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Professor Karl Schmidt
Teams and roles for Karl Schmidt
Overview
Research interests
My research falls into several different areas of pure and applied mathematics, loosely held together by the guiding idea of spectral analysis of matrices and linear operators.
- Spectral theory - differential operators, functional analysis, operator theory, asymptotic analysis, differential equations, computational spectral theory
- Multivariate statistics - singular spectrum analysis of time series, mathematical methods in medical genetics
- Combinatorial number theory and linear algebra - matrix symmetries, sum systems of integers, generalised divisor functions, Dirichlet series
Teaching
I have taught a wide range of mathematical subjects, ranging from mathematical analysis, ordinary and partial differential equations, functional analysis, spectral theory, topology to numerical analysis, quantum mechanics and classical mechanics.
My current teaching comprises a suite of modules on functional and harmonic analysis (Year 3) and spectral theory of self-adjoint operators (Year 4 / MSc) as well as a PhD-level module on spectral theory of ordinary differential equations for the MAGIC consortium.
I regularly supervise undergraduate (Year 3 and Year 4) projects on topics in spectral theory, mathematical analysis and the mathematics of musical temperaments.
Administrative functions
Postgraduate Research Studies Lead, COMAT
Publication
2026
- Yao, K. et al. 2026. 78. Joint modelling of polygenic risk and rare variants enhances liability-based disease prediction [Abstract]. European Neuropsychopharmacology 111 (Supple) 113105. (10.1016/j.euroneuro.2026.113105)
- Escott-Price, V. , Leonenko, G. and Schmidt, K. M. 2026. Positioning personal polygenic risk score against the population background. Research in Statistics 4 (1) 2685935. (10.1080/27684520.2026.2685935)
2025
- Law, A. D. , Lettington, M. C. and Schmidt, K. M. 2025. Multifactorisations and divisor functions. UTILITAS MATHEMATICA 125 , pp.43-60. (10.61091/um125-03)
- Schmidt, K. M. and Umeda, T. 2025. On the resolvent convergence of discrete Dirac operators on 3D cubic lattices. Journal of Mathematical Analysis and Applications 546 (2) 129247. (10.1016/j.jmaa.2025.129247)
- Jameel, G. S. and Schmidt, K. M. 2025. Eigenvalue bounds for perturbed periodic Dirac operators. Journal of Spectral Theory 15 (3), pp.1279-1303. (10.4171/jst/556)
- Higham, N. J. , Lettington, M. C. and Schmidt, K. M. 2025. Symmetry decomposition and matrix multiplication. Linear Algebra and its Applications 710 , pp.310-335. (10.1016/j.laa.2025.01.041)
2023
- Escott-Price, V. and Schmidt, K. M. 2023. Pitfalls of predicting age related traits by polygenic risk scores. Annals of Human Genetics 87 (5), pp.203-209. (10.1111/ahg.12520)
- Schmidt, K. M. and Umeda, T. 2023. Continuum limits for discrete Dirac operators on 2D square lattices. Analysis and Mathematical Physics 13 (46)(10.1007/s13324-023-00809-7)
- Simmonds, E. et al. 2023. What does heritability of Alzheimer’s disease represent?. PLoS ONE 18 (4) e0281440. (10.1371/journal.pone.0281440)
- Lettington, M. and Schmidt, K. 2023. On the sum of left and right circulant matrices. Linear Algebra and its Applications 658 , pp.62-85. (10.1016/j.laa.2022.10.024)
2021
- Brown, B. M. et al. 2021. The inverse problem for a spectral asymmetry function of the Schrödinger operator on a finite interval. Mathematika 67 (4), pp.788-806. (10.1112/mtk.12105)
- Escott-Price, V. and Schmidt, K. M. 2021. Correction to: Probability of Alzheimer's disease based on common and rare genetic variants. Alzheimer's Research and Therapy 13 (1) 156. (10.1186/s13195-021-00898-1)
- Escott-Price, V. and Schmidt, K. M. 2021. Probability of Alzheimer’s disease based on common and rare genetic variants. Alzheimer's Research & Therapy 13 140. (10.1186/s13195-021-00884-7)
- Higham, N. J. , Lettington, M. C. and Schmidt, K. M. 2021. Integer matrix factorisations, superalgebras and the quadratic form obstruction. Linear Algebra and its Applications 622 , pp.250-267. (10.1016/j.laa.2021.03.028)
- Escott-Price, V. and Schmidt, K. M. 2021. Challenges of adjusting Single-Nucleotide Polymorphism effect sizes for linkage disequilibrium. Human Heredity 85 (1), pp.24-34. (10.1159/000513303)
2020
- Lettington, M. C. and Schmidt, K. M. 2020. Divisor functions and the number of sum systems. Integers 20 A61.
