E. Brian Davies
(Edward) Brian Davies is Professor of Mathematics, King's College London (1981–), and author of the popular science book Science in the Looking Glass: What do Scientists Really Know.
Education
Davies was schooled in Wales where his father was a teacher of Mathematics. He went on to graduate with a first class degree from University of Oxford where he stayed to complete his D.Phil Some Problems in Functional Analysis in 1968.
Research Interests
Operator theory/functional analysis
Non-self-adjoint operators Elliptic partial differential operators Computational aspects of spectral theory Schrodinger operators/quantum theory Philosophy of science
Professor E B Davies' research has concentrated on the use of methods from functional analysis and operator theory to study the behaviour of differential operators. The focus is usually on problems involving the spectral theory of elliptic differential operators, for example the Laplacian or Schroedinger operators. Much of his research over the last twenty years has either developed or used estimates (upper and lower bounds) on heat kernels, and he is regarded as a world authority on this topic. Since 1995, much of his work has related to the spectral theory of non-selfadjoint operators and to the theory of one-parameter semigroups and most of his students since 1995 have been studying the spectral theory of non-self-adjoint Schrodinger operators or the properties of resonances sometimes undertaking bits of numerical computation to support their theoretical work. This has often been motivated by numerical discoveries and his interest in computation is as a source of ideas for theorems which can be proved in an entirely rigorous manner.
Teaching
Davies has spent his teaching career at King's College teaching Mathematical Analysis to undergraduate, postgraduate and his long line of Doctoral students. He was appointed the Head of the Department of Mathematics at the college from 1990-1993. He has edited a number of mathematical journals and book series and has been Visiting Professor at Cornell University and California Institute of Technology.
Career
Professor Davies was Tutorial Fellow at St John's College, Oxford in 1970 and one of two editors for the Quarterly Journal of Mathematics, Oxford. He developed the theory of open quantum systems, writing a monograph on the subject, which became the standard text. Appointed to the Chair of Pure Mathematics at King's College, London in 1981 he researched heat kernels and spectral theory. His monograph on heat kernels, involving several hundred publications and having applications in many parts of mathematics, is the authoritative text on the subject. He founded the London Mathematical Society Student Text Series in the 1980s, and became Senior Editor for the LMS Monographs. Subsequently he was Head of Kings's College Mathematics Department from 1990-3. He has edited a number of mathematical journals and book series and has been Visiting Professor at Cornell University and California Institute of Technology. He was elected a Fellow of the Royal Society in 1995 and a Fellow of King's College London the following year. He served as a panel member for the 2001 Research Assessment Exercise and has been a member of the National Advisory Board of the Isaac Newton Institute. He was elected President of the London Mathematical Society for two years from November 2007.
Publications
Books
Quantum Theory of Open Systems. Academic Press, 1976. ISBN 0 12 206150 0.
One-Parameter Semigroups. London Math. Soc. Monograph series, No.15. Academic Press, 1980. ISBN 0 12 206280 9.
Heat Kernels and Spectral Theory. Cambridge Tracts in Mathematics, Vol. 92. Cambridge University Press 1989. ISBN 0 521 36136 2.
Spectral Theory and Differential Operators. Cambridge Studies in Advanced Mathematics, Vol. 42. 1995 Cambridge University Press. ISBN 0 521 47250 4. 2001: contract signed for translation into Japanese.
Spectral Theory and Geometry. eds. E B Davies and Y Safarov. London Math. Soc. Lecture Note Series no. 273. Cambridge University Press, Cambridge, 1999. ISBN 0 521 77749 6
Science in the Looking Glass. Oxford Univ.Press, Oxford, 2003. ISBN 0 19 852543 5
Linear Operators and their Spectra. Cambridge Univ. Press, 2007 to appear.
Mathematical Research
1960s
1. A generalised theory of convexity. Proc. London Math. Soc. (3) 17 (1967)644-652.
2. The existence of characters on topological lattices. J. London Math. Soc. 43(1968) 217-220.
3. The structure and ideal theory of the pre-dual of a Banach lattice Trans. Amer.Math. Soc. 131 (1968) 544-555.
4. On the Banach duals of certain spaces with the Riesz decomposition property.Quart. J. Math. Oxford (2) 18, (1967) 109-111.
