the interior Newtonian potential quadratic, so a converse to Newton's theorem identifies $\Omega$ as an ellipsoid. Equality in the spectral estimate also makes the Hessian of this potential isotropic, which forces the ellipsoid to be a ball.

math.AP↗
arXiv2026

A computer-assisted counterexample to the planar Berenstein conjecture

Recent work of Colbrook and Stepaniants produced the first counterexamples to the planar Pompeiu and Schiffer conjectures and introduced the conformal fixed-disc, disk-polynomial, and validated-tail machinery used here. By adapting this framework to the complementary Dirichlet endpoint, we disprove the unrestricted planar Berenstein conjecture. Specifically, we construct a bounded simply connected domain $\Omega$ with real-analytic Jordan boundary, which is not a disc and for which there exist $k\in(27.4381178838,27.4381198839)$ and a nonzero real-valued function $u\in C^\omega(\overline\Omega)$ satisfying $(\Delta+k^2)u=0$ in $\Omega$, with $u=0, \partial_\nu u=\text{constant}\ne0$ on $\partial\Omega$. Thus the overdetermined Dirichlet--Neumann data do not characterize the disc without an additional sign assumption on $u$. The domain has dihedral symmetry of order $26$, but is neither a disc nor centrally symmetric, and the corresponding eigenfunction changes sign. Equivalently, its boundary arclength measure satisfies $\widehat{\sigma_{\partial\Omega}}(k\omega)=0$ for $\omega\in\mathbb S^1$. After conformally transferring to the unit disc, exact support identities and quantitative disk-polynomial estimates yield rigorous control of the infinite-dimensional tail. A Newton--Kantorovich argument then reduces existence to finitely many explicit inequalities, which are certified using interval arithmetic. The extension from the Pompeiu--Schiffer problem is not formal. The earlier construction absorbs both boundary conditions into a single inverse-Laplacian equation. At the Dirichlet endpoint considered here, the nonzero Neumann datum forces the harmonic source modes to remain, producing a coupled interior--boundary system involving the full zero-Dirichlet inverse and its Neumann trace, together with a separate sign-recovery problem.

math.AP↗
arXiv2026

A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures

The planar Pompeiu problem, originating in 1929, and the associated Schiffer conjecture are long-standing rigidity questions linking rigid-motion integral transforms and Fourier zero sets to overdetermined Neumann eigenvalue problems. We construct a bounded simply connected noncircular domain $\Omega\subset\mathbb{R}^2$ with real-analytic Jordan boundary and a nonconstant function $u$ such that $(\Delta+k^2)u=0$ in $\Omega$, $u=1,\partial_\nu u=0$ on $\partial\Omega $ for some $k\in(31.967007261,31.967007293)$. Thus $u$ is a Neumann eigenfunction which is constant on the boundary, and $\Omega$ is a counterexample to Schiffer's conjecture. Green's identity also gives $\widehat{\mathbf 1_\Omega}(k\omega)=0$ $(\omega\in\mathbb S^1)$, so $\Omega$ fails the Pompeiu property and is also a counterexample to the planar Pompeiu conjecture for bounded simply connected Lipschitz domains. We obtain the domain as $\Omega=\phi(\mathbb{D})$, where $\phi$ is a ten-fold symmetric conformal map close to an explicitly listed polynomial of degree $301$. On the unit disc, the analytic problem becomes a cubic operator equation on real coefficient spaces, $F(g,p)=g+|p|^2(1+Kg)=0$, where $K$, expressed in a disk-polynomial basis, is an explicit inverse of the Laplacian on the range compatible with zero Dirichlet and Neumann traces, and $p=k\phi'$. Positivity of the disk-polynomial linearisation coefficients, sharp bounds for $K$, and monotone control of the infinite tails establish an a posteriori contraction near the listed polynomial in a weighted coefficient algebra, and hence an exact zero of $F$.