- Hill, S. L. et al. 2020. Some properties and applications of non-trivial divisor functions. Ramanujan Journal 51 (10.1007/s11139-018-0093-9)
2019
- Huxley, M. N. , Lettington, M. C. and Schmidt, K. M. 2019. On the structure of additive systems of integers. Periodica Mathematica Hungarica 78 , pp.178-199. (10.1007/s10998-018-00275-w)
- Phillips, T. R. L. , Schmidt, K. M. and Zhigljavsky, A. 2019. Extension of the Schoenberg theorem to integrally conditionally positive definite functions. Journal of Mathematical Analysis and Applications 470 (1), pp.659-678. (10.1016/j.jmaa.2018.10.032)
- Vivian-Griffiths, T. et al. 2019. Predictive modeling of schizophrenia from genomic data: Comparison of polygenic risk score with kernel support vector machines approach. American Journal of Medical Genetics Part B: Neuropsychiatric Genetics 180 (1), pp.80-85. (10.1002/ajmg.b.32705)
2018
- Baker, E. et al. 2018. POLARIS: polygenic LD-adjusted risk score approach for set-based analysis of GWAS data. Genetic Epidemiology 42 (4), pp.366-377. (10.1002/gepi.22117)
- Hill, S. , Lettington, M. C. and Schmidt, K. M. 2018. Block representations and spectral properties of constant sum matrices. Electronic Journal of Linear Algebra 34 , pp.170-190. (10.13001/1081-3810.3530)
- Phillips, T. and Schmidt, K. M. 2018. On unbounded positive definite functions. Mathematica Pannonica 26 (2), pp.33-51.
2017
- Lettington, M. C. , Schmidt, K. M. and Hill, S. 2017. On superalgebras of matrices with symmetry properties. Linear and Multilinear Algebra 66 (8), pp.1538-1563. (10.1080/03081087.2017.1363153)
- Vronskaya-Robl, M. and Schmidt, K. M. 2017. On perturbation stability of SSA and MSSA forecasts and the supportiveness of time series. Statistics and Its Interface 10 (1), pp.33-46.
2016
- Schmidt, K. M. 2016. On the asymptotics of the spectral density of radial Dirac operators with divergent potential. Integral Equations and Operator Theory 85 , pp.137-149. (10.1007/s00020-015-2276-8)
2015
- Schmidt, K. M. and Umeda, T. 2015. Schnol's theorem and spectral properties of massless Dirac operators with scalar potentials. Letters in Mathematical Physics 105 , pp.1479-1497. (10.1007/s11005-015-0799-1)
2014
- Brunnock, R. , Lettington, M. C. and Schmidt, K. M. 2014. On square roots and norms of matrices with symmetry properties. Linear Algebra and its Applications 459 , pp.175-207. (10.1016/j.laa.2014.06.054)
- Hughes, D. and Schmidt, K. M. 2014. Absolutely continuous spectrum of Dirac operators with square-integrable potentials. Proceedings of the Royal Society of Edinburgh: Section A Mathematics 144 (3), pp.533-555. (10.1017/S0308210512001187)
- Escott-Price, V. , Ghodsi, M. and Schmidt, K. M. 2014. How allele frequency and study design affect association test statistics with misrepresentation errors. Biostatistics 15 (2), pp.311-326. (10.1093/biostatistics/kxt048)
- Schmidt, K. M. and Umeda, T. 2014. Spectral properties of massless Dirac operators with real-valued potentials. Presented at: Spectral and scattering theory and related topics Kyoto University, Japan 14-16 December 2011. RIMS Kôkyûroku Bessatsu B45: Spectral and Scattering Theory and Related Topics. Kyoto, Japan: , pp.25-30.
2013
- Brown, B. M. , Eastham, M. S. P. and Schmidt, K. M. 2013. Periodic differential operators. Operator Theory: Advances and Applications Vol. 230Basel: Birkhäuser.
- Kerfriden, P. et al. 2013. Statistical extraction of process zones and representative subspaces in fracture of random composites. International Journal for Multiscale Computational Engineering 11 (3), pp.253-287. (10.1615/IntJMultCompEng.2013005939)
- Schmidt, K. M. and Zhigljavsky, A. A. 2013. An extremal property of the generalized arcsine distribution. Metrika 76 (3), pp.347-355. (10.1007/s00184-012-0391-y)
2012
- Brown, B. M. and Schmidt, K. M. 2012. On the HELP Inequality for Hill Operators on Trees. In: Brown, B. M. , Lang, J. and Wood, I. G. eds. Spectral Theory, Function Spaces and Inequalities: New Techniques and Recent Trends. Operator Theory: Advances and Applications Vol. 219.Basel: Springer. , pp.21-36. (10.1007/978-3-0348-0263-5_2)
- Escott-Price, V. et al. 2012. Permutation-based approaches do not adequately allow for linkage disequilibrium in gene-wide multi-locus association analysis. European Journal of Human Genetics 20 (8), pp.890-896. (10.1038/ejhg.2012.8)
2011
- Korotyaev, E. L. and Schmidt, K. M. 2011. On the resonances and eigenvalues for a 1D half-crystal with localised impurity. Journal für die reine und angewandte Mathematik (Crelles Journal) 2012 (670), pp.217-248. (10.1515/CRELLE.2011.153)
2010
- Schmidt, K. M. 2010. Spectral properties of rotationally symmetric massless dirac operators. Letters in Mathematical Physics 92 (3), pp.231-241. (10.1007/s11005-010-0393-5)
2009
- Brown, B. M. , Christiansen, J. S. and Schmidt, K. M. 2009. Spectral properties of a q-Sturm–Liouville Operator. Communications in Mathematical Physics 287 (1), pp.259-274. (10.1007/s00220-008-0623-1)
- Evans, W. D. and Schmidt, K. M. 2009. A discrete Hardy-Laptev-Weidl type inequality and associated Schrödinger-type operators. Revista Matemática Complutense 22 (1), pp.75-90.
