Our mathematical research focus on linear and nonlinear partial differential equations (PDEs), functional analysis, and operator theory, and here mostly operator theoretic methods in differential equations, motivated by problems in quantum physics. Analogous to Newton’s equations, describing the motion of macroscopic bodies, from falling apples to space crafts orbiting the Earth, the Schrödinger equation describes microscopic phenomena, from the simplest atoms to processes inside stars. The differential operator appearing in the Schrödinger equation, the so-called Schrödinger operator, characterizes the physical system and it is therefore one of the most interesting objects in mathematical physics; its spectral theory has deep roots in non-relativistic quantum mechanics. A number of mathematical notions and theories were inspired by the needs of the theory of the Schrödinger operator and its generalizations.
Few-body atoms and molecules
Maximum ionization
H. Cox, M. Melgaard, V.J.J. Syrjanen, Maximum Ionization in Restricted and Unrestricted Hartree-Fock Theory, Atoms 9 (2021), no. 1, article 13.
M. Melgaard and T. Johnson, On the maximal ionization for the atomic Pauli operator, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), no. 2063, 3355-3364.
Stability of matter in external fields (Auxiliary mathematical tools: Cwikel-Lieb-Rozenblum inequality and Lieb-Thirring inequalities)
M. Melgaard, E.-M. Ouhabaz, and G. Rozenblum, Negative discrete spectrum of perturbed multivortex Aharonov-Bohm Hamiltonians, Annals Henri Poincare 5 (2004), 979-1012.
M. Melgaard and G. Rozenblum, Spectral estimates for magnetic operators, Mathematica Scandinavica 79 (1996), 237-254.
Hartree-Fock method
Hartree-Fock method with magnetic fields
Hartree-Fock methods with relativistic effects
Multi configurative quantum chemistry
DFT with magnetic fields
DFT with relativistic effects
C. Argaez, M. Melgaard, Minimizers for open-shell, spin-polarised Kohn-Sham equations for non-relativistic and quasi-relativistic molecular systems, Methods and Applications in Analysis 23 (2016), no 3, 269-292
C. Argaez, M. Melgaard, Existence of a minimizer for the quasi-relativistic Kohn-Sham model, Electronic Journal of Differential Equations, Volume 2012 (2012), no. 18, 1-20.
Dynamics of quasi-relativistic boson stars in the mean-field limit
Quasi-relativistic Choquard equation arising in cosmology
M. Melgaard, Confinement effects on scattering for a nanoparticle, Acta Phys. Polon. B 38 (2007), 197-214.
M. Melgaard, Quantum collisions in semi-constrainted structures, Modern Physics Letters B 21 (2007), no. 13, 767-779.
M. Melgaard, Bound states for the three-dimensional Aharonov-Bohm quantum wire, Few-Body Systems (Formerly Acta Physica Austriaca) 45 (2004), Nos. 1-2, 77-97.
Our mathematical research focus on linear and nonlinear partial differential equations (PDEs), functional analysis, and operator theory, and here mostly operator theoretic methods in differential equations, motivated by problems in quantum physics. Analogous to Newton’s equations, describing the motion of macroscopic bodies, from falling apples to space crafts orbiting the Earth, the Schrödinger equation describes microscopic phenomena, from the simplest atoms to processes inside stars. The differential operator appearing in the Schrödinger equation, the so-called Schrödinger operator, characterizes the physical system and it is therefore one of the most interesting objects in mathematical physics; its spectral theory has deep roots in non-relativistic quantum mechanics. A number of mathematical notions and theories were inspired by the needs of the theory of the Schrödinger operator and its generalizations.