math.AP↗
arXiv2026

No Spectral Invisibility for Dissipative Barrier Truncations in Any Dimension

The dissipative barrier method suppresses spectral pollution, but whether it can itself conceal genuine spectral points has remained open. Known as the graveyard problem in computational spectral theory, the higher-dimensional case has remained unresolved for more than a decade. We resolve it for Schr\"{o}dinger operators in dimensions $d\geq2$; together with the known one-dimensional theorem, this settles the no-invisibility problem in all dimensions. Let $A=-\Delta+V$ be a Dirichlet Schr\"odinger operator on a connected open set $\Omega\subseteq\mathbb R^d$ with $V\in L^1_{\mathrm{loc}}(\Omega)$ bounded below, and set $H=A+iS$, where $S\geq0$ and $S\in L^p(\Omega)$, with $1 0}$ be a nested family of nonempty connected bounded open sets such that $\Omega_R\nearrow\Omega$ as $R\nearrow+\infty$. Denote by $H_R$ the Dirichlet truncation of $H$ to $\Omega_R$. We prove that every spectral point of $H$ is detected by the truncations: for every $\lambda\in\sigma(H)$ and every neighborhood $U$ of $\lambda$, $\sigma(H_R)\cap U\neq\emptyset$ for sufficiently large $R$. Equivalently, $\sigma(H)\subseteq\liminf_{R\to\infty}\sigma(H_R)$. Thus the barrier method does not trade suppression of spectral pollution for spectral invisibility. No regularity of $\partial\Omega$ is required, and the assumptions reach the critical Sobolev scale. The proof combines compactness of the dissipative form perturbation, Cwikel-type Schatten estimates for Birman--Schwinger operators, generalized strong resolvent convergence, and a reverse Hansmann--Weyl spectral-variation inequality due to Gil'. A two-dimensional numerical example illustrates the absence of spectral invisibility for the truncated dissipative operators.

math.NA↗
arXiv2026

Nystr\"om Error Beyond $M$-Matrices: A Minimal Diagonally Dominant Obstruction

We study the nuclear-norm error of a column-selected Nystr\"om approximation to $K=(L+\gamma I)^{-1}$, where $L$ is symmetric diagonally dominant and $\gamma>0$. Our central question is whether this error has diminishing returns. A Schur-complement identity reduces the question to traces of inverses of principal submatrices. Existing $M$-matrix results settle the case in which $L$ is a symmetric diagonally dominant $M$-matrix (SDDM). However, diagonal dominance alone is not enough: failure occurs already in dimension three. We construct an exact one-parameter SDD family and determine its sharp failure interval. A $2\times2$ identity proves that dimension three is minimal within the SDD class. We then show that failure persists under strict diagonal dominance; with a nonempty selected base set, dimension four is minimal. Finally, we prove invariance under signature switching, derive a three-dimensional formula showing how a signed triangle causes failure, and give an example in which greedy column selection misses the optimal pair. Together, these findings complete the answer to Problem 4.6 in a recent Simons workshop report.

math.NA↗
arXiv2026

Residual-Guided Dictionary Learning for Spectrally Accurate Koopman Approximation

Koopman theory promises linear structure in nonlinear dynamics, but numerical Koopman spectra are easy to compute and hard to trust. A finite EDMD matrix always has eigenvalues; the problem is that many of them may have nothing to do with the infinite-dimensional operator. In this paper we make spectral reliability the objective of dictionary learning. We train neural-network dictionaries not merely to predict the next snapshot, but to minimize Residual Dynamic Mode Decomposition residuals: operator-level a posteriori errors that test whether computed eigenvalues and modes are genuine Koopman spectral objects. To keep the learned observables from collapsing into an unstable coordinate system, the loss also penalizes the condition number of the lifted data matrix. Thus the method couples two requirements that should not be separated: small Koopman residuals and a well-conditioned representation. The result is a learned dictionary that is expressive, numerically stable, and spectrally disciplined. Across conservative and dissipative benchmark systems, the method sharply reduces spectral pollution, improves residual pseudospectral inclusion, and lowers forecast error relative to standard fixed dictionaries. On sea-surface temperature data, it gives cleaner Koopman diagnostics and substantially better one-step forecasts from noisy observations with no governing equations. The message is simple: neural Koopman learning should be judged not by prediction alone, but by whether its spectral claims can be certified. Residuals provide the certificate; conditioning makes it computable.