5. Tensor products, infnite products and projective limits of Choquet simplexes.With G. Vincent Smith.Math. Scand. 22 (1968) 145-164.
6. The Choquet theory and representation of ordered Banach spaces. Illinois J.Math 13 ( 1969) 176-187.
7. On the Borel structure of C*-alegbras. Commun. Math. Phys. 8, 147-163(1968).
8. The structure of Sigma*-algebras. Quart. J. Math. Oxford (2) 20 (1969)351-366.
9. Decomposition of traces on separable C* algebras. Quart. J. Math. Oxford (2),20 (1969) 97-111.
1970s
10. An operational approach to quantum probability. With J.T. Lewis. Commun.Math. Phys. 17, 239-260 (1970).
11. On the repeated measurement of continuous observables in quantum mechanics.J. Functional Anal. 6 (1970) 318-346.
12. Quantum stochastic processes. Commun. Math. Phys. 15, 277-306 (1970).
13. Involutary automorphisms of operator algebras. Trans. Amer. Math. Soc.158 (1971) 115-142.
14. Quantum stochastic processes II. Commun. Math. Phys. 19, 83-105 (1970).
15. A generalisation of Kaplansky's theorem. J. London Math. Soc. 4 (1972)435-436.
16. Quantum stochastic processes III. Commun. Math. Phys. 22, 51-70 (1971).
17. An example related to the foundations of quantum theory. J. Math. Phys. 13(1972) 39-41.
18. Some contraction semigroups in quantum probability. Z. Wahrscheinlichkeits-theorie 23 (1972) 261-273.
19. Hilbert space representations of Lie algebras. Commun. Math. Phys. 23,159-168 (1971).
20. Diffusion for weakly coupled quantum oscillators. Commun. Math. Phys. 27,309-325 (1972).
21. The thermodynamic limit for an imperfect Boson gas. Commun. Math, Phys.28, 69-86 (1972).
22. Properties of the Green's functions of some Schrödinger operators. J. LondonMath. Soc. (2) 7 (1973) 483-491.
23. The ideal Boson gas in an external scalar potential. Commun. Math. Phys.30, 229-247 (1972).
24. The harmonic oscillator in a heat bath. Commun. Math, Phys. 33, 171-186(1973).
25. Exact dynamics of an infnite-atom Dicke maser model. Commun. Math.Phys. 33, 187-205 (1973).
26. Dynamics of a multi-level Wigner-Weisskopf atom. J. Math. Phys. 15 (1974)2036-2041.
27. The infnite atom Dicke maser model II. Commun. Math. Phys. 34, 237-249( 1973).
28. Symmetries of compact convex sets. Quart. J. Math., Oxford (2) 25 (1974)323-328.
29. Parametric interactions and scattering. Commun. Math. Phys. 37 (1974)161-174.
30. Time dependent scattering theory. Math. Ann. 210 (1974) 149-162.
31. Markovian master equations. Commun. Math. Phys. 39 (1974) 91-110.
32. Markovian master equations II. Math. Ann. 219 (1976) 147-158.
34. A model for absorption or decay. Helv. Phys. Acta 48 (1975) 365-382.
35. The use of master equations in quantum mechanics. IASPS Conference Proceedings. Warsaw 1975.
36. Resonances, spectral concentration and exponential decay. Lett. Math. Phys.1 (1975) 31-35.
37. Time decay for fermion systems with persistent vacuum, with J.P. Eckmann.Helv. Phys. Acta 48 (1975) 731-742. 38. Asymptotic analysis of some abstract evolution equations. J. Functional Anal. 25 (1977) 81-101.
