# Publications

## Recent group highlights

(For a full list see below).

Regularity, symmetry and asymptotic behaviour of solutions for some Stein-Weiss type integral systems

We consider the positive solutions of some integral systems related to the static Hartree type equations. Firstly, assuming that the exponent p, see paper abstract for details, belongs to some suitable interval depending the parameters mu, tau, alpha, and beta, we are able to prove some nonexistence results for the positive solution. Secondly, we also establish some qualitative results for the integrable solution of the system like regularity, symmetry and asymptotic behaviour. As a corollary, we deduce the corresponding results for the equivalent weighted Hartree type nonlocal equations. The results obtained in this paper generalize and complement the existing results in the literature.

M. Melgaard, M. Yang, Minbo, X. Zhou

Pacific Journal of Mathematics (2022), to appear

Solitary waves and excited states for Boson stars

We study the nonlinear, nonlocal, time-dependent partial differential equation $i \pd_{t}\vf=(\sqrt{ -\D +m^{2} } -m) \vf -\left( \frac{1}{|x|} \ast | \vf |^{2} \right)\vf$ on $\R^{3}$, which is known to describe the dynamics of quasi-relativistic boson stars in the mean-field limit. For positive mass parameter $m>0$ we establish existence of infinitely many (corresponding to distinct energies $\l_{k}$) travelling solitary waves, $\vf_{k}(x,t) = e^{i \l_{k} t } \f_{k}(x-vt)$, with speed $|v| <1$, where $c=1$ corresponds to the speed of light in our choice of units. These travelling solitary waves cannot be obtained by applying a Lorentz boost to a solitary wave at rest (with $v=0$) because Lorentz covariance fails. Instead we study a suitable variational problem for which the functions $\f_{k} \in \Hb^{1/2}(\R^{3})$ arise as solutions (called boosted excited states) to a Choquard type equation in $\R^{3}$, where the negative Laplacian is replaced by the pseudo-differential operator $\sqrt{ -\D +m^{2} } -m$ and an additional term $i (v \cdot \nabla)$ enters. Moreover, we give a new proof for existence of boosted ground states. The results are based on perturbation methods in critical point theory.

M. Melgaard, F. Y. Zongo

Analysis and Applications 20 (2022), no. 2, 285-302

Maximum Ionization in Restricted and Unrestricted Hartree-Fock Theory

In this paper we investigate the maximum number of electrons that can be bound to a system of nuclei modelled by Hartree-Fock theory. We consider both the Restricted and Unrestricted Hartree-Fock models. We are taking a non-existence approach (necessary but not sufficient), in other words we are finding an upper bound on the maximum number of electrons. In giving a detailed account of the proof of the bound by Lieb [Theorem 1, Phys. Rev. A 29 (1984), 3018] for the Hartree-Fock models we establish several new auxiliary results, furthermore we propose a condition that, if satisfied, will give an improved upper bound on the maximum number of electrons within the Restricted Hartree-Fock model. For two-electron atoms we show that the latter condition holds.

H. Cox, M. Melgaard, V.J.J. Syrjanen

Atoms 9 (2021), no. 1, article 13

Poisson wave trace formula for Dirac resonances at spectrum edges and applications

We study the self-adjoint Dirac operators $D=\ham{D}_0 + V(x)$, where $D_0$ is the free three-dimensional Dirac operator and $V(x)$ is a smooth compactly supported Hermitian matrix potential. We define resonances of $D$ as poles of the meromorphic continuation of its cut-off resolvent. By analyzing the resolvent behaviour at the spectrum edges $\pm m$, we establish a generalized Birman-Krein formula, taking into account possible resonances at $\pm m$. As an application of the new Birman-Krein formula we establish the Poisson wave trace formula in its full generality. The Poisson wave trace formula links the resonances with the trace of the difference of the wave groups. The Poisson wave trace formula, in conjunction with asymptotics of the scattering phase, allows us to prove that, under certain natural assumptions on $V$, the perturbed Dirac operator has infinitely many resonances; a result similar in nature to Melrose’s classic 1995 result for Schrödinger operators.

B. Cheng, M. Melgaard

Asian Journal of Mathematics 25 (2021), no. 2, 243-276

The bound-state stability of the hydride ion in HF theory

This review brings together mathematical proofs and high-accuracy quantum chemical calculations on the hydride ion, the anion of hydrogen. Our discussion is confined to Hartree-Fock theory and the non-relativistic time-independent Schroedinger equation, within the fixed-nucleus approximation. It is written so as to be accessible to both the mathematics and physical chemistry communities.