math.NA↗
arXiv2026

PRONE: Petrov-Galerkin Operator Learning Unifies DMD, SINDy & Koopmanism

Data-driven dynamics often asks how to linearize a nonlinear system. We ask instead: which observables should be advanced, and where should their futures live? This leads to Petrov Regression Of Nonlinear Evolution (PRONE), a Petrov--Galerkin regression framework based on $ \Psi(\mathbf{X})K \approx \Phi(\mathbf{Y}), $ with distinct trial and test dictionaries. In this form, DMD, EDMD, SINDy, Koopman regression, sparse regression, and low-rank regression become variants of one construction: different dictionaries, weights, and constraints. We keep the linear algebra of Koopman learning, but drop the artificial requirement that a finite model map a dictionary into itself. With this asymmetry, eigenmodes are no longer the right objects. Instead, we use singular modes, which identify the observable combinations captured by the data, their projected futures, and the strength of the coupling between the two spaces. We identify the limiting projected operator and prove $L^2$ convergence of the resulting nonlinear predictor. We give examples from chaotic maps, the double gyre, a pitching-airfoil wake, and Lorenz--63, where PRONE outperforms DeepONets, Fourier neural operators, and reservoir computers with considerably fewer parameters. These examples show the same message: lift once, regress once, and let the singular structure reveal statistics, transport, prediction, and dimension.

math.DS↗
arXiv2026

Ten Digits on a Train: AI-Assisted Verification of Two Eigenvalue Problems

Accurate numerical eigenvalues are often difficult to certify, especially in singular or non-normal settings. This article reports a human--AI collaboration on two such computations. For a singular self-adjoint Schr\"odinger operator, a verified zero count and Dirichlet--Neumann bracketing certify the complete negative spectrum to ten decimal places. For a delicate non-normal atom--molecule benchmark, a previously unresolved resonance pair is separated, with each member enclosed to ten digits. The second result is achieved not by increasing the precision of one-way shooting, but by reformulating the problem as a global matching system for projective solution lines. The infinite tail is encoded as uncertainty in the terminal projective data, and a componentwise, tail-robust Krawczyk--Brouwer inclusion supplies the certificate. This gives a reusable architecture for analytic boundary-value systems with ill-conditioned propagation and uncertain asymptotic data. The collaboration also exposes the strengths and limits of AI assistance. AI rapidly produced accurate candidates and plausible proof strategies, but several failed, including one apparently complete tail argument that omitted the componentwise check required by a nonuniform polydisc. Validated computation is a stringent test of AI-assisted mathematics: the output is not merely a number, but a number with a proof. These examples show why the proof object matters, and why human mathematical judgment remained decisive. More broadly, as AI makes code, exposition, and plausible numerical claims inexpensive, standards for verification, attribution, peer review, and training must adapt. The implications are unsettling; the opportunity is extraordinary.

math.NA↗
arXiv2026

Residual Pseudospectra Reveal a Physics-Informed Koopman Backbone for Tropical Pacific Variability and ENSO Prediction

Tropical Pacific sea-surface-temperature (SST) variability spans interacting timescales, with the ENSO as its dominant interannual expression. Yet the dynamical structure organizing this variability and underpinning extended-range predictability remains difficult to extract from high-dimensional observations. Koopman operator learning offers spectral coordinates for nonlinear dynamics, yet finite geophysical records often produce dense, sampling-sensitive spectra whose physical content is ambiguous. We show that this apparent redundancy reflects coherent operator-level structure. Combining kernel Extended Dynamic Mode Decomposition with residual minimization and pseudospectral analysis, we use the Koopman eigenvalue relation as a physics-informed consistency test to organize learned spectra. Applied to ERA5 and HadISST tropical Pacific SST anomalies, the residual landscape identifies 19 robust residual-minimum frequencies with coherent spatial modes that persist across products and sampling realizations. Together, these modes define a compact Koopman backbone spanning low-frequency modulation through quasi-biennial components, including ENSO-band variability. The surrounding spectral cloud is structured by integer powers and nonlinear combinations of this backbone, forming a residual-ordered Koopman hierarchy. The backbone reconstructs substantial Nino3.4 variance and enables skillful out-of-sample forecasts, with greatest gains at 8-18-month leads. By embedding dynamical consistency into physics-informed operator learning, the framework turns opaque spectra into robust, interpretable and predictive representations of tropical Pacific variability.