- Schmidt, K. M. and Zhigljavsky, A. A. 2009. A characterization of the arcsine distribution. Statistics & Probability Letters 79 (24), pp.2451-2455. (10.1016/j.spl.2009.08.018)
2008
- Escott-Price, V. and Schmidt, K. M. 2008. On multiple-testing correction in genome-wide association studies. Genetic Epidemiology 32 (6), pp.567-573. (10.1002/gepi.20331)
- Aumann, S. , Brown, B. M. and Schmidt, K. M. 2008. A Hardy-Littlewood-type inequality for the p-Laplacian. Bulletin of the London Mathematical Society 40 (3), pp.525-531. (10.1112/blms/bdn035)
- Brown, B. M. , Langer, M. and Schmidt, K. M. 2008. The HELP inequality on trees. Presented at: Isaac Newton Institute for Mathematical Sciences Cambridge, UK 8 January - 29 June 2007. Published in: Exner, P. et al., Analysis on Graphs and Its Applications. Proceedings of Symposia in Pure Mathematics Vol. 77. Providence, RI: American Mathematical Society. , pp.337-354.
- Eastham, M. S. P. and Schmidt, K. M. 2008. Asymptotics of the spectral density for radial dirac operators with divergent potentials. Publications of the Research Institute for Mathematical Sciences 44 (1), pp.107-129. (10.2977/prims/1207921078)
2007
- Monaquel, S. J. and Schmidt, K. M. 2007. On M-functions and operator theory for non-self-adjoint discrete Hamiltonian systems. Journal of Computational and Applied Mathematics 208 (1), pp.82-101. (10.1016/j.cam.2006.10.043)
2006
- Schmidt, K. M. 2006. Exceptional coupling constants for the Coulomb-Dirac operator with anomalous magnetic moment. Journal of Computational and Applied Mathematics 194 (1), pp.17-25. (10.1016/j.cam.2005.06.022)
- Escott-Price, V. and Schmidt, K. M. 2006. Individual SNP Allele Reconstruction from Informative Markers Selected by a Non-Linear Gauss-Type Algorithm. Human Heredity 62 (2), pp.97-106. (10.1159/000096097)
- Escott-Price, V. and Schmidt, K. M. 2006. Susceptibility of Biallelic Haplotype and Genotype Frequencies to Genotyping Error. Biometrics 62 (4), pp.1116-1123. (10.1111/j.1541-0420.2006.00563.x)
2005
- Eastham, M. S. P. and Schmidt, K. M. 2005. Absence of high-energy spectral concentration for Dirac systems with divergent potentials. Proceedings of the Royal Society of Edinburgh: Section A Mathematics 135 (4), pp.689-702. (10.1017/S0308210500004078)
- Escott-Price, V. et al. 2005. Design of Case-controls Studies with Unscreened Controls. Annals of Human Genetics 69 (5), pp.566-576. (10.1111/j.1529-8817.2005.00175.x)
2004
- Kalf, H. and Schmidt, K. M. 2004. Spectral stability of the Coulomb-Dirac Hamiltonian with anomalous magnetic moment. Journal of Differential Equations 205 (2), pp.408-423. (10.1016/j.jde.2004.04.009)
- Brown, B. M. et al. 2004. Distribution of eigenvalues in gaps of the essential spectrum of Sturm-Liouville operators - a numerical approach. Journal Of Computational Analysis And Applications 6 (1), pp.85-95.
- Schmidt, K. M. 2004. On the integrability of differential equations by quadratures according to Maximovič. Journal of Computational and Applied Mathematics 171 (1-2), pp.393-410. (10.1016/j.cam.2004.01.021)
2003
- Escott-Price, V. and Schmidt, K. M. 2003. Approximate projectors in singular spectrum analysis. SIAM Journal on Matrix Analysis and Applications (SIMAX) 24 (4), pp.932-942. (10.1137/S0895479801398967)
- Schmidt, K. M. 2003. Eigenvalue asymptotics of perturbed periodic Dirac systems in the slow-decay limit. Proceedings Of The American Mathematical Society 131 (4), pp.1205-1214.
