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E. B. Davies

Publications and source records attributed to E. B. Davies.

At least 19 recordsLinked to original sources
arXiv2015

Level sets of the resolvent norm of a linear operator revisited

It is proved that the resolvent norm of an operator with a compact resolvent on a Banach space $X$ cannot be constant on an open set if the underlying space or its dual is complex strictly convex. It is also shown that this is not the case for an arbitrary Banach space: there exists a separable, reflexive space $X$ and an unbounded, densely defined operator acting in $X$ with a compact resolvent whose norm is constant in a neighbourhood of zero; moreover $X$ is isometric to a Hilbert space on a subspace of co-dimension $2$. There is also a bounded linear operator acting on the same space whose resolvent norm is constant in a neighbourhood of zero. It is shown that similar examples cannot exist in the co-dimension $1$ case.

math.SP↗
arXiv2012

Sectorial perturbations of self-adjoint matrices and operators

This paper considers $N\times N$ matrices of the form $A_γ=A+ γB$, where $A$ is self-adjoint, $γ\in C$ and $B$ is a non-self-adjoint perturbation of $A$. We obtain some monodromy-type results relating the spectral behaviour of such matrices in the two asymptotic regimes $|γ|\to\infty$ and $|γ|\to 0$ under certain assumptions on $B$. We also explain some properties of the spectrum of $A_γ$ for intermediate sized $γ$ by considering the limit $N\to\infty$, concentrating on properties that have no self-adjoint analogue. A substantial number of the results extend to operators on infinite-dimensional Hilbert spaces.

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arXiv2011

Singular Schrodinger operators in one dimension

We consider a class of singular Schrödinger operators $H$ that act in $L^2(0,\infty)$, each of which is constructed from a positive function $ϕ$ on $(0,\infty)$. Our analysis is direct and elementary. In particular it does not mention the potential directly or make any assumptions about the magnitudes of the first derivatives or the existence of second derivatives of $ϕ$. For a large class of $H$ that have discrete spectrum, we prove that the eigenvalue asymptotics of $H$ does not depend on rapid oscillations of $ϕ$ or of the potential. Similar comments apply to our treatment of the existence and completeness of the wave operators.

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arXiv2011

Archimedes' calculations of square roots

We reconsider Archimedes' evaluations of several square roots in 'Measurement of a Circle'. We show that several methods proposed over the last century or so for his evaluations fail one or more criteria of plausibility. We also provide internal evidence that he probably used an interpolation technique. The conclusions are relevant to the precise calculations by which he obtained upper and lower bounds on pi.

math.HO↗
arXiv2010

An Inverse Spectral Theorem

We prove a substantial extension of an inverse spectral theorem of Ambarzumyan, and show that it can be applied to arbitrary compact Riemannian manifolds, compact quantum graphs and finite combinatorial graphs, subject to the imposition of Neumann (or Kirchhoff) boundary conditions.

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arXiv2010

Algebraic aspects of spectral theory

We describe some aspects of spectral theory that involve algebraic considerations but need no analysis. Some of the important applications of the results are to the algebra of $n\times n$ matrices with entries that are polynomials or more general analytic functions.

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arXiv2010

Non-Weyl Resonance Asymptotics for Quantum Graphs

We consider the resonances of a quantum graph $\mathcal G$ that consists of a compact part with one or more infinite leads attached to it. We discuss the leading term of the asymptotics of the number of resonances of $\mathcal G$ in a disc of a large radius. We call $\mathcal G$ a \emph{Weyl graph} if the coefficient in front of this leading term coincides with the volume of the compact part of $\mathcal G$. We give an explicit topological criterion for a graph to be Weyl. In the final section we analyze a particular example in some detail to explain how the transition from the Weyl to the non-Weyl case occurs.

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arXiv2008

Decomposing the Essential Spectrum

We use C*-algebra theory to provide a new method of decomposing the eseential spectra of self-adjoint and non-self-adjoint Schrodinger operators in one or more space dimensions.

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arXiv2005

Triviality of the Peripheral Point Spectrum

If $T_t=\rme^{Zt}$ is a positive one-parameter contraction semigroup acting on $l^p(X)$ where $X$ is a countable set and $1\leq p <\infty$, then the peripheral point spectrum $P$ of $Z$ cannot contain any non-zero elements. The same holds for Feller semigroups acting on $L^p(X)$ if $X$ is locally compact.

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arXiv2004

Semi-classical Analysis and Pseudospectra

We prove an approximate spectral theorem for non-self-adjoint operators and investigate its applications to second order differential operators in the semi-classical limit. This leads to the construction of a twisted FBI transform. We also investigate the connections between pseudospectra and boundary conditions in the semi-classical limit.

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arXiv2003

Asymptotic behaviour of quasi-orthogonal polynomials

We obtain explicit upper and lower bounds on the norms of the spectral projections of the non-self-adjoint harmonic oscillator. Some of our results apply to a variety of other families of orthogonal polynomials.

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arXiv2003

Approximating semigroups by using pseudospectra

We study evolution equations with non-self-adjoint generators, for example the convection-diffusion equation. Spectral expansions are not a reliable method of solving such equations, because they are so ill-conditioned. We introduce a new method using pseusospectra to produce an approximated spectral expansion, and explain its theoretical status. We also give some simple numerical examples to show that it is much better conditioned.

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arXiv2001

Eigenvalues of an elliptic system

We describe the spectrum of a non-self-adjoint elliptic system on a finite interval. Under certain conditions we find that the eigenvalues form a discrete set and converge asymptotically at infinity to one of several straight lines. The eigenfunctions need not generate a basis of the relevant Hilbert space, and the larger eigenvalues are extremely sensitive to small perturbations of the operator. We show that the leading term in the spectral asymptotics is closely related to a certain convex polygon, and that the spectrum does not determine the operator up to similarity. Two elliptic systems which only differ in their boundary conditions may have entirely different spectral asymptotics. While our study makes no claim to generality, the results obtained will have to be incorporated into any future general theory.

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arXiv2001

Spectral Stability of the Neumann Laplacian

We prove the equivalence of Hardy- and Sobolev-type inequalities, certain uniform bounds on the heat kernel and some spectral regularity properties of the Neumann Laplacian associated with an arbitrary region of finite measure in Euclidean space. We also prove that if one perturbs the boundary of the region within a uniform Hölder category then the eigenvalues of the Neumann Laplacian change by a small and explicitly estimated amount. AMS subject classifications: 35P15, 35J25, 47A75, 47B25, 26D10, 46E35. Keywords: Neumann Laplacian, Sobolev inequalities, Hardy inequalities, spectral stability, Hölder continuity.

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arXiv2000

Separation of variables in perturbed cylinders

We study the Laplace operator subject to Dirichlet boundary conditions in a two-dimensional domain that is one-to-one mapped onto a cylinder (rectangle or infinite strip). As a result of this transformation the original eigenvalue problem is reduced to an equivalent problem for an operator with variable coefficients. Taking advantage of the simple geometry we separate variables by means of the Fourier decomposition method. The ODE system obtained in this way is then solved numerically yielding the eigenvalues of the operator. The same approach allows us to find complex resonances arising in some non-compact domains. We discuss numerical examples related to quantum waveguide problems.

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