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About
Ian is Deputy Examinations Officer for undergraduate Mathematics programmes and serves on the School's Research Postgraduate Staff Student Consultative Committee.
Contact Information
Address
Room 132
Office hours: Tu 10:30-11:30/Th 14:30-15:30
Publications
Also view these in the Kent Academic Repository
Article
Brown, B. et al. (2017). Gap Localization of TE-Modes by Arbitrarily Weak Defects. Journal of the London Mathematical Society [Online] 95:942-962. Available at: http://dx.doi.org/10.1112/jlms.12046.
Abstract | View in KAR | View Full Text
This paper considers the propagation of TE-modes in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a periodic background medium. Both the periodic background problem and the perturbed problem are modelled by a divergence type equation. A feature of our analysis is that we allow discontinuities in the coefficients of the operator, which is required to model many photonic crystals. It is shown that arbitrarily weak perturbations introduce spectrum into the spectral gaps of the background operator.
Brown, B. et al. (2017). Inverse problems for boundary triples with applications. Studia Mathematica [Online]. Available at: http://dx.doi.org/10.4064/sm8613-11-2016.
Abstract | View in KAR | View Full Text
This paper discusses the inverse problem of how much information on an operator can be determined/detected from `measurements on the boundary'. Our focus is on non-selfadjoint operators and their detectable subspaces (these determine the part of
the operator `visible' from `boundary measurements').
We show results in an abstract setting, where we consider the relation between the M-
function (the abstract Dirichlet to Neumann map or the transfer matrix in system theory)
and the resolvent bordered by projections onto the detectable subspaces. More specifically, we investigate questions of unique determination, reconstruction, analytic continuation and jumps across the essential spectrum.
The abstract results are illustrated by examples of Schr�odinger operators, matrix differential operators and, mostly, by multiplication operators perturbed by integral operators(the Friedrichs model), where we use a result of Widom to show that the detectable subspace can be characterized in terms of an eigenspace of a Hankel-like operator.
Judge, E., Naboko, S. and Wood, I. (2016). Eigenvalues for perturbed periodic Jacobi matrices by the Wigner-von Neumann approach. Integral Equations and Operator Theory [Online] 85:427-450. Available at: http://dx.doi.org/10.1007/s00020-016-2302-5.
Abstract | View in KAR | View Full Text
The Wigner-von Neumann method, which has previously been used for perturbing continuous Schrödinger operators, is here applied to their discrete counterparts. In particular, we consider perturbations of arbitrary T-periodic Jacobi matrices. The asymptotic behaviour of the subordinate solutions is investigated, as too are their initial components, together giving a general technique for embedding eigenvalues, λ, into the operator's absolutely continuous spectrum. Introducing a new rational function, C(λ;T), related to the periodic Jacobi matrices, we describe the elements of the a.c. spectrum for which this construction does not work (zeros of C(λ;T)); in particular showing that there are only finitely many of them.
Fischbacher, C., Naboko, S. and Wood, I. (2016). The Proper Dissipative Extensions of a Dual Pair. Integral Equations and Operator Theory [Online] 85:573-599. Available at: http://dx.doi.org/10.1007/s00020-016-2310-5.
Abstract | View in KAR | View Full Text
Let A and B be dissipative operators on a Hilbert space H and let (A,B) form a dual pair, i.e. A ⊂ B*, resp. B ⊂ A*. We present a method of determining the proper dissipative extensions C of this dual pair, i.e. A ⊂ C ⊂ B* provided that D(A) ∩ D(B) is dense in H. Applications to symmetric operators, symmetric operators perturbed
by a relatively bounded dissipative operator and more singular differential operators are discussed. Finally, we investigate the stability of the numerical range of the different dissipative extensions.
Brown, B. et al. (2016). Detectable subspaces and inverse problems for Hain-Luest-type operators. Mathematische Nachrichten [Online]. Available at: http://dx.doi.org/10.1002/mana.201500231.
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We examine the extent to which a block operator matrix of Hain-Luest type can be reconstructed from its Titchmarsh-Weyl coefficients.
The detectable subspace of the operator is determined in a variety of cases and the question of unique determination of the coefficients is considered for both first and second order operators.
Brown, B. et al. (2015). On the spectrum of waveguides in planar photonic bandgap structures. Proceedings of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences [Online]. Available at: http://rspa.royalsocietypublishing.org/content/471/2176/20140673.abstract?ijkey=FLRWoEIYzrPaXy6&keytype=ref.
We study a Helmholtz-type spectral problem related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a two-dimensional periodic medium. The defect is infinitely extended and aligned with one of the coordinate axes. The perturbation is expected to introduce guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. In the first part of the paper, we prove that, somewhat unexpectedly, guided mode spectrum can be created by arbitrarily "small" perturbations. Secondly we show that, after performing a Floquet decomposition in the axial direction of the waveguide, for any fixed value of the quasi-momentum $k_x$ the perturbation generates at most finitely many new eigenvalues inside the gap.