- Schmidt, K. M. 2003. Spectral properties of Dirac systems with coefficients infinite at infinity. RIMS Kokyuroku 1315 , pp.24-31.
2002
- Schmidt, K. M. 2002. Eigenvalues in gaps of perturbed periodic Dirac operators: numerical evidence. Journal of Computational and Applied Mathematics 148 (1), pp.169-181. (10.1016/S0377-0427(02)00580-0)
- Schmidt, K. M. 2002. A short proof for Bargmann-type inequalities. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 458 (2027), pp.2829-2832. (10.1098/rspa.2002.1021)
2001
- Schmidt, K. M. 2001. A remark on a paper by Evans and Harris on the point spectra of Dirac operators. Proceedings Of The Royal Society Of Edinburgh Section A-Mathematics 131 (A), pp.1237-1243. (10.1017/S0308210500001360)
- Schmidt, K. M. 2001. An Application of Gesztesy–Simon–Teschl Oscillation Theory to a Problem in Differential Geometry. Journal of Mathematical Analysis and Applications 261 (1), pp.61-71. (10.1006/jmaa.2001.7471)
2000
- Schmidt, K. M. 2000. A Remark on The Essential Spectra of Dirac Systems. Bulletin of the London Mathematical Society 32 (1), pp.63-70. (10.1112/S0024609399006463)
- Schmidt, K. M. 2000. Critical Coupling Constants and Eigenvalue Asymptotics of Perturbed Periodic Sturm-Liouville Operators. Communications in Mathematical Physics 211 (2), pp.465-485. (10.1007/s002200050822)
- Schmidt, K. M. 2000. Relative Oscillation–Non-Oscillation Criteria for Perturbed Periodic Dirac Systems. Journal of Mathematical Analysis and Applications 246 (2), pp.591-607. (10.1006/jmaa.2000.6821)
1999
- Schmidt, K. M. 1999. Oscillation of the perturbed Hill equation and the lower spectrum of radially periodic Schrodinger operators in the plane. Proceedings of the American Mathematical Society 127 (8), pp.2367-2374.
1998
- Brown, B. M. et al. 1998. Welsh Eigenvalues of radially periodic Schrödinger operators. Journal of Mathematical Analysis and Applications 225 (1), pp.347-357. (10.1006/jmaa.1998.6048)
- Schmidt, K. M. and Yamada, O. 1998. Spherically symmetric dirac operators with variable mass and potentials infinite at infinity. Publications Of The Research Institute For Mathematical Sciences 34 (3), pp.211-227.
1997
- Schmidt, K. M. 1997. Absolutely continuous spectrum of Dirac systems with potentials infinite at infinity. Mathematical Proceedings of the Cambridge Philosophical Society 122 (2), pp.377-384.
1996
- Schmidt, K. M. 1996. Dense point spectrum for the one-dimensional Dirac operator with an electrostatic potential. Proceedings of the Royal Society of Edinburgh, Section: A Mathematics 126 (5), pp.1087-1096. (10.1017/S0308210500023271)
1995
- Schmidt, K. M. 1995. A remark on boundary value problems for the Dirac operator. Quarterly Journal of Mathematics 46 (184), pp.509-516.
- Schmidt, K. M. 1995. Dense point spectrum and absolutely continuous spectrum in spherically symmetric Dirac operators. Forum Mathematicum 7 (7), pp.459-475. (10.1515/form.1995.7.459)
1993
- Schmidt, K. M. 1993. On the essential spectrum of Dirac operators with spherically symmetric potentials. Mathematische Annalen 297 (1), pp.117-131. (10.1007/BF01459491)
- Schmidt, K. M. 1993. On the genericity of nonvanishing instability intervals in periodic Dirac systems. Annales De L'Institut Henri Poincare-Physique Theorique 59 (3), pp.315-326.