H. Cox, A. L. Baskerville, V. J .J. Syrjanen, M. Melgaard

Advances in Quantum Chemistry 81 (2020), 167-189

Ground state solutions to Hartree-Fock equations with magnetic fields

Within the Hartree-Fock theory of atoms and molecules we prove existence of a ground state in the presence of an external magnetic field when:(1) the diamagnetic effect is taken into account; (2) both the diamagnetic effect and the Zeeman effect are taken into account. For both cases the ground state exists provided the total charge $Z_{\rm tot}$ of the nuclei $K$ exceeds $N-1$, where $N$ is the number of electrons. For the first case, the Schrödinger case, we complement prior results by Esteban-Lions (1989) and Enstedt-Melgaard (2008) by allowing a wide class of magnetic potentials. In the second case, the Pauli case, we include the magnetic field energy in order to obtain a stable problem and we assume $Z_{\rm tot} \a^{2} \leq 0.041$, where $\a$ is the fine structure constant.the diamagnetic.

C. Argaez, M. Melgaard

Applicable Analysis 97 (2018), no. 14, 2377-2403

## Full List

Regularity, symmetry and asymptotic behaviour of solutions for some Stein-Weiss type integral systems
M. Melgaard, M. Yang, Minbo, X. Zhou
Pacific Journal of Mathematics (2022), to appear

Solitary waves and excited states for Boson stars
M. Melgaard, F. Y. Zongo
Analysis and Applications 20 (2022), no. 2, 285-302

Maximum Ionization in Restricted and Unrestricted Hartree-Fock Theory
H. Cox, M. Melgaard, V.J.J. Syrjanen
Atoms 9 (2021), no. 1, article 13

Poisson wave trace formula for Dirac resonances at spectrum edges and applications
B. Cheng, M. Melgaard
Asian Journal of Mathematics 25 (2021), no. 2, 243-276

The bound-state stability of the hydride ion in HF theory
H. Cox, A. L. Baskerville, V. J .J. Syrjanen, M. Melgaard
Advances in Quantum Chemistry 81 (2020), 167-189

Fractional magnetic Sobolev inequalities with two variables
Z. Guo, M. Melgaard
Mathematical Inequalities and Applications 22 (2019), no. 2, 703-718

Ground state solutions to Hartree-Fock equations with magnetic fields
C. Argaez, M. Melgaard
Applicable Analysis 97 (2018), no. 14, 2377-2403

Schrödinger equations with magnetic fields and Hardy-Sobolev critical exponents
Z. Guo, M. Melgaard, W. Zou
Electron. J. Differential Equations, Vol. 2017 (2017), No. 199, pp. 1-18

Poisson wave trace formula for perturbed Dirac operators
J. Kungsman and M. Melgaard
J. Operator Theory 77 (2017), no. 1, 133-147

Minimizers for open-shell, spin-polarised Kohn-Sham equations for non-relativistic and quasi-relativistic molecular systems
C. Argaez and M. Melgaard
Methods and Applications in Analysis 23 (2016), no 3, 269-292

Complex absorbing potential method for the perturbed Dirac operator
J. Kungsman and M. Melgaard
Communications in Partial Differential Equations 39 (2014), no. 8, 1451-1478

Existence of Dirac resonances in the semi-classical limit
J. Kungsman and M. Melgaard
Dynamics of Partial Differential Equations 11 (2014), no. 4, 381-395

Complex absorbing potential method for the perturbed Dirac operator. Clusters of resonances
J. Kungsman and M. Melgaard
Journal of Operator Theory 71 (2014), issue 1, 259-283

Abstract criteria for multiple solutions to nonlinear coupled equations involving magnetic Schrödinger
M. Enstedt and M. Melgaard
Journal of Differential Equations 253 (2012), no. 6, 1729-1743

Stiefel and Grassmann manifolds in quantum chemistry
E. Chiumiento and M. Melgaard
Journal of Geometry and Physics 62 (2012), no. 8, 1866-1881

Multiple solutions of the quasi relativistic Choquard equation
M. Melgaard, F. D. Y. Zongo
Journal of Mathematical Physics 53 (2012), 033709 (12 pp)

Existence of a minimizer for the quasi-relativistic Kohn-Sham model
C. Argaez and M. Melgaard
Electronic Journal of Differential Equations, Volume 2012 (2012), no. 18, 1-20