Abels, H., Grubb, G. and Wood, I. (2014). Extension Theory and Krein-type Resolvent Formulas for Nonsmooth Boundary Value Problems
. Journal of Functional Analysis [Online] 266:4037-4100. Available at: http://www.sciencedirect.com/science/article/pii/S0022123614000342.
For a strongly elliptic second-order operator $A$ on a bounded domain $\Omega\subset \mathbb{R}^n$ it has been known for many years how to interpret the general closed $L_2(\Omega)$-realizations of $A$ as representing boundary conditions (generally nonlocal), when the domain and coefficients are smooth. The purpose of the present paper is to extend this representation to nonsmooth domains and coefficients, including the case of H\"older $C^{\frac32+\varepsilon}$-smoothness, in such a way that pseudodifferential methods are still available for resolvent constructions and ellipticity considerations. We show how it can be done for domains with $B^\frac32_{2,p}$-smoothness and operators with $H^1_q$-coefficients, for suitable $p>2(n-1)$ and $q>n$. In particular, Kre\u\i{}n-type resolvent formulas are established in such nonsmooth cases. Some unbounded domains are allowed.
Brown, B. et al. (2014). Spectrum created by line defects in periodic structures. Mathematische Nachrichten [Online] 287:1972-1985. Available at: http://dx.doi.org/10.1002/mana.201300165.
We study a Helmholtz-type spectral problem related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a three-dimensional periodic medium; the defect is infinitely extended in one direction, but compactly supported in the remaining two.
This perturbation introduces guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem.
We will show that even small perturbations lead to additional spectrum in the spectral gaps of the unperturbed operator and investigate some properties of the spectrum that is created.
Brown, B., Grubb, G. and Wood, I. (2009). M -functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems. Mathematische Nachrichten [Online] 282:314-347. Available at: http://dx.doi.org/10.1002/mana.200810740.
In this paper, we combine results on extensions of operators with recent results on the relation between the M -function and the spectrum, to examine the spectral behaviour of boundary value problems. M -functions are defined for general closed extensions, and associated with realisations of elliptic operators. In particular, we consider both ODE and PDE examples where it is possible for the operator to possess spectral points that cannot be detected by the M -function
Brown, B. et al. (2009). The Abstract TITCHMARSH-WEYL M-Function for adjoint operator pairs and its relation to the Spectrum. Integral Equations and Operator Theory [Online] 63:297-320. Available at: http://dx.doi.org/10.1007/s00020-009-1668-z.
In the setting of adjoint pairs of operators we consider the question: to what extent does the Weyl M-function see the same singularities as the resolvent of a certain restriction AB of the maximal operator? We obtain results showing that it is possible to describe explicitly certain spaces S and S such that the resolvent bordered by projections onto these subspaces is analytic everywhere that the M-function is analytic. We present three examples – one involving a Hain-Lüst type operator, one involving a perturbed Friedrichs operator and one involving a simple ordinary differential operators on a half line – which together indicate that the abstract results are probably best possible.
Brown, B., Eastham, M. and Wood, I. (2009). Conditions for the spectrum associated with an
asymptotically straight leaky wire to comprise
the interval (−∞, ∞). Journal of Physics A: Mathematical and Theoretical [Online] 42. Available at: http://dx.doi.org/10.1088/1751-8113/42/5/055207.
We consider a quantum (or leaky) wire in the plane, and the wire supports a singular attraction which becomes large at distant points on the wire. An analogous regular potential arises from the motion of a hydrogen atom in an electric field. We prove that, as in the regular case, the spectrum is the whole of (−∞, ∞).
Brown, B., Eastham, M. and Wood, I. (2009). Estimates for the lowest eigenvalue of a star graph. Journal of Mathematical Analysis and Applications [Online] 354:24-30. Available at: http://dx.doi.org/10.1016/j.jmaa.2008.12.014.
We derive new estimates for the lowest eigenvalue of the Schrödinger operator associated
with a star graph in R^2. We achieve this by a variational method and a procedure for
identifying test functions which are sympathetic to the geometry of the star graph.
Brown, B. et al. (2008). Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDES and block operator matrices. Journal of the London Mathematical Society [Online] 77:700-718. Available at: http://dx.doi.org/10.1112/jlms/jdn006.
Starting with an adjoint pair of operators, under suitable abstract versions of standard PDE hypotheses,
we consider the Weyl M-function of extensions of the operators. The extensions are determined by abstract
boundary conditions and we establish results on the relationship between the M-function as an analytic function
of a spectral parameter and the spectrum of the extension. We also give an example where the M-function does
not contain the whole spectral information of the resolvent, and show that the results can be applied to elliptic
PDEs where the M-function corresponds to the Dirichlet to Neumann map.
