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Dr Karl Michael Schmidt

Overview

Dr Karl Michael Schmidt Position: Reader Email: schmidtkm@cardiff.ac.uk
Telephone: +44(0)29 208 76778
Fax: +44(0)29 208 74199
Extension: 76778
Location: M/2.49

Research Interests

Ordinary and Partial Differential Operators

Spectral Theory and Mathematical Physics

Asymptotic Analysis of Differential Equations

Analysis in Probability Theory

Mathematical Methods in Medical Genetics

Research Group

Analysis and Differential Equations

Recent Significant Publications

Schmidt K M, Spectral properties of rotationally symmetric massless Dirac operators. Lett. Math. Phys. 92 (2010) 231-241

Brown B M , Christiansen J S, Schmidt K M, Spectral properties of a q-Sturm-Liouville operator. Commun. Math. Phys. 287 (2009) 259-274

Aumann S, Brown B M, Schmidt K M, A Hardy-Littlewood-type inequality for the p-Laplacian. Bull. London Math. Soc. 40 (2008) 525-531

Eastham M S P, Schmidt K M, Asymptotics of the spectral density for radial Dirac operators with divergent potentials. Publications RIMS, Kyoto (2008)

Moskvina V, Schmidt K M, On multiple testing correction in genome-wide association studies.Genetic Epidemiology 32 (2008) 567-573

Schmidt K M, Exceptional coupling constants for the Coulomb-Dirac operator with anomalous magnetic moment, J. of Computational and Applied Mathematics, 2006, 17-25, 194 (1).

Schmidt K M and Eastham M S P, Absence of high-energy spectral concentration for Dirac systems with divergent potentials, Proceedings of the Royal Society of Edinburgh Section A Mathematics, 2005, 689-702, 135 (4).

Schmidt K M and Kalf H, Spectral stability of the Coulomb-Dirac Hamiltonian with anomalous magnetic moment, J. of Diff. Eq., 2004, 408-423, 205 (2).

Teaching

Undergraduate

MA0221 Analysis III

MA2005 Ordinary Differential Equations

MA0328 Linear Spaces and Operators 

Postgraduate

MAGIC057 Spectral Theory of Ordinary Differential Operators

Administrative Duties

Director of Postgraduate Research Studies

Isaac Newton Institute Correspondent

Member of School Research Committee

Member of School Learning and Teaching Committee

Member of Graduate School in Physical Sciences & Engineering Management Group

Personal Website

Dr Karl Michael Schmidt's personal pages

Publications

On the essential spectrum of Dirac operators with spherically symmetric potentials. Math. Ann. 297 (1993) 117-131

On the genericity of nonvanishing instability intervals in periodic Dirac systems. Ann. Inst. Henri Poincaré, Phys, Théor. 59 (1993) 315-326

Dense point spectrum and absolutely continuous spectrum in spherically symmetric Dirac operators. Forum Math. 7 (1995) 459-475

A remark on boundary value problems for the Dirac operator. Quart. J. Math. Oxford (2) 46 (1995) 509-516

Dense point spectrum for the one-dimensional Dirac operator with an electrostatic potential. Proc. Roy. Soc. Edinburgh 126A (1996) 1087-1096

Absolutely continuous spectrum of Dirac systems with potentials infinite at infinity. Math. Proc. Camb. Phil. Soc. 122 (1997) 377-384

(with O. Yamada) Spherically symmetric Dirac operators with variable mass and potentials infinite at infinity. Publ. RIMS 34 (1998) 211-227

(with B.M. Brown, M.S.P. Eastham, A.M. Hinz, T. Kriecherbauer, D.K.R. McCormack) Welsh eigenvalues of radially periodic Schrödinger operators. J. Math. Anal. Appl. 225 (1998) 347-357

Oscillation of the perturbed Hill equation and the lower spectrum of radially periodic Schrödinger operators in the plane. Proc. Amer. Math. Soc. 127 (1999) 2367-2374

A remark on the essential spectra of Dirac systems. Bull. London Math. Soc. 32 (2000) 63-70

Critical coupling constants and eigenvalue asymptotics of perturbed periodic Sturm-Liouville operators. Commun. Math. Phys. 211 (2000) 465-485

Relative oscillation - non-oscillation criteria for perturbed periodic Dirac systems. J. Math. Anal. Appl. 246 (2000) 591-607

An application of Gesztesy-Simon-Teschl oscillation theory to a problem in differential geometry. J. Math. Anal. Appl. 261 (2001) 61-71

A remark on a paper by Evans and Harris on the point spectra of Dirac operators. Proc. Roy. Soc. Edinburgh 131A (2001) 1237-1243

Eigenvalues in gaps of perturbed periodic Dirac operators : numerical evidence. J. Comput. Appl. Math. 148 (2002) 169-181

A short proof for Bargmann-type inequalities. Proc. R. Soc. Lond. A 458 (2002) 2829-2832

Eigenvalue asymptotics of perturbed periodic Dirac systems in the slow-decay limit. Proc. Amer. Math. Soc. 131 (2003) 1205-1214

(with V. Moskvina) Approximate projectors in singular spectrum analysis. SIAM J. Matrix Anal. Appl. 24 (2003) 932-942

Spectral properties of Dirac systems with coefficients infinite at infinity. Wave phenomena and symptotic analysis, RIMS Kokyuroku, Kyoto Univ. 1315 (2003) 24-31

On the integrability of differential equations by quadratures according to Maximovic. J. Comput. Appl. Math. 171 (2004) 393-410