39. The classical limit for quantum dynamical semigroups. Commun. Math. Phys.49 (1976) 113-129.
40. Eigenfunction expansions for singular Schrödinger operators. Arch. Rat.Mech. Anal. 63 (1977) 261-272.
41. Quantum dynamical semigroups and the neutron diffusion equation. Rep.Math. Phys. 11 (1977) 169-188.
42. First and second quantised neutron diffusion equations. Commun. Math.Phys. 52 (1977) 111-126.
43. Quantum communication systems. I.E.E.E. Trans. Inf. Theory 23 (1977)530-534.
44. Information and quantum measurements. I.E.E.E. Trans. Int. Theory 24(1978) 596-599.
45. Dilations of completely positive maps. J. London Math. Inf. Theory 24 (1978)330-338.
46. A model of atomic radiation. Ann. Inst. H. Poincare 28A (1978) 91-110.
47. Scattering from infnite sheets. Math. Proc. Camb. Phil. Soc. 82 (1977)327-334.
48. Irreversible dynamics of infnite fermion systems. Commun. Math. Phys. 55(1977) 231- 258.
49. A model of heat conduction. J. Stat. Phys. 18 (1978) 161-170.
50. Generators of dynamical semigroups. J. Functional Anal. 34 (1979) 421-432.
51. Master equations: a survey of rigorous results. Rend. Sem. Mat. Fis. Milano47 (1977) 165-173.
52. Particle-boson interactions and the weak coupling limit. J. Math. Phys. 20(1979) 345-351.
53. With M Aizenmann and E.H. Lieb: Positive linear maps which are orderbounded on C*-subalgebras. Adv. Math. 28 (1978) 84-86.
54. With H. Spohn: Open quantum systems with time-dependent Hamiltonians and their linear response. J. Stat. Phys. 19 (1978) 511-523.
55. Uniqueness of the standard form of the generator of a quantum dynamicalsemigroup. Rep. Math. Phys. 17 (1980) 249-255.
56. Dynamical semigroups, generators and measurement theory. p. 85-98 in"Mathematical Problems in the Quantum Theory of Irreversible Processes", Proceedings of Arco Felice Symposium, April, 1978.
57. With B. Simon: Scattering theory for systems with different spatial asymptotics on the left and right. Commun. Math. Phys. 63 (1978) 277-301.
58. Two-channel Hamiltonians and the optical model of nuclear scattering. Ann.Inst. H. Poincare 29A (1978) 395-413.
59. Symmetry breaking for a non-linear Schrödinger equation. Commun. Math.Phys. 64 (1979) 191-210.
60. Scattering from a spatially restricted atom I. Adv. Math. 34 (1979) 28-45.
61. Classical approximations to Gibbs states. Math. Proc. Camb. Phil. Soc. 86(1979) 521-527.
62. Non-unitary scattering and capture I. Commun. Math. Phys. 71 (1980)277-288.
63. Some time-dependent Hartree equations. Ann. Inst. H. Poincare 31A (1979)319-337.
1980s
64. Irreversible and non-linear dynamics of open systems. Proc. NATO AdvancedStudy Institute, Cambridge 1979, ed. R.G. Woolley, Plenum Press, 1980.
65. Asymptotic completeness for a quark model of meson scattering. Ann. Phys.126 (1980) 171-185.
66. On Enss' approach to scattering theory. Duke Math. J. 47 (1980) 171-185.
67. Non-unitary scattering and capture II. Ann. Inst. H. Poincare 32A (1980)361-375.
68. Energy dependence of the scattering operator. Adv. Applied Math. 1 (1980)300-323.
69. Metastable states of molecules. Commun. Math. Phys. 75 (1980) 263-283.
70. Non-linear functionals in quantum mechanics. Lecture Notes, E.P.F. Lausanne,1980.
71. With M.D. Srinivas: Photon counting probabilities in quantum optics. OpticaActa 28 (1981) 981-996.
72. Asymptotic modifications of dynamical semigroups on C* algebras. J. LondonMath. Soc. 24 (1981) 537-547.
73. With Ph. A Martin: An approach to metastability in some ferromagneticsystems. Helv. Phys Acta 54 (1981) 106-130.
74. Symmetry breaking for molecular open systems. Ann. Inst. H. Poincare 35A(1981) 149-171.
75. Metastable states of symmetric Markov semigroups I. Proc. London Math.Soc. 45 (1982) 133-150.
76. Metastability and the Ising model. J. Stat. Phys. 27 (1982) 657-675.
77. Metastable states of symmetric Markov semigroups II. J. London Math. Soc.26 (1982) 541-556.
78. Energy dependence of the scattering operator II. Commun. Math. Phys. 86(1982) 149-160.
79. Stability of molecular eigenstates. Proc. Berlin IAMP Conference, 1981.
80. The twisting trick for double well Hamiltonians. Commun. Math. Phys. 85(1982) 471-479.
81. JWKB and related bound on Schrödinger eigenfunctions. Bull. London Math.Soc. 14 (1982) 273-284.
82. With M.D. Srinivas: What are the photon counting probabilities for open systems - a reply to Mandel's comments. Optica Acta 29 (1982) 235-238.