Solutions to quasi-relativistic multi-configurative Hartree-Fock equations in quantum chemistry
C. Argaez and M. Melgaard
Nonlinear Analysis TMA (theory, methods and applications) 75 (2012), 384-404

Complex absorbing potential method for systems
J. Kungsman and M. Melgaard
Dissertationes Mathematicae 469 (2010), 58 pp (Polish Academy of Sciences)

Existence of infinitely many distinct solutions to the quasi-relativistic Hartree-Fock equations
M. Enstedt and M. Melgaard
International Journal of Mathematics and Mathematical Sciences, vol. 2009 (2009), article ID 651871, 20 pages

Non-existence of a minimizer to the magnetic Hartree-Fock functional
M. Enstedt and M. Melgaard
Positivity 12 (2008), 653-666

Existence of a solution to Hartree-Fock equations with decreasing magnetic field
M. Enstedt and M. Melgaard
Nonlinear Analysis TMA (Theory, Methods and Applications) 69 (2008), 2125-2141

Confinement effects on scattering for a nanoparticle
M. Melgaard
Acta Phys. Polon. B 38 (2007), 197-214

Scattering properties for a pair of Schrödinger type operators on cylindrical domain
M. Melgaard
Cent. Eur. J. Math. 5 (2007), 134-153

Quantum collisions in semi-constrainted structures
M. Melgaard
Modern Physics Letters B 21 (2007), no. 13, 767-779

Thresholds properties for matrix-valued Schrödinger operators, II. Resonances
M. Melgaard
Journal of Differential Equations 226 (2006), no. 2, 687-703

The Friedrichs extension of the Aharonov-Bohm Hamiltonian on a disk
J. F. Brasche and M. Melgaard
Integral Equations Operator Theory 52 (2005), no. 3, 419-436

Optimal limiting absorption principle for a Schr&#246dinger type operator on a Lipschitz cylinder
M. Melgaard
Manuscripta Mathematica 118 (2005), no. 2, 253-270

On the maximal ionization for the atomic Pauli operator
M. Melgaard and T. Johnson
Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), no. 2063, 3355-3364

Thresholds properties for matrix-valued Schrödinger operators

Journal of Mathematical Physics 46 (2005), p 83507

Bound states for the three-dimensional Aharonov-Bohm quantum wire
M. Melgaard
Few-Body Systems (Formerly Acta Physica Austriaca) 45 (2004), Nos. 1-2, 77-97

Negative discrete spectrum of perturbed multivortex Aharonov-Bohm Hamiltonians
M. Melgaard, E.-M. Ouhabaz, and G. Rozenblum
Annals Henri Poincare 5 (2004), 979-1012

Eigenvalue asymptotics for weakly perturbed Dirac and Schrödinger operators with constant magnetic fields of full rank
M. Melgaard and G. Rozenblum
Communications in Partial Differential Equations 28 (2003), Nos. 3 and 4, 697-736

Quantum scattering near the lowest Landau threshold for a Schrödinger operator with a constant magnetic field
M. Melgaard
Central European Journal of Mathematics 1 (2003), no. 4, 477-509

On bound states for systems of weakly coupled Schrödinger equations in one space
M. Melgaard
Journal of Mathematical Physics 43 (2002), no. 11, pp 5 365-385

New approach to quantum scattering near the lowest Landau threshold for a Schr&#246dinger operator with a constant magnetic field
M. Melgaard
Few-Body Systems (Formerly Acta Physica Austriaca) 32 (2002), 1-22.

Spectral properties in the low-energy limit of one-dimensional Schrödinger operators $-d^{2}/dx^{2}+V$. The case $<1,V1> eq 0$
M. Melgaard
Mathematische Nachrichten 238 (2002), 113-143.

Perturbation of eigenvalues embedded at a threshold
A. Jensen and M. Melgaard
Proceedings of the Royal Society of Edinburgh Section A 131 (2002), 163-179.

Spectral properties at a threshold for two-channel Hamiltonians. II. Applications to scattering theory
M. Melgaard
Journal of Mathematical Analysis and Applications 256 (2001), no. 2, 568-586

Spectral properties at a threshold for two-channel Hamiltonians. I. Abstract theory
M. Melgaard
Journal of Mathematical Analysis and Applications 256 (2001), no. 1, 281-303.

Spectral estimates for magnetic operators<
M. Melgaard and G. Rozenblum
Mathematica Scandinavica 79 (1996), 237-254