(with H. Kalf) Spectral stability of the Coulomb-Dirac Hamiltonian with anomalous magnetic moment. J. Diff. Eq. 205 (2004) 408-423

(with B.M. Brown, M.S.P. Eastham, A.M. Hinz) Distribution of eigenvalues in gaps of the essential spectrum of Sturm-Liouville operators - a numerical approach. Journal of Computational Analysis and Applications. Vol 6 No 1 (2004) 85-95

(with V. Moskvina, P. Holmans, N. Craddock) Design of case-controls studies with unscreened controls. Ann. Hum. Genet. 69 (2005) 566-576

(with M.S.P. Eastham) Absence of high-energy spectral concentration for Dirac systems with divergent potentials. Proc. Roy. Soc. Edinburgh 135A (2005) 689-702

Exceptional coupling constants for the Coulomb-Dirac operator with anomalous magnetic moment. J. Comput. Appl. Math. 194 (2006) 17-25

(with V. Moskvina) Susceptibility of biallelic haplotype and genotype frequencies to genotyping error. Biometrics 62 (2006) 1116-1123

(with V. Moskvina) Individual SNP allele reconstruction from informative markers selected by a non-linear Gauss-type algorithm. Human Heredity 62 (2006) 97-106

(with S. J. Monaquel) On M-functions and operator theory for non-selfadjoint discrete Hamiltonian systems. J. Comput. Appl. Math. 208 (2007) 82-101

(with M.S.P. Eastham) Asymptotics of the spectral density for radial Dirac operators with divergent potentials. Publications RIMS, Kyoto (2008)

(with V. Moskvina) On multiple testing correction in genome-wide association studies.Genetic Epidemiology 32 (2008) 567-573

(with B.M. Brown, M. Langer) The HELP inequality on trees,in: Analysis on Graphs and its Applications, Proceedings of Symposia in Pure Mathematics 77 (2008) 337-354

(with S. Aumann, B.M. Brown) A Hardy-Littlewood-type inequality for the p-Laplacian. Bull. London Math. Soc. 40 (2008) 525-531

(with B.M. Brown, J.S. Christiansen) Spectral properties of a q-Sturm-Liouville operator. Commun. Math. Phys. 287 (2009) 259-274

(with W.D. Evans) A discrete Hardy-Laptev-Weidl type inequality and associated Schrödinger-type operators. Rev. Mat. Complut. 22 (2009) 75-90

(with A.A. Zhigljavsky) A characterization of the arcsine distribution. Statistics and Probability Letters 79 (2009) 2451-2455

Spectral properties of rotationally symmetric massless Dirac operators. Letters in Mathematical Physics 92 (2010) 231-241

(with E.L. Korotyaev) On the resonances and eigenvalues for a 1D half-crystal with localised impurity. J. Reine Angew. Math.(submitted)

Research

Research Interests

  • Spectral properties of Dirac and Schroedinger operators: fine structure and numerics of spectra; massless Dirac operator; Klein paradox; asymptotic spectral density; anomalous magnetic moment; graph operators
  • Inequalities: Bargmann-, Hardy-, HELP-type inequalities
  • Applications of spectral analysis: stability problems in differential geometry; operators in orthogonal polynomial theory; singular spectrum analysis of time series; approximate spectral projectors in signal processing
  • Mathematical methods in genetics: dynamics of genotyping error models; stochastic analysis of tests in medical genetics; influence of correlations (linkage disequilibrium)

Current Research Funding

  • Isaac Newton Institute programme 'SRO: Spectral Theory of Relativistic Operators' (Cambridge)
  • Richard Whipp studentship `Analysis of dynamical processes for test assessment in Medical Genetics' (Cardiff University)

Major Conference Talks Since 2004

  • Himeji Conference on Partial Differential Equations (Univ. Hyogo & Ritsumeikan Univ., 2/2011)
  • UK-China Workshop on Singular Spectrum Analysis and its Applications (Cardiff, 9/2010)
  • Analysis on Graphs and its Applications (INI Cambridge, 7/2010)
  • IWOTA 2010 (Berlin, 7/2010)
  • Gregynog Workshop on Analytical and Computational Problems in Spectral Theory (Gregynog, 7/2008)
  • EPSRC Symposium, The Mathematics of Quantum Systems, Spectral Theory Workshop (Warwick, 4/2005)
  • Spectral Theory Network Conference (Cardiff, 7/2004)

Postgraduate Students

Graduated (Since 2000)

S Monaquel (joint supervision with W D Evans) - Some Problems Associated with Differential and Difference Equations, PhD 2004

S Aumann (joint supervision with B M Brown) - A Hardy-Littlewood-Type Inequality for the p-Laplacian, MPhil 2007

Current

M Ghodsi (joint supervision with V Moskvina) - Analysis of Dynamical Processes for Test Assessment in Medical Genetics

D Hughes - Spectral Analysis of Dirac Operators under Integrability Conditions

Biography

Qualifications

1990 Dipl. phys. (Ludwig-Maximilians-Universitaet Muenchen)

1992 Dr. rer. nat. (Ludwig-Maximilians-Universitaet Muenchen)

1999 Habilitation (Ludwig-Maximilians-Universitaet Muenchen)

Previous Positions

1992-1993 Research fellow (Mathematisches Institut, LMU Muenchen)

1993-2000 Lecturer (Mathematisches Institut, LMU Muenchen)

1994-1995 Visiting Associate (Dept Mathematics, California Institute of Technology, Pasadena)

2000-2003 Lecturer (School of Mathematics, Cardiff)

2003-2009 Senior Lecturer (School of Mathematics, Cardiff)