83. Some norm bounds and quadratic form inequalities for Schrödinger operators. J. Oper. Theory 9 (1983) 147-162.
84. Dynamical stability of metastable states. J. Func. Anal. 46 (1982) 373-386.
85. The harmonic functions of mean ergodic Markov semigroups. Math. Zeit. 181(1982) 543-552.
86. With C.J.K. Batty: Positive semigroups and resolvents. J. Oper. Theory 10(1983) 357-363.
87. Hypercontractive and related bounds for double well Schrödinger operators.Quart. J. Math. (2), 34 (1983) 407-421.
88. An adiabatic theorem applicable to the Stark effect. Commun. Math. Phys.89 (1983) 329-339.
89. Spectral properties of metastable Markov semigroups. J. Funct. Anal. 52(1983) 315-329.
90. Double well hamiltonians. Proc. London Math. Soc. 49 (1984) 193-206.
91. Some norm bounds and quadratic form inequalities for Schrödinger operators II. J. Oper. Theory 12 (1984) 177-196.
92. The rate of resolvent and semigroup convergence. Quart. J. Math. Oxford(2), 35 (1984) 121-130.
93. With B. Simon: Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. Journal of Funct. Anal. 59 (1984) 335-395.
94. A generation theorem for operators commuting with group actions. MathProc. Camb. Phil. Soc. 96 (1984) 313-320.
95. With B. Simon: Ultracontractive semigroups and some problems in analysis.Aspects of Math. and its Appl., A. Festschrift in honour of L. Nachbin Vol. 34 ed.J.A. Barroso: Elsevier Science Publ. B.V. 1986, 265-280.
96. Trace properties of the Dirichlet Laplacian. Math. Zeit. 188 (1985) 245-251.
97. With Pl. Muthuramalingam: Trace properties of some highly anisotropicoperators. Jour. of London Math. Soc. (2), 31 (1985) 137-149.
98. Structural isomers, double wells, resonances and Dirichlet decoupling. Annals of Physics 157 (1984) 166-182.
99. The eigenvalue distribution of degenerate Dirichlet forms. J. Dif. Eqns. 60(1985) 103-130.
100. The asymptotic distribution and lower bounds to the spectrum for some singular elliptic operators. CIRM Colloquium, Luminy 1984, eds. B. Helfer and D. Robert, Publ. de l'Universite de Nantes, 1985, 37-42.
101. With B. Simon: L1-properties of intrinsic Schrfodinger semigroups. J. Funct. Anal. 65 (1986) 126-146.
102. Criteria for ultracontractivity. Ann. Inst. H. Poincare 43A (1985) 181-194.
103. L1-properties of second order elliptic operators. Bull. London Math. Soc. 17(1985) 417-436.
104. Heat kernel bounds for second order elliptic operators on Riemannian manifolds. Amer. J. Math. 109 (1987) 545-570.
105. Perturbations of ultracontractive semigroups. Quart. J. Math. (2), 37 (1986)167-176.
106. Spectral properties of some second order elliptic operators on LP spaces.Proc. Symp on Aspects of Positivity in Functional Analysis, 1985. Eds. M. Wolf,R. Nagel. and U Schlotterbeck. North-Holland, 1986, 41-48.
107. Explicit constants for Gaussian upper bounds on heat kernels. Amer. J.Math. 109 (1987) 319-334.108. With E.M. Harrell II. Conformally fat Riemannian metrics, Schrödinger operators and semiclassical approximations. J. Dif. Eqns. 66 (1987) 165-188.
109. The equivalence of certain heat kernel and Green function bounds. J. Functional Anal. 71 (1987) 88-103.
110. With M.M.H. Pang. The Cauchy Problem and a generalization of the Hille-Yosida theorem. Proc. London Math. Soc. (3) 55 (1987) 181-208.
111. With N. Mandouvalos. Heat kernel bounds on manifolds with cusps. J.Functional Analysis 75 (1987) 311-322.
112. Kernel estimates for functions of second order elliptic operators. Quart. J.Math. Oxford (2) 39 (1988) 37-46.
113. With B. Simon and M. Taylor. Lp spectral theory of Kleinian groups. J.Functional Anal. 78 (1988) 116-136.
114. Lipschitz continuity of functions of operators in the Schatten classes. J.London Math. Soc. 37 (1988) 148-157.
115. Gaussian upper bounds for the heat kernels of some second order operatorson Riemannian manifolds. J. Functional Anal. 80 (1988) 16-32.
