arXiv subjects

Lyonell Boulton

Publications and source records attributed to Lyonell Boulton.

At least 19 recordsLinked to original sources

arXiv2026

Linear completeness of trajectories in Sobolev spaces and the symmetrised polydisk

We establish a framework to determine the linear completeness of families of non-linear trajectories in Hilbert spaces, which relies on an infinite analytic block Toeplitz operator formulation. By means of this approach, we show the linear completeness in Sobolev spaces of two families of classical functions. One is the moving family of dilated Weierstrass functions. The other is the family of eigenfunctions of the Gross-Pitaevskii equation with trapping potential in an infinite square well. Our results provide a new insight on the formulation of general methods to examine this intriguing concept, bridging classical non-linear analysis and linear approximation theory.

math.FA↗

arXiv2025

Jumps, cusps and fractals in the solution of the periodic linear Benjamin-Ono equation

We establish two complementary results about the regularity of the solution of the periodic initial value problem for the linear Benjamin-Ono equation. We first give a new simple proof of the statement that, for a dense countable set of the time variable, the solution is a finite linear combination of copies of the initial condition and of its Hilbert transform. In particular, this implies that discontinuities in the initial condition are propagated in the solution as logarithmic cusps. We then show that, if the initial condition is of bounded variation (and even if it is not continuous), for almost every time the graph of the solution in space is continuous but fractal, with upper Minkowski dimension equal to 3/2. In order to illustrate this striking dichotomy, in the final section we include accurate numerical evaluations of the solution profile, as well as estimates of its box-counting dimension for two canonical choices of irrational time.

math.AP↗

arXiv2025

Spectral analysis of Dirac operators for dislocated potentials with a purely imaginary jump

In this paper we present a complete spectral analysis of Dirac operators with non-Hermitian matrix potentials of the form $i\operatorname{sgn}(x)+V(x)$ where $V\in L^1$. For $V=0$ we compute explicitly the matrix Green function. This allows us to determine the spectrum, which is purely essential, and its different types. It also allows us to find sharp enclosures for the pseudospectrum and its complement, in all parts of the complex plane. Notably, this includes the instability region, corresponding to the interior of the band that forms the numerical range. Then, with the help of a Birman-Schwinger principle, we establish in precise manner how the spectrum and pseudospectrum change when $V\not=0$, assuming the hypotheses $\|V\|_{L^1}<1$ or $V\in L^1\cap L^p$ where $p>1$. We show that the essential spectra remain unchanged and that the $\varepsilon$-pseudospectrum stays close to the instability region for small $\varepsilon$. We determine sharp asymptotic for the discrete spectrum, whenever $V$ satisfies further conditions of decay at infinity. Finally, in one of our main findings, we give a complete description of the weakly-coupled model.

math.SP↗

arXiv2024

The pseudospectrum of an operator with Bessel-type singularities

In this paper we examine the asymptotic structure of the pseudospectrum of the singular Sturm-Liouville operator $L=\partial_x(f\partial_x)+\partial_x$ subject to periodic boundary conditions on a symmetric interval, where the coefficient $f$ is a regular odd function that has only a simple zero at the origin. The operator $L$ is closely related to a remarkable model examined by Davies in 2007, which exhibits surprising spectral properties balancing symmetries and strong non-self-adjointness. In our main result, we derive a concrete construction of classical pseudo-modes for $L$ and give explicit exponential bounds of growth for the resolvent norm in rays away from the spectrum.

math.SP↗

arXiv2024

The phenomenon of revivals on complex potential Schrödinger's equation

The mysterious phenomena of revivals in linear dispersive periodic equations was discovered first experimentally in optics in the 19th century, then rediscovered several times by theoretical and experimental investigations. While the term has been used systematically and consistently by many authors, there is no consensus on a rigorous definition. In this paper, we describe revivals modulo a regularity condition in a large class of Schrödinger's equations with complex bounded potentials. As we show, at rational times the solution is given explicitly by finite linear combinations of translations and dilations of the initial datum, plus an additional continuous term.

math.AP↗

arXiv2024

Jumps and cusps: a new revival effect in local dispersive PDEs

We study the presence of a non-trivial revival effect in the solution of linear dispersive boundary value problems for two benchmark models which arise in applications: the Airy equation and the dislocated Laplacian Schr{ö}dinger equation. In both cases, we consider boundary conditions of Dirichlet-type. We prove that, at suitable times, jump discontinuities in the initial profile are revived in the solution not only as jump discontinuities but also as logarithmic cusp singularities. We explicitly describe these singularities and show that their formation is due to interactions between the symmetries of the underlying spatial operators with the periodic Hilbert transform.