math.NA↗
arXiv2026

Deep Embedded Multiplicative DMD for Algebra-Preserving Koopman Learning

Koopman theory turns nonlinear dynamics into a linear spectral problem. In computation, however, everything depends on a hard finite-dimensional choice: the observables must be expressive, nearly invariant under the dynamics, and, ideally, compatible with composition. Deep Koopman methods learn flexible coordinates, whereas structure-preserving methods enforce operator identities on fixed dictionaries. We combine these ideas by introducing Deep Embedded Multiplicative Dynamic Mode Decomposition (DeepMDMD), a method that learns a latent space and a partition of it, while enforcing the Koopman product rule as an exact algebraic constraint. Training alternates between an exact multiplicative operator update and a differentiable latent-clustering step that promotes Koopman closure. The result is a finite transition map on learned latent cells. Its nonzero spectrum lies on the unit circle, its dictionary is shaped by the dynamics rather than by ambient geometry, and forecasts are made in latent coordinates before being decoded to physical space. Across Hamiltonian, chaotic, and fluid examples, DeepMDMD learns dictionaries that are far more compact and dynamically coherent than those produced by geometric MDMD partitions. It reduces spectral pollution, reveals richer continuous-spectrum structure, and gives stable forecasts under severe noise. In high-dimensional flows, including a 158,624-dimensional cylinder wake and a noisy $Re=20,000$ lid-driven cavity, it preserves coherent structures and long-time spectral statistics where state-space MDMD fails. These results suggest a practical rule for Koopman learning: learn the coordinates, constrain the algebra.

cs.LG↗
arXiv2026

Finding Koopman Invariant Subspaces via Personalized PageRank

Selecting a finite dictionary of observables whose span is Koopman-invariant is a central challenge in data-driven Koopman operator approximation. We address this problem by exploiting zero-block structure in Extended Dynamic Mode Decomposition (EDMD) matrices. We show that any sub-dictionary whose span is Koopman-invariant induces an exact zero block in the EDMD matrix, even for finite data. We then show that such blocks can be detected by applying PageRank to a row-normalized EDMD matrix constructed from a large initial dictionary. The theory extends to approximately invariant subspaces and yields stronger guarantees for personalized PageRank (PPR) when the seed observables lie inside the target block and reach all observables in that block. Combining EDMD concentration bounds with PageRank perturbation theory gives end-to-end detection guarantees with $O(1/\sqrt{M})$ finite-sample scaling and explicit constants. More generally, without assuming an invariant subspace exists, high PPR mass on a sub-dictionary controls discounted multi-step leakage from the seed observables. Numerical experiments on the Duffing oscillator, Van der Pol oscillator, Lorenz system, and a three-well Ramachandran potential suggest that the method identifies compact, interpretable dictionaries with accurate predictions.

math.DS↗
arXiv2026

Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds

Koopman operator theory provides a global linear representation of nonlinear dynamics and underpins many data-driven methods. In practice, however, finite-dimensional feature spaces induced by a user-chosen dictionary are rarely invariant, so closure failures and projection errors lead to spurious eigenvalues, misleading Koopman modes, and overconfident forecasts. This paper addresses a central validation problem in data-driven Koopman methods: how to quantify invariance and projection errors for an arbitrary feature space using only snapshot data, and how to use these diagnostics to produce actionable guarantees and guide dictionary refinement? A unified a posteriori methodology is developed for certifying when a Koopman approximation is trustworthy and improving it when it is not. Koopman invariance is quantified using principal angles between a subspace and its Koopman image, yielding principal observables and a principal angle decomposition (PAD), a dynamics-informed alternative to SVD truncation with significantly improved performance. Multi-step error bounds are derived for Koopman and Perron--Frobenius mode decompositions, including RKHS-based pointwise guarantees, and are complemented by Gaussian process expected error surrogates. The resulting toolbox enables validated spectral analysis, certified forecasting, and principled dictionary and kernel learning, demonstrated on chaotic and high-dimensional benchmarks and real-world datasets, including cavity flow and the Pluto--Charon system.

math.NA↗
arXiv2025

Weighted Birkhoff Averages Accelerate Data-Driven Methods

Many data-driven algorithms in dynamical systems rely on ergodic averages that converge painfully slowly. One simple idea changes this: taper the ends. Weighted Birkhoff averages can converge much faster (sometimes superpolynomially, even exponentially) and can be incorporated seamlessly into existing methods. We demonstrate this with five weighted algorithms: weighted Dynamic Mode Decomposition (wtDMD), weighted Extended DMD (wtEDMD), weighted Sparse Identification of Nonlinear Dynamics (wtSINDy), weighted spectral measure estimation, and weighted diffusion forecasting. Across examples ranging from fluid flows to El Ni\~no data, the message is clear: weighting costs nothing, is easy to implement, and often delivers markedly better results from the same data.

math.DS↗