Articles
- Yao, K. et al. 2026. 78. Joint modelling of polygenic risk and rare variants enhances liability-based disease prediction [Abstract]. European Neuropsychopharmacology 111 (Supple) 113105. (10.1016/j.euroneuro.2026.113105)
- Escott-Price, V. , Leonenko, G. and Schmidt, K. M. 2026. Positioning personal polygenic risk score against the population background. Research in Statistics 4 (1) 2685935. (10.1080/27684520.2026.2685935)
- Law, A. D. , Lettington, M. C. and Schmidt, K. M. 2025. Multifactorisations and divisor functions. UTILITAS MATHEMATICA 125 , pp.43-60. (10.61091/um125-03)
- Schmidt, K. M. and Umeda, T. 2025. On the resolvent convergence of discrete Dirac operators on 3D cubic lattices. Journal of Mathematical Analysis and Applications 546 (2) 129247. (10.1016/j.jmaa.2025.129247)
- Jameel, G. S. and Schmidt, K. M. 2025. Eigenvalue bounds for perturbed periodic Dirac operators. Journal of Spectral Theory 15 (3), pp.1279-1303. (10.4171/jst/556)
- Higham, N. J. , Lettington, M. C. and Schmidt, K. M. 2025. Symmetry decomposition and matrix multiplication. Linear Algebra and its Applications 710 , pp.310-335. (10.1016/j.laa.2025.01.041)
- Escott-Price, V. and Schmidt, K. M. 2023. Pitfalls of predicting age related traits by polygenic risk scores. Annals of Human Genetics 87 (5), pp.203-209. (10.1111/ahg.12520)
- Schmidt, K. M. and Umeda, T. 2023. Continuum limits for discrete Dirac operators on 2D square lattices. Analysis and Mathematical Physics 13 (46)(10.1007/s13324-023-00809-7)
- Simmonds, E. et al. 2023. What does heritability of Alzheimer’s disease represent?. PLoS ONE 18 (4) e0281440. (10.1371/journal.pone.0281440)
- Lettington, M. and Schmidt, K. 2023. On the sum of left and right circulant matrices. Linear Algebra and its Applications 658 , pp.62-85. (10.1016/j.laa.2022.10.024)
- Brown, B. M. et al. 2021. The inverse problem for a spectral asymmetry function of the Schrödinger operator on a finite interval. Mathematika 67 (4), pp.788-806. (10.1112/mtk.12105)
- Escott-Price, V. and Schmidt, K. M. 2021. Correction to: Probability of Alzheimer's disease based on common and rare genetic variants. Alzheimer's Research and Therapy 13 (1) 156. (10.1186/s13195-021-00898-1)
- Escott-Price, V. and Schmidt, K. M. 2021. Probability of Alzheimer’s disease based on common and rare genetic variants. Alzheimer's Research & Therapy 13 140. (10.1186/s13195-021-00884-7)
- Higham, N. J. , Lettington, M. C. and Schmidt, K. M. 2021. Integer matrix factorisations, superalgebras and the quadratic form obstruction. Linear Algebra and its Applications 622 , pp.250-267. (10.1016/j.laa.2021.03.028)
- Escott-Price, V. and Schmidt, K. M. 2021. Challenges of adjusting Single-Nucleotide Polymorphism effect sizes for linkage disequilibrium. Human Heredity 85 (1), pp.24-34. (10.1159/000513303)
- Lettington, M. C. and Schmidt, K. M. 2020. Divisor functions and the number of sum systems. Integers 20 A61.
- Hill, S. L. et al. 2020. Some properties and applications of non-trivial divisor functions. Ramanujan Journal 51 (10.1007/s11139-018-0093-9)
- Huxley, M. N. , Lettington, M. C. and Schmidt, K. M. 2019. On the structure of additive systems of integers. Periodica Mathematica Hungarica 78 , pp.178-199. (10.1007/s10998-018-00275-w)
- Phillips, T. R. L. , Schmidt, K. M. and Zhigljavsky, A. 2019. Extension of the Schoenberg theorem to integrally conditionally positive definite functions. Journal of Mathematical Analysis and Applications 470 (1), pp.659-678. (10.1016/j.jmaa.2018.10.032)
- Vivian-Griffiths, T. et al. 2019. Predictive modeling of schizophrenia from genomic data: Comparison of polygenic risk score with kernel support vector machines approach. American Journal of Medical Genetics Part B: Neuropsychiatric Genetics 180 (1), pp.80-85. (10.1002/ajmg.b.32705)
- Baker, E. et al. 2018. POLARIS: polygenic LD-adjusted risk score approach for set-based analysis of GWAS data. Genetic Epidemiology 42 (4), pp.366-377. (10.1002/gepi.22117)
- Hill, S. , Lettington, M. C. and Schmidt, K. M. 2018. Block representations and spectral properties of constant sum matrices. Electronic Journal of Linear Algebra 34 , pp.170-190. (10.13001/1081-3810.3530)
- Phillips, T. and Schmidt, K. M. 2018. On unbounded positive definite functions. Mathematica Pannonica 26 (2), pp.33-51.
- Lettington, M. C. , Schmidt, K. M. and Hill, S. 2017. On superalgebras of matrices with symmetry properties. Linear and Multilinear Algebra 66 (8), pp.1538-1563. (10.1080/03081087.2017.1363153)
- Vronskaya-Robl, M. and Schmidt, K. M. 2017. On perturbation stability of SSA and MSSA forecasts and the supportiveness of time series. Statistics and Its Interface 10 (1), pp.33-46.
