## Dr Karl Michael Schmidt

### Overview

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**

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

### 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)