116. With N. Mandouvalos. heat kernel bounds on hyperbolic space and Kleiniangroups. Proc. London Math. Soc. (3) 57 (1988) 182-208.
117. With M.M.H. Pang: Sharp heat kernel bounds for some Laplace operators.Quart. J. Math, Oxford (2), 40 (1989) 281-290.
118. With O.S. Rothaus: Markov semigroups on C*-bundles. J. Functional Anal.85 (1989) 264-286.
119. Pointwise bounds on the space and time derivatives of heat kernels. J. Oper.Theory 21 (1989) 367-378.
120. An eigenvalue bound for compact manifolds with varying curvature. Ann.Global Anal. and Geom. 7 (1989) 107-114.
121. With O.S. Rothaus: A BLW inequality for vector bundles and applicationsto spectral bounds. J. Functional Anal. 86 (1989) 390-410.
122. With A. Abdessemed: Some commutator estimates in the Schatten classes.J. London Math. Soc. (2) 39 (1989) 299-308.
123. With M. van den Berg: Heat fow out of regions in Rm. Math. Zeit 202(1989) 463-482.
124. Pointwise inequalities for heat kernels, pp 361–364, in IXth Inter. Congressin Math. Physics, Swansea, 1988, eds. B. Simon, A. Truman, I.M. Davies; IOP Publishing Ltd. 1989.
1990s
125. Spectral properties of compact manifolds and changes of metric. Amer. J.Math. 112 (1990) 15-39.
126. The hyperbolic geometry and spectrum of irregular domains. Nonlinearity 3(1990) 913-945.
127. With L. Gross and B. Simon: Hypercontractivity: A Bibliographic Review.Ideas and Methods in Quantum and Statistical Mechanics, p370-389 in eds. S.Albeverio et al., Volume 2, Camb. Univ. Press, 1992.
128. With M.M.H. Pang: The eigenvalue gap for second order elliptic operators with Dirichlet boundary conditions. J. Dif. Eqns. 88 (1990) 46-90.
129. With B. Simon: Spectral properties of the Neumann Laplacian of Horns.Geometric and Functional Analysis. 2 (1992) 105-117.
130. Heat kernel bounds, conservation of probability and the Feller property. J.D'Analyse Math. 58 (1992) 99-119.
131. with B. Simon: Lp norms of non-critical Schrödinger semigroups. J. Functional Anal. 102 (1991) 95-115.
132. Heat kernels in one dimension. Quart. J. Math. Oxford (2) 44 (1993).283-299.
133. The state of the art for heat kernel bounds on negatively curved manifolds.Bull. London Math. Soc. 25 (1993) 289-292.
134. with J.M. Lindsay : Non-commutative symmetric Markov semigroups. Math.Zeit. 210 (1992) 379-411.
135. Large deviations for heat kernels on graphs. J. London Math. Soc. 47 (1993)65-72.
136. Analysis on graphs and noncommutative geometry. J. Functional Analysis,111 (1993) 398-430.
137. Two-dimensional Riemannian manifolds with fractal boundaries. J. LondonMath. Soc. (2) 49 (1994) 343-356.
138. with J.M. Lindsay: Superderivations and symmetric Markov semigroups. Commun. Math. Phys. 157 (1993) 359-370.
139. Eigenvalue stability bounds via weighted Sobolev spaces. Math. Zeit. 214(1993) 357-371.
140. The Functional Calculus. J. London Math. Soc. (2) 55 (1995) 166-176.
141. with M. Lianantonakis: Heat kernel and Hardy estimates for locally Euclidean manifolds with fractal boundaries. Geom. and Funct. Anal. 3 (1993) 527-563.
142. The Hardy constant. Quart. J. Math. Oxford (2), 46 (1995) 417-431.
143. Lp spectral independence and L1 analyticity. J. London Math. Soc. (2) 52 (1995) 177-184.
144. Uniformly elliptic operators with measurable coefcients. J Functional Anal.
132 (1995)141-169.