- Schmidt, K. M. 2016. On the asymptotics of the spectral density of radial Dirac operators with divergent potential. Integral Equations and Operator Theory 85 , pp.137-149. (10.1007/s00020-015-2276-8)
- Schmidt, K. M. and Umeda, T. 2015. Schnol's theorem and spectral properties of massless Dirac operators with scalar potentials. Letters in Mathematical Physics 105 , pp.1479-1497. (10.1007/s11005-015-0799-1)
- Brunnock, R. , Lettington, M. C. and Schmidt, K. M. 2014. On square roots and norms of matrices with symmetry properties. Linear Algebra and its Applications 459 , pp.175-207. (10.1016/j.laa.2014.06.054)
- Hughes, D. and Schmidt, K. M. 2014. Absolutely continuous spectrum of Dirac operators with square-integrable potentials. Proceedings of the Royal Society of Edinburgh: Section A Mathematics 144 (3), pp.533-555. (10.1017/S0308210512001187)
- Escott-Price, V. , Ghodsi, M. and Schmidt, K. M. 2014. How allele frequency and study design affect association test statistics with misrepresentation errors. Biostatistics 15 (2), pp.311-326. (10.1093/biostatistics/kxt048)
- Kerfriden, P. et al. 2013. Statistical extraction of process zones and representative subspaces in fracture of random composites. International Journal for Multiscale Computational Engineering 11 (3), pp.253-287. (10.1615/IntJMultCompEng.2013005939)
- Schmidt, K. M. and Zhigljavsky, A. A. 2013. An extremal property of the generalized arcsine distribution. Metrika 76 (3), pp.347-355. (10.1007/s00184-012-0391-y)
- Escott-Price, V. et al. 2012. Permutation-based approaches do not adequately allow for linkage disequilibrium in gene-wide multi-locus association analysis. European Journal of Human Genetics 20 (8), pp.890-896. (10.1038/ejhg.2012.8)
- Korotyaev, E. L. and Schmidt, K. M. 2011. On the resonances and eigenvalues for a 1D half-crystal with localised impurity. Journal für die reine und angewandte Mathematik (Crelles Journal) 2012 (670), pp.217-248. (10.1515/CRELLE.2011.153)
- Schmidt, K. M. 2010. Spectral properties of rotationally symmetric massless dirac operators. Letters in Mathematical Physics 92 (3), pp.231-241. (10.1007/s11005-010-0393-5)
- Brown, B. M. , Christiansen, J. S. and Schmidt, K. M. 2009. Spectral properties of a q-Sturm–Liouville Operator. Communications in Mathematical Physics 287 (1), pp.259-274. (10.1007/s00220-008-0623-1)
- Evans, W. D. and Schmidt, K. M. 2009. A discrete Hardy-Laptev-Weidl type inequality and associated Schrödinger-type operators. Revista Matemática Complutense 22 (1), pp.75-90.
- Schmidt, K. M. and Zhigljavsky, A. A. 2009. A characterization of the arcsine distribution. Statistics & Probability Letters 79 (24), pp.2451-2455. (10.1016/j.spl.2009.08.018)
- Escott-Price, V. and Schmidt, K. M. 2008. On multiple-testing correction in genome-wide association studies. Genetic Epidemiology 32 (6), pp.567-573. (10.1002/gepi.20331)
- Aumann, S. , Brown, B. M. and Schmidt, K. M. 2008. A Hardy-Littlewood-type inequality for the p-Laplacian. Bulletin of the London Mathematical Society 40 (3), pp.525-531. (10.1112/blms/bdn035)
- Eastham, M. S. P. and Schmidt, K. M. 2008. Asymptotics of the spectral density for radial dirac operators with divergent potentials. Publications of the Research Institute for Mathematical Sciences 44 (1), pp.107-129. (10.2977/prims/1207921078)
- Monaquel, S. J. and Schmidt, K. M. 2007. On M-functions and operator theory for non-self-adjoint discrete Hamiltonian systems. Journal of Computational and Applied Mathematics 208 (1), pp.82-101. (10.1016/j.cam.2006.10.043)
- Schmidt, K. M. 2006. Exceptional coupling constants for the Coulomb-Dirac operator with anomalous magnetic moment. Journal of Computational and Applied Mathematics 194 (1), pp.17-25. (10.1016/j.cam.2005.06.022)
- Escott-Price, V. and Schmidt, K. M. 2006. Individual SNP Allele Reconstruction from Informative Markers Selected by a Non-Linear Gauss-Type Algorithm. Human Heredity 62 (2), pp.97-106. (10.1159/000096097)
- Escott-Price, V. and Schmidt, K. M. 2006. Susceptibility of Biallelic Haplotype and Genotype Frequencies to Genotyping Error. Biometrics 62 (4), pp.1116-1123. (10.1111/j.1541-0420.2006.00563.x)
- Eastham, M. S. P. and Schmidt, K. M. 2005. Absence of high-energy spectral concentration for Dirac systems with divergent potentials. Proceedings of the Royal Society of Edinburgh: Section A Mathematics 135 (4), pp.689-702. (10.1017/S0308210500004078)
- Escott-Price, V. et al. 2005. Design of Case-controls Studies with Unscreened Controls. Annals of Human Genetics 69 (5), pp.566-576. (10.1111/j.1529-8817.2005.00175.x)
- Kalf, H. and Schmidt, K. M. 2004. Spectral stability of the Coulomb-Dirac Hamiltonian with anomalous magnetic moment. Journal of Differential Equations 205 (2), pp.408-423. (10.1016/j.jde.2004.04.009)
- Brown, B. M. et al. 2004. Distribution of eigenvalues in gaps of the essential spectrum of Sturm-Liouville operators - a numerical approach. Journal Of Computational Analysis And Applications 6 (1), pp.85-95.