Brown, B., Eastham, M. and Wood, I. (2008). Conditions for the spectrum associated with an asymptotically straight leaky wire to contain an interval [− α^2/4, ∞). Archiv der Mathematik [Online] 90:554-558. Available at: http://dx.doi.org/10.1007/s00013-008-2612-1.
The method of singular sequences is used to provide a simple and, in some respects, a more general proof of a known spectral result for leaky wires. The method introduces a new concept of asymptotic straightness.
Wood, I. (2007). Maximal L^p -regularity for the Laplacian on Lipschitz domains
. Mathematische Zeitschrift [Online] 255:855-875. Available at: http://dx.doi.org/10.1007/s00209-006-0055-6.
We consider the Laplacian with Dirichlet or Neumann boundary
conditions on bounded Lipschitz domains �, both with the following two domains of
definition:D1(�) = {u ∈ W1,p(�) : �u ∈ Lp(�), Bu = 0}, orD2(�) = {u ∈ W2,p(�) :
Bu = 0}, where B is the boundary operator.We prove that, under certain restrictions
on the range of p, these operators generate positive analytic contraction semigroups
on Lp(�) which implies maximal regularity for the corresponding Cauchy problems.
In particular, if � is bounded and convex and 1 < p ≤ 2, the Laplacian with domain
D2(�) has the maximal regularity property, as in the case of smooth domains. In the
last part,we construct an example that proves that, in general, the Dirichlet–Laplacian
with domain D1(�) is not even a closed operator.
Hieber, M. and Wood, I. (2007). The Dirichlet problem in convex bounded domains for operators in non-divergence form with L∞-coefficients. Differential Integral Equations 20:721-734.
Consider the Dirichlet problem for elliptic and parabolic equations in nondivergence form
with variable coefficients in convex bounded domains of $\R^n$. We prove solvability of the
elliptic problem and maximal
$L^q$-$L^p$-estimates for the solution of the parabolic problem provided
the coefficients $a_{ij} \in L^\infty$ satisfy a Cordes condition and $p \in (1,2]$ is close to $2$.
This implies that in two dimensions, i.e. $n=2$, the elliptic Dirichlet problem is always solvable if the associated operator is uniformly strongly elliptic, and $p \in (1,2]$ is close to $2$, for maximal $L^q$-$L^p$-regularity in the parabolic case an additional assumption on the growth of the coefficients is needed.
Geissert, M. et al. (2005). The Ornstein-Uhlenbeck semigroup in exterior domains. Archiv der Mathematik [Online] 85:554-562. Available at: http://link.springer.com.chain.kent.ac.uk/article/10.1007%2Fs00013-005-1400-4.
Let Omega be an exterior domain in R^n. It is shown that Ornstein-Uhlenbeck operators L
generate C_0-semigroups on L^p(Omega) for p in (1, \infty) provided Omega is
smooth. The method presented also allows to determine the domain D(L)
of L and to prove L^p-L^q smoothing
properties of e^{tL}. If Omega is only Lipschitz, results
of this type are shown to be true for p close to 2.
Book section
Brown, B., Evans, W. and Wood, I. (2013). Some spectral properties of Rooms and Passages domains and their skeletons. in: Holden, H., Simon, B. and Teschl, G. eds. Spectral analysis, differential equations and mathematical physics: a festschrift in honor of Fritz Gesztesy's 60th birthday. Providence, RI, USA: AMS, pp. 69-85. Available at: http://www.ams.org.chain.kent.ac.uk/books/pspum/087/.
In this paper we investigate spectral properties of Laplacians on Rooms and Passages domains. In the first part, we use Dirichlet-Neumann bracketing techniques to show that for the Neumann Laplacian in certain Rooms and Passages domains the second term of the asymptotic expansion of the counting function is of order $\sqrt{\lambda}$. For the Dirichlet Laplacian our methods only give an upper estimate of the form $\sqrt{\lambda}$. In the second part of the paper, we consider the relationship between Neumann Laplacians on Rooms and Passages domains and Sturm-Liouville operators on the skeleton.
Brown, B. et al. (2011). Floquet-Bloch Theory for Elliptic Problems with Discontinuous Coefficients. in: Janas, J. et al. eds. Spectral Theory and Analysis: Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP) 2008. Poland: Springer, pp. 1-20. Available at: http://dx.doi.org/10.1007/978-3-7643-9994-8_1.
We study spectral properties of elliptic problems of order 2m with periodic coefficients in L∞. Our goal is to obtain a Floquet-Bloch type representation of the spectrum in terms of the spectra of associated operators acting on the period cell. Our approach using bilinear forms and operators in H−m-type spaces easily handles discontinuous coefficients and has the merit of being rather direct. In addition, the cell of periodicity is allowed to be unbounded, i.e. periodicity is not required in all spatial directions.