Schrödinger’s theory of quantum mechanics distinguishes bound states and scattering states, the former being associated with point spectrum and the latter with the continuous spectrum. These two states are linked by resonances associated with meta-stable (or quasi-stationary) states for certain types of quantum systems and since the beginning of quantum theory such resonances have posed some of the most notoriously difficult challenges. Resonances are pseudo-eigenvalues of certain classes of Schrödinger operators and the pseudo-eigenvalues are characterized by having nonzero imaginary parts which determine the decay rates of the corresponding unstable states. A priori, this sounds as a contradiction because a Schrödinger operator H acting on a Hilbert space is self-adjoint and, as a consequence, its spectrum is real. For this reason, some mathematicians think that the foundations of resonance theory is still not on a firm ground. However, a resonance z is associated with a “resonance eigenfunction” satisfying the equation H f = z f where the function f satisfies a condition at infinity that implies that it does not belong to the afore-mentioned Hilbert space. This approach has lead to the computation of resonances that correspond to quantities that can be measured experimentally, and the understanding that any definition of resonance must depend not only on the Schrödinger operator itself, but also on some additional background data.
B. Cheng, M. Melgaard, Poisson wave trace formula for Dirac resonances at spectrum edges and applications, Asian Journal of Mathematics 25 (2021), no. 2, 243-276.
J. Kungsman and M. Melgaard, Poisson wave trace formula for perturbed Dirac operators, J. Operator Theory 77 (2017), no. 1, 133-147.
J. Kungsman and M. Melgaard, Complex absorbing potential method for the perturbed Dirac operator, Communications in Partial Differential Equations 39 (2014), no. 8, 1451-1478.
J. Kungsman and M. Melgaard, Existence of Dirac resonances in the semi-classical limit, Dynamics of Partial Differential Equations 11 (2014), no. 4, 381-395.
J. Kungsman and M. Melgaard, Complex absorbing potential method for the perturbed Dirac operator. Clusters of resonances, Journal of Operator Theory 71 (2014), issue 1, 259-283.
J. Kungsman and M. Melgaard, Complex absorbing potential method for systems,Dissertationes Mathematicae 469 (2010), 58 pp (Polish Academy of Sciences).
M. Melgaard, Thresholds properties for matrix-valued Schrödinger operators, II. Resonances, Journal of Differential Equations 226 (2006), no. 2, 687-703.
Z. Chen, M. Melgaard, Semiclassical Egorov theorem for a matrix-valued Schrödinger operator without hyperbolicity condition, submitted manuscript, 35 pages, 2024.
Z. Chen, M. C. Dominguez, M. Melgaard, Egorov type theorem for a matrix-valued Schrödinger operator without hyperbolicity condition, submitted manuscript, 33 pages, 2024.
M. Melgaard, Thresholds properties for matrix-valued Schrödinger operators, Journal of Mathematical Physics 46 (2005), p 83507.
M. Melgaard, On bound states for systems of weakly coupled Schrödinger equations in one space, Journal of Mathematical Physics 43 (2002), no. 11, pp 5 365-385.
A. Jensen, M. Melgaard, Perturbation of eigenvalues embedded at a threshold, Proceedings of the Royal Society of Edinburgh Section A 131 (2002), 163-179.
M. Melgaard, E.-M. Ouhabaz, and G. Rozenblum, Negative discrete spectrum of perturbed multivortex Aharonov-Bohm Hamiltonians
M. Melgaard and G. Rozenblum, Spectral estimates for magnetic operators, Mathematica Scandinavica 79 (1996), 237-254.
M. Melgaard, Spectral properties in the low-energy limit of one-dimensional Schrödinger operators $-d^{2}/dx^{2}+V$. The case $<1,V1> \neq 0$, Mathematische Nachrichten 238 (2002), 113-143.
M. Melgaard, Spectral properties at a threshold for two-channel Hamiltonians. II. Applications to scattering theory, Journal of Mathematical Analysis and Applications 256 (2001), no. 2, 568-586.
M. Melgaard, Spectral properties at a threshold for two-channel Hamiltonians. I. Abstract theory, Journal of Mathematical Analysis and Applications 256 (2001), no. 1, 281-303.
M. Melgaard and G. Rozenblum, Eigenvalue asymptotics for weakly perturbed Dirac and Schrödinger operators with constant magnetic fields of full rank, Communications in Partial Differential Equations 28 (2003), Nos. 3 and 4, 697-736.