- Schmidt, K. M. 2004. On the integrability of differential equations by quadratures according to Maximovič. Journal of Computational and Applied Mathematics 171 (1-2), pp.393-410. (10.1016/j.cam.2004.01.021)
- Escott-Price, V. and Schmidt, K. M. 2003. Approximate projectors in singular spectrum analysis. SIAM Journal on Matrix Analysis and Applications (SIMAX) 24 (4), pp.932-942. (10.1137/S0895479801398967)
- Schmidt, K. M. 2003. Eigenvalue asymptotics of perturbed periodic Dirac systems in the slow-decay limit. Proceedings Of The American Mathematical Society 131 (4), pp.1205-1214.
- Schmidt, K. M. 2003. Spectral properties of Dirac systems with coefficients infinite at infinity. RIMS Kokyuroku 1315 , pp.24-31.
- Schmidt, K. M. 2002. Eigenvalues in gaps of perturbed periodic Dirac operators: numerical evidence. Journal of Computational and Applied Mathematics 148 (1), pp.169-181. (10.1016/S0377-0427(02)00580-0)
- Schmidt, K. M. 2002. A short proof for Bargmann-type inequalities. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 458 (2027), pp.2829-2832. (10.1098/rspa.2002.1021)
- Schmidt, K. M. 2001. A remark on a paper by Evans and Harris on the point spectra of Dirac operators. Proceedings Of The Royal Society Of Edinburgh Section A-Mathematics 131 (A), pp.1237-1243. (10.1017/S0308210500001360)
- Schmidt, K. M. 2001. An Application of Gesztesy–Simon–Teschl Oscillation Theory to a Problem in Differential Geometry. Journal of Mathematical Analysis and Applications 261 (1), pp.61-71. (10.1006/jmaa.2001.7471)
- Schmidt, K. M. 2000. A Remark on The Essential Spectra of Dirac Systems. Bulletin of the London Mathematical Society 32 (1), pp.63-70. (10.1112/S0024609399006463)
- Schmidt, K. M. 2000. Critical Coupling Constants and Eigenvalue Asymptotics of Perturbed Periodic Sturm-Liouville Operators. Communications in Mathematical Physics 211 (2), pp.465-485. (10.1007/s002200050822)
- Schmidt, K. M. 2000. Relative Oscillation–Non-Oscillation Criteria for Perturbed Periodic Dirac Systems. Journal of Mathematical Analysis and Applications 246 (2), pp.591-607. (10.1006/jmaa.2000.6821)
- Schmidt, K. M. 1999. Oscillation of the perturbed Hill equation and the lower spectrum of radially periodic Schrodinger operators in the plane. Proceedings of the American Mathematical Society 127 (8), pp.2367-2374.
- Brown, B. M. et al. 1998. Welsh Eigenvalues of radially periodic Schrödinger operators. Journal of Mathematical Analysis and Applications 225 (1), pp.347-357. (10.1006/jmaa.1998.6048)
- Schmidt, K. M. and Yamada, O. 1998. Spherically symmetric dirac operators with variable mass and potentials infinite at infinity. Publications Of The Research Institute For Mathematical Sciences 34 (3), pp.211-227.
- Schmidt, K. M. 1997. Absolutely continuous spectrum of Dirac systems with potentials infinite at infinity. Mathematical Proceedings of the Cambridge Philosophical Society 122 (2), pp.377-384.
- Schmidt, K. M. 1996. Dense point spectrum for the one-dimensional Dirac operator with an electrostatic potential. Proceedings of the Royal Society of Edinburgh, Section: A Mathematics 126 (5), pp.1087-1096. (10.1017/S0308210500023271)
- Schmidt, K. M. 1995. A remark on boundary value problems for the Dirac operator. Quarterly Journal of Mathematics 46 (184), pp.509-516.
- Schmidt, K. M. 1995. Dense point spectrum and absolutely continuous spectrum in spherically symmetric Dirac operators. Forum Mathematicum 7 (7), pp.459-475. (10.1515/form.1995.7.459)
- Schmidt, K. M. 1993. On the essential spectrum of Dirac operators with spherically symmetric potentials. Mathematische Annalen 297 (1), pp.117-131. (10.1007/BF01459491)
- Schmidt, K. M. 1993. On the genericity of nonvanishing instability intervals in periodic Dirac systems. Annales De L'Institut Henri Poincare-Physique Theorique 59 (3), pp.315-326.