Wood, I. (2009). The Ornstein-Uhlenbeck Semigroup in Bounded and Exterior Lipschitz Domains . in: Janas, J. et al. eds. Methods of Spectral Analysis in Mathematical Physics. Basel: Birkhaeuser, pp. 415-435. Available at: http://dx.doi.org/10.1007/978-3-7643-8755-6_21.
We consider bounded Lipschitz domains Ω in ℝ n . It is shown that the Dirichlet-Laplacian generates an analytic C 0-semigroup on L p (Ω) for p in an interval around 2 and that the corresponding Cauchy problem has the maximal L q -regularity property. We then prove that for bounded or exterior Lipschitz domains Ornstein-Uhlenbeck operators A generate C 0-semigroups in the same p-interval. The method, also allows to determine the domain D(A) of A and, if Ω satisfies an outer ball condition, allows to show L p -L q -smoothing properties of the semigroups.
Brown, B. et al. (2009). On Spectral Bounds for Photonic Crystal Waveguides . in: Bandle, C. et al. eds. Inequalities and Applications. Basel: Birkhaeuser, pp. 23-30. Available at: http://dx.doi.org/10.1007/978-3-7643-8773-0_3.
For a (d + 1)-dimensional photonic crystal with a linear defect strip (waveguide), we calculate real intervals containing spectrum of the associated spectral problem. If such an interval falls completely into a spectral gap of the unperturbed problem (without defect), this will prove the existence of additional spectrum induced by the waveguide.
Hieber, M. and Wood, I. (2003). Asymptotics of perturbations to the wave equation. in: Evolution equations. New York: Marcel Dekker, pp. 243-252.
Conference or workshop item
Brown, B., Eastham, M. and Wood, I. (2008). An example on the discrete spectrum of a star graph. in: Exner, P. et al. eds. Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, January 8-June 29, 2007. American Mathematical Society, pp. 331-336.
Book
Wood, I. (2005). Elliptic and Parabolic Problems in Non-Smooth Domains. [Online]. Berlin: Logos-Verlag. Available at: http://www.logos-verlag.de/cgi-bin/engbuchmid?isbn=1059&lng=deu&id=.
Regularity of solutions is an important part of the theory of partial differential equations. In this text, the regularity of solutions to elliptic and parabolic problems in Lipschitz domains is investigated.
Maximal regularity estimates are useful when dealing with nonlinear parabolic problems. However, the known maximal regularity results for smooth domains no longer hold in Lp-spaces over Lipschitz domains for the whole range of exponents p. Here, maximal regularity estimates are shown for the Laplacian with suitable domain in Lp-spaces for a restricted range of p.
Operators with L∞-coefficients in convex domains and Ornstein-Uhlenbeck operators in exterior Lipschitz domains are also discussed.
Edited book
Brown, B.M., Lang, J. and Wood, I. eds. (2012). Spectral Theory, Function Spaces and Inequalities: New Techniques and Recent Trends. [Online]. Basel: Birkhaeuser. Available at: http://www.springer.com/birkhauser/mathematics/book/978-3-0348-0262-8.
This is a collection of contributed papers which focus on recent results in areas of differential equations, function spaces, operator theory and interpolation theory. In particular, it covers current work on measures of non-compactness and real interpolation, sharp Hardy-Littlewood-Sobolev inequalites, the HELP inequality, error estimates and spectral theory of elliptic operators, pseudo differential operators with discontinuous symbols, variable exponent spaces and entropy numbers. These papers contribute to areas of analysis which have been and continue to be heavily influenced by the leading British analysts David Edmunds and Des Evans. This book marks their respective 80th and 70th birthdays.
Forthcoming
Judge, E., Naboko, S. and Wood, I. (2018). Spectral results for perturbed periodic Jacobi matrices using the discrete Levinson technique. Studia Mathematica.
Abstract | View in KAR | View Full Text
For an arbitrary Hermitian period-T Jacobi operator, we assume a perturbation by a Wigner-von Neumann type potential to devise subordinate solutions to the formal spectral equation for a (possibly infinite) real set, S, of the spectral parameter. We employ discrete Levinson type techniques to achieve this, which allow the analysis of the asymptotic behaviour of the solutions. This enables us to construct infinitely many spectral singularities on the absolutely continuous spectrum of the periodic Jacobi operator, which are stable with respect to an l^1-perturbation. An analogue of the quantisation conditions from the continuous case appears, relating the frequency of the oscillation of the potential to the quasi-momentum associated with the purely periodic operator.
Research Interests
Analysis of PDEs and spectral theory, in particular
- study of spectral properties of non-selfadjoint operators via boundary triples and M-functions (generalised Dirichlet-to-Neumann maps)
- regularity to solutions of PDEs in Lipschitz domains and
- waveguides in periodic structures.