M. Melgaard, Quantum scattering near the lowest Landau threshold for a Schrödinger operator with a constant magnetic field, Central European Journal of Mathematics 1 (2003), no. 4, 477-509.
M. Melgaard, New approach to quantum scattering near the lowest Landau threshold for a Schrödinger operator with a constant magnetic field, Few-Body Systems (Formerly Acta Physica Austriaca) 32 (2002), 1-22.
J. F. Brasche and M. Melgaard, The Friedrichs extension of the Aharonov-Bohm Hamiltonian on a disk, Integral Equations Operator Theory 52 (2005), no. 3, 419-436. (Using singular Sturm-Liouville theory for ODEs)
M. Melgaard, E.-M. Ouhabaz, and G. Rozenblum, Negative discrete spectrum of perturbed multivortex Aharonov-Bohm Hamiltonians, Annals Henri Poincare 5 (2004), 979-1012.
M. Melgaard, Scattering properties for a pair of Schrödinger type operators on cylindrical domain, Cent. Eur. J. Math. 5 (2007), 134-153.
M. Melgaard, Optimal limiting absorption principle for a Schrödinger type operator on a Lipschitz cylinder, Manuscripta Mathematica 118 (2005), no. 2, 253-270.
M. Melgaard, Confinement effects on scattering for a nanoparticle, Acta Phys. Polon. B 38 (2007), 197-214.
M. Melgaard, Quantum collisions in semi-constrainted structures, Modern Physics Letters B 21 (2007), no. 13, 767-779.
M. Melgaard, Bound states for the three-dimensional Aharonov-Bohm quantum wire, Few-Body Systems (Formerly Acta Physica Austriaca) 45 (2004), Nos. 1-2, 77-97.
C. Argaez, M. Melgaard, Solutions to quasi-relativistic multi-configurative Hartree-Fock equations in quantum chemistry, Nonlinear Analysis TMA (theory, methods and applications) 75 (2012), 384-404.
M. Melgaard, F. Y. Zongo, Multiple solutions of the quasi relativistic Choquard equation, Journal of Mathematical Physics 53 (2012), 033709 (12 pp)
M. Enstedt and M. Melgaard, Existence of infinitely many distinct solutions to the quasi-relativistic Hartree-Fock equations, International Journal of Mathematics and Mathematical Sciences, vol. 2009 (2009), article ID 651871, 20 pages.
C. Argaez, M. Melgaard, Minimizers for open-shell, spin-polarised Kohn-Sham equations for non-relativistic and quasi-relativistic molecular systems, Methods and Applications in Analysis 23 (2016), no 3, 269-292
C. Argaez, M. Melgaard, Existence of a minimizer for the quasi-relativistic Kohn-Sham model, Electronic Journal of Differential Equations, Volume 2012 (2012), no. 18, 1-20.
M. Melgaard, M. Yang, Minbo, X. Zhou, Regularity, symmetry and asymptotic behaviour of solutions for some Stein-Weiss type integral systems, Pacific Journal of Mathematics 317 (2022), no. 1, 153-186.
Z. Guo, M. Melgaard, Fractional magnetic Sobolev inequalities with two variables, Mathematical Inequalities and Applications 22 (2019), no. 2, 703-718.
C. Argaez, M. Melgaard, Ground state solutions to Hartree-Fock equations with magnetic fields, Applicable Analysis 97 (2018), no. 14, 2377-2403.
Z. Guo, M. Melgaard, W. Zou, Schrödinger equations with magnetic fields and Hardy-Sobolev critical exponents, Electron. J. Differential Equations, Vol. 2017 (2017), No. 199, pp. 1-18.
M. Enstedt and M. Melgaard, Existence of a solution to Hartree-Fock equations with decreasing magnetic field, Nonlinear Analysis TMA (Theory, Methods and Applications) 69 (2008), 2125-2141.