145. Non-Gaussian aspects of heat kernel behaviour. J. London Math. Soc. (2) 55 (1997) 105-125.
146. Long time asymptotics of fourth order parabolic equations. J. d'Analyse Math. 67 (1995) 323-345.
147. Nonlinear Schrdinger operators and molecular structure. J. Phys. A, Math. Gen. 28 (1995) 4025-4041.
148. Heat kernel bounds for higher order elliptic operators. pp III.1-III.11,Journes "Equations aux drives partielles", Proceedings Saint Jean de Monts, May/June 1995. Groupement de Recherche CNRS, no.1151, Rennes, 1995. ISBN 2-73-02-0331.
149. with G Barbatis. Sharp bounds on heat kernels of higher order uniformlyelliptic operators. J. Oper. Theory 36 (1996) 179-198.
150. Lp spectral independence for certain uniformly elliptic operators. pp122–125in `Partial Differential Equations in Mathematical Physics', eds. L Hrmander andA Melin. Birkhauser, 1996.
152. with A M Hinz. Explicit constants for Rellich inequalities in Lp(f). Math.Zeit. 227 (1998) 511-523.
153. with A M Hinz. Kato class potentials for higher order elliptic operators. J.London Math. Soc. (2) 58 (1998) 669-678.
153. with L Parnovski. Trapped modes in acoustic waveguides. Quart. J. Mech.and Appl. Math. 51 (1998) 477-492.
154. Limits on Lp regularity of self-adjoint elliptic operators. J. Dif. Eqns. 135(1997) 83-102.
155. Lp spectral theory of higher order elliptic differential operators. Bull. LondonMath. Soc. 29 (1997) 513-546.
156. Spectral enclosures and complex resonances for general self-adjoint operators.LMS J. Comput. Math. 1 (1998) 42-74.
157. Pointwise lower bounds on the heat kernels of higher order elliptic operators.Math. Proc. Camb. Phil. Soc. 125 (1999) 105-111.
158. Ground state energy of almost periodic Schrdinger operators. Ergodic Theoryand Dynamical Systems 19 (1999) 591-609.
159. A hierarchical method for obtaining eigenvalue enclosures. Math. of Comp.69 (2000) 1435-1455.
160. Pseudospectra of differential operators. J. Operator Theory 43 (2000) 243-262.
161. Pseudospectra, the harmonic oscillator and complex resonances. Proc. R.Soc. Lond. A, 455 (1999) 585-599.
162. Semi-classical states for non-self-adjoint Schrdinger operators. Commun.Math. Phys. 200 (1999) 35-41.
163. Sharp boundary estimates for elliptic operators. Math. Proc. Camb. Phil.Soc. 129 (2000) 165-178.
164. A review of Hardy inequalities. in 'Conference in honour of V G Maz'ya,Rostock, 1998'. Operator Theory: Advances and Applications, vol.110, Birkhauser Verlag, Basel, 1999.
165. The computation of thresholds for Schrdinger operators. LMS J. Comput.Math. 2 (1999) 139-154.
166. Computational spectral theory. pp 76–94 in 'Spectral Theory and Geometry',eds. E B Davies and Y Safarov. London Math. Soc. Lecture Note Series no. 273.Cambridge University Press, Cambridge, 1999.
2000s
167. with A Aslanyan: Spectral instability for some Schrdinger operators. Numerische Mathematik 85 (2000) 525-552.
168. Wild spectral behaviour of anharmonic oscillators. Bull. London Math. Soc.32 (2000) 432-438.
169. with B M Brown, P J Jimack, M D Mihajlovic: A numerical investigation of the solution of a class of fourth order eigenvalue problems. Proc. Roy. Soc.London A 456 (2000) 1505-1521.
170. with A Aslanyan: On eigenfunction approximations for typical non-self-adjoint Schrdinger operators. Proc. Roy. Soc. London A 456 (2000) 1291-1303.
171. with A A Abramov and A Aslanyan: Bounds on complex eigenvalues andresonances. J. Phys. A, 34 (2001) 57-72.
172. Spectral properties of random non-self-adjoint matrices and operators. Proc.Roy. Soc. London A 457 (2001) 191-206.
173 Spectral Theory of Pseudo-ergodic Operators. Commun. Math. Phys. 216(2001) 687-704.
174 with V I Burenkov: Spectral stability of the Neumann Laplacian. J. Dif.Eqns. 186 (2002) 485-508.