Book sections
- Brown, B. M. and Schmidt, K. M. 2012. On the HELP Inequality for Hill Operators on Trees. In: Brown, B. M. , Lang, J. and Wood, I. G. eds. Spectral Theory, Function Spaces and Inequalities: New Techniques and Recent Trends. Operator Theory: Advances and Applications Vol. 219.Basel: Springer. , pp.21-36. (10.1007/978-3-0348-0263-5_2)
Books
- Brown, B. M. , Eastham, M. S. P. and Schmidt, K. M. 2013. Periodic differential operators. Operator Theory: Advances and Applications Vol. 230Basel: Birkhäuser.
Conferences
- Schmidt, K. M. and Umeda, T. 2014. Spectral properties of massless Dirac operators with real-valued potentials. Presented at: Spectral and scattering theory and related topics Kyoto University, Japan 14-16 December 2011. RIMS Kôkyûroku Bessatsu B45: Spectral and Scattering Theory and Related Topics. Kyoto, Japan: , pp.25-30.
- Brown, B. M. , Langer, M. and Schmidt, K. M. 2008. The HELP inequality on trees. Presented at: Isaac Newton Institute for Mathematical Sciences Cambridge, UK 8 January - 29 June 2007. Published in: Exner, P. et al., Analysis on Graphs and Its Applications. Proceedings of Symposia in Pure Mathematics Vol. 77. Providence, RI: American Mathematical Society. , pp.337-354.
Research
Having started my academic life in mathematical physics, specifically in quantum mechanics, I have developed research interests in a variety of areas of mathematics. If there is a common theme, it will be spectral analysis - unlocking the information hidden in the eigenvalues and more general spectral properties of matrices and linear operators.
Spectral theory
I am interested in analysing the properties of linear operators, typically differential operators - such as the Schrödinger and Dirac operators of quantum mechanics and other operators of mathematical physics -, integral operators and infinite matrices, combining the powerful conceptualisations of functional analysis and operator theory with the more technical study of differential equations, especially the qualitative and asymptotic properties of their solutions (which are usually not accessible in explicit form), and with general mathematical analysis.
Multivariate statistics
I bring my knowledge of mathematical analysis to different applications in the study of real data. One such area is singular spectrum analysis, a method of time series analysis based on singular value decomposition of associated matrices.
In long-standing collaboration with Prof. Valentina Escott-Price, I contribute to the use of mathematical techniques in medical genetics, developing novel approaches to the analysis of correlated statistical data in very large and high-dimensional datasets and challenging existing metodology in the field.
Combinatorial number theory and linear algebra
With Dr Matthew Lettington, I study symmetry decompositions of matrices and ensuing questions of additive combinatorics, generalised divisor functions and corresponding Dirichlet series.
Research group
Teaching
Undergraduate / taught postgraduate - autumn semester
- MA4010/MAT020 Spectral Theory and Applications
Undergraduate - spring semester
- MA3005 Functional and Fourier Analysis
Postgraduate research level - autumn semester
- MAGIC057 Spectral Theory of Ordinary Differential Operators
Biography
Qualifications
- 1990 Dipl. phys. (Ludwig-Maximilians-Universität München)
- 1992 Dr. rer. nat. (Ludwig-Maximilians-Universität München)
- 1999 Habilitation (Ludwig-Maximilians-Universität München)
Academic positions
- 1992 - 1993 Research fellow (Mathematisches Institut, LMU München)
- 1993 - 2000 Lecturer (Mathematisches Institut, LMU München)
- 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)
- 2009 - 2017 Reader (School of Mathematics, Cardiff)
- 2017 - present Professor (School/Department of Mathematics, Cardiff)
Honours
- 2020 Elected Fellow of the Learned Society of Wales
Supervisions
Current supervision
Past projects
- 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
- M Ghodsi (joint supervision with V Moskvina) - Analysis of Dynamical Processes for Test Assessment in Medical Genetics, PhD 2012
- D Hughes - Spectral Analysis of Dirac Operators under Integrability Conditions, PhD 2012
- M Vronskaya (joint supervision with A Zhigljavsky) - Stability of Singular Spectrum Analysis and Causality in Time Series, Phd 2014
- S Hill (joint supervision with M C Lettington) - Problems Related to Number Theory: Sum-and-Distance Systems, Reversible Squares Matrices and Divisor Functions, PhD 2018
- E Baker (jointly with V Escott-Price (main supervisor) et al, MEDIC) - Investigation of New Strategies for Identifying Causal Mechanisms in Alzheimer's Disease Taking Bioinformatics Approaches Beyond GWAS, PhD 2018
- T Phillips (joint supervision with A Zhigljavsky) - On Positive and Conditionally Negative Definite Functions with a Singularity at Zero, and their Applications in Potential Theory, PhD 2018
- M Lewis (joint supervision with K Cherednichenko, Bath) - Stability of Solutions of a One-dimensional p-Laplace Equation with Periodic Potential, PhD 2019
- A Law (joint supervision with M C Lettington) - On Topics Related to Sum Systems, PhD 2023
- G S Jameel - Complex Analysis of Non-Selfadjoint Differential Operators, PhD 2026