math.AP↗

arXiv2023

The eigenvalues and eigenfunctions of the non-linear equation associated to second order Sobolev embeddings

We consider the non-linear eigenvalue equations characterizing $L^p$ into $L^q$ Sobolev embeddings of second order for Navier boundary conditions at both ends of a line segment. We give a complete description of the s-numbers and the extremal functions in the general case $(p,q)\in(1,\infty)^2$. Among other results, we show that these can be expressed in terms of those of related first order embeddings, if and only if $\frac{1}{p}+\frac{1}{q}=1$. Our findings shed new light on the surprising nature of higher order Sobolev spaces in the Banach space setting.

math.CA↗

arXiv2023

Completeness of trajectories associated to Appell hypergeometric functions

We examine the linear completeness of trajectories of eigenfunctions associated to non-linear eigenvalue problems, subject to Dirichlet boundary conditions on a segment. We pursue two specific goals. On the one hand, we establish that linear completeness persists for the non-linear Schrödinger equation, even when the trajectories lie far from those of the linear equation where bifurcations occur. On the other hand, we show that this is also the case for a fully non-linear version of this equation which is naturally associated with Appell hypergeometric functions. Both models shed new light on a framework for completeness in the non-linear setting, considered by L.E. Fraenkel over 40 years ago, that may have significant potential but which does not seem to have received much attention.

math.SP↗

arXiv2022

Norm-ideal perturbations of one-parameter semigroups and applications

The notion of equivalence classes of generators of one-parameter semigroups based on the convergence of the Dyson expansion can be traced back to the seminal work of Hille and Phillips, who in Chapter XIII of the 1957 edition of their Functional Analysis monograph, developed the theory in minute detail. Following their approach of regarding the Dyson expansion as a central object, in the first part of this paper we examine a general framework for perturbation of generators relative to the Schatten-von Neumann ideals on Hilbert spaces. This allows us to develop a graded family of equivalence relations on generators, which refine the classical notion and provide stronger-than-expected properties of convergence for the tail of the perturbation series. We then show how this framework realises in the context of non-self-adjoint Schrodinger operators.

math.FA↗

arXiv2022

Spectral Geometry of Nuts and Bolts

We study the spectrum of Laplace operators on a one-parameter family of gravitational instantons of bi-axial Bianchi IX type coupled to an abelian connection with self-dual curvature. The family of geometries includes the Taub-NUT, Taub-bolt and Euclidean Schwarzschild geometries and interpolates between them. The interpolating geometries have conical singularities along a submanifold of co-dimension two, but we prove that the associated Laplace operators have natural self-adjoint extensions and study their spectra. In particular, we determine the essential spectrum and prove that its complement, the discrete spectrum, is infinite. We compute these eigenvalues numerically and compare the numerical results with an analytical approximation derived from the asymptotic Taub-NUT form of each of the geometries in our family.

hep-th↗

arXiv2021

Beyond periodic revivals for linear dispersive PDEs

We study the phenomenon of revivals for the linear Schrödinger and Airy equations over a finite interval, by considering several types of non-periodic boundary conditions. In contrast with the case of the linear Schrödinger equation examined recently (which we develop further), we prove that, remarkably, the Airy equation does not generally exhibit revivals even for boundary conditions very close to periodic. We also describe a new, weaker form of revival phenomena, present in the case of certain Robin-type boundary conditions for the linear Schrödinger equation. In this weak revival, the dichotomy between the behaviour of the solution at rational and irrational times persists, but in contrast with the classical periodic case, the solution is not given by a finite superposition of copies of the initial condition.

math.AP↗

arXiv2020

New Revival Phenomena for Linear Integro-Differential Equations

We present and analyse a novel manifestation of the revival phenomenon for linear spatially periodic evolution equations, in the concrete case of three nonlocal equations that arise in water wave theory and are defined by convolution kernels. Revival in these cases is manifested in the form of dispersively quantised cusped solutions at rational times. We give an analytic description of this phenomenon, and present illustrative numerical simulations.

math.AP↗

arXiv2019

Computation of sharp estimates of the Poincaré constant on planar domains with piecewise self-similar boundary