175 with G M L Gladwell, J Leydold and P F Stadler: Discrete nodal domaintheorems. Linear Alg. and Applic. 336 (2001) 51-60.
176 Building infnite machines. Brit. J. Phil. Sci. 52 (2001) 671-682.
177 Empiricism in Arithmetic and Analysis. Philosophia Mathematica (3) 11(2003) 53-66.
178 with M Lambrou: On the law of implication. Crux Mathematicorum with Mathematical Mayhem. 25 no. 8 (1999) 525-528.
179 with Jiban Nath: Schrdinger operators with slowly decaying potentials. J.Comp. Appl. Math. 148 (2002) 1-28.
180 The Newtonian myth.Stud. Hist. Phil. Sci. 34 (2003) 763-780.Errata, Stud. Hist. Phil. Sci. 35 (2004) 413.
181 with A Aslanyan: Separation of variables in deformed cylinders.J. Comput. Phys. 174 (2001) 327-344.
182 Eigenvalues of an elliptic system.Math. Zeit. 243 (2003) 719-743.
183 Non-self-adjoint differential operators.Bull. London Math. Soc. 34 (2002) 513-532.
184 Quantum mechanics does not require the continuity of space.Stud. Hist. Phil. Mod. Phys. 34 (2003) 319-328.
186 with M Plum: Spectral pollution.IMA J. Numer. Anal. 24 (2004) 417-438.
187 Semigroup growth bounds. J. Oper. Theory 53:2 (2005) 225-249.
188 Some remarks on the foundations of quantum mechanics.Brit. J. Phil. Sci. 56 (2005) 521-539.
189 with A Kuijlaars: Spectral asymptotics of the non-self-adjoint harmonic oscillator.J. London Math. Soc. (2) 70 (2004) 420-426.
190 Computing the decay of a simple reversible sub-Markov semigroup.London Math. Soc. J. Comp. Math. 7 (2004) 1-20.
191 Semi-classical analysis and pseudospectra.J. Dif. Eqns. 216 (2005) 153-187.
192 A defence of mathematical pluralism.Phil.Math. 13(3) (2005) 252-276.
193 Spectral bounds using higher order numerical ranges.LMS J. Comput. Math. 8 (2005) 17-45.
194 Pluralism in Mathematics.Phil. Trans. Royal Soc. A 363 (2005) 2449-2460.
196 Triviality of the peripheral point spectrum.J. Evol. Eqns. 5 (2005) 407-415.
197 Some refections on Newton's 'Principia'.Preprint Feb 2005. British J. Hist. Sci., to appear.
198 With B Simon: Eigenvalue Estimates for non-normal matrices and the zeros ofrandom orthogonal polynomials on the unit circle.J. Approx. Theory 141 (2006) 189-213.
199 Non-Self-Adjoint Operators and Pseudospectra in `Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday', eds. P. Deift, F. Gesztesy, P. Perry, and W. Schlag. Proc.Symp. Pure Math., Amer. Math. Soc., Providence, RI, to appear.
200 Approximate Diagonalization. Preprint, 2006.
202 Epistemological Pluralism. Preprint 2006.
203 with M Hager: Perturbations of Jordan Matrices. Preprint, 2006.
History and Philosophy of Science
The Mechanical Philosophy 1660 - 1675. Preprint August 2007.
Some Reflections on Newton's 'Principia' Preprint February 2005. To appear in British J. Hist. Sci.
Whither Mathematics? Notices Amer. Math. Soc. 52 (no. 11) (2005) 1350-1356.
Pluralism in mathematics Phil. Trans. Royal Soc. A 363 (2005) 2449-2460.
A defence of mathematical pluralism. Phil. Math. 13(3) (2005) 252-276.
Some remarks on the foundations of quantum mechanics. Brit. J. Phil. Sci. 56 (2005) 521-539.
Science in the Looking Glass. Published by Oxford University Press in August 2003, and in paperback in May 2007.
Building infinite machines. British Journal for the Philosophy of Science 52 (2001) 671-582.
Empiricism in Arithmetic and Analysis. Philosophia Mathematica (3) 11 (2003) 53-66.
The Newtonian Myth Studies Hist. Phil. Sci, part A, 34 (2003) 763-780.
Quantum mechanics does not require the continuity of space Stud. Hist. Phil. Modern Phys. 34 (2003) 319-328.