We establish a strategy for finding sharp upper and lower numerical bounds of the Poincaré constant on a class of planar domains with piecewise self-similar boundary. The approach consists of four main components: W1) tight inner-outer shape interpolation, W2) conformal mapping of the approximate polygonal regions, W3) grad-div system formulation of the spectral problem and W4) computation of the eigenvalue bounds. After describing the method, justifying its validity and determining general convergence estimates, we show concrete evidence of its effectiveness by computing lower and upper bound estimates for the constant on the Koch snowflake.

math.NA↗

arXiv2018

Quantum Bound States in Yang-Mills-Higgs Theory

We give rigorous proofs for the existence of infinitely many (non-BPS) bound states for two linear operators associated with the Yang-Mills-Higgs equations at vanishing Higgs self-coupling and for gauge group SU(2): the operator obtained by linearising the Yang-Mills-Higgs equations around a charge one monopole and the Laplace operator on the Atiyah-Hitchin moduli space of centred charge two monopoles. For the linearised system we use the Riesz-Galerkin approximation to compute upper bounds on the lowest 20 eigenvalues. We discuss the similarities in the spectrum of the linearised system and the Laplace operator, and interpret them in the light of electric-magnetic duality conjectures.

math-ph↗

arXiv2018

Perturbations of Gibbs semigroups and the non-selfadjoint harmonic oscillator

Let $T$ be the generator of a $C_0$-semigroup $e^{-Tt}$ which is of finite trace for all $t>0$ (a Gibbs semigroup). Let $A$ be another closed operator, $T$-bounded with $T$-bound equal to zero. In general $T+A$ might not be the generator of a Gibbs semigroup. In the first half of this paper we give sufficient conditions on $A$ so that $T+A$ is the generator of a Gibbs semigroup. We determine these conditions in terms of the convergence of the Dyson-Phillips expansion corresponding to the perturbed semigroup in suitable Schatten-von Neumann norms. In the second half of the paper we consider $T=H_\vartheta=-e^{-i\vartheta}\partial_x^2+e^{i\vartheta}x^2$, the non-selfadjoint harmonic oscillator, on $L^2(\mathbb{R})$ and $A=V$, a locally integrable potential growing like $|x|^α$ for $0\leq α<2$ at infinity. We establish that the Dyson-Phillips expansion converges in this case in an $r$ Schatten-von Neumann norm for $r>\frac{4}{2-α}$ and show that $H_\vartheta+V$ is the generator of a Gibbs semigroup $\mathrm{e}^{-(H_\vartheta+V)τ}$ for $|\argτ|\leq \fracπ{2}-|\vartheta|$. From this we determine asymptotics for the eigenvalues and for the resolvent norm of $H_\vartheta+V$.

math.SP↗

arXiv2017

A multi-term basis criterion for families of dilated periodic functions

In this paper we formulate a concrete method for determining whether a system of dilated periodic functions forms a Riesz basis in $L^2(0,1)$. This method relies on a general framework developed by Hedenmalm, Lindqvist and Seip about 20 years ago, which turns the basis question into one about the localisation of the zeros and poles of a corresponding analytic multiplier. Our results improve upon various criteria formulated previously, which give sufficient conditions for invertibility of the multiplier in terms of sharp estimates on the Fourier coefficients. Our focus is on the concrete verification of the hypotheses by means of analytical or accurate numerical approximations. We then examine the basis question for profiles in a neighbourhood of a non-basis family generated by periodic jump functions. For one of these profiles, the $p$-sine functions, we determine a threshold for positive answer to the basis question which improves upon those found recently.

math.CA↗

arXiv2015

Generalised cosine functions, basis and regularity properties

We examine regularity and basis properties of the family of rescaled $p$-cosine functions. We find sharp estimates for their Fourier coefficients. We then determine two thresholds, $p_0<2$ and $p_1>2$, such that this family is a Schauder basis of $L_s(0,1)$ for all $s>1$ and $p\in[p_0,p_1]$.

math.CA↗

arXiv2015

Sharp eigenvalue enclosures for the perturbed angular Kerr-Newman Dirac operator

A certified strategy for determining sharp intervals of enclosure for the eigenvalues of matrix differential operators with singular coefficients is examined. The strategy relies on computing the second order spectrum relative to subspaces of continuous piecewise linear functions. For smooth perturbations of the angular Kerr-Newman Dirac operator, explicit rates of convergence due to regularity of the eigenfunctions are established. Existing benchmarks are validated and sharpened by several orders of magnitude in the unperturbed setting.

math.NA↗