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Refereed articles in journals

Year	2021
Authors	H. Cox, M. Melgaard, V.J.J. Syrjanen
Title	Maximum Ionization in Restricted and Unrestricted Hartree-Fock Theory
Reference	Atoms, to appear (30 pp).

Abstract 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 Lieb's bound [Theorem 1, Phys. Rev. A \textbf{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.

Year	2021
Authors	B. Cheng, M. Melgaard
Title	Poisson wave trace formula for Dirac resonances at spectrum edges and applications
Reference	Asian J. Math., to appear (34 pp).

Abstract 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.

Year	2020
Authors	H. Cox, A. L. Baskerville, V. J .J. Syrjanen, M. Melgaard
Title	The bound-state stability of the hydride ion in HF theory,
Reference	Adv. Quantum Chem. 81, 167-189

Abstract 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 Schrödinger equation, within the fixed-nucleus approximation. It is written so as to be accessible to both the mathematics and physical chemistry communities.

Year	2019
Authors	Z. Guo, M. Melgaard
Title	Fractional magnetic Sobolev inequalities with two variables
Reference	Math. Ineqn. Appl 22, no. 2, 703-718

Abstract A fractional magnetic Sobolev inequality with two variables and critical exponents is considered in this paper, and the best constant in the inequality is determined. As an application of the inequality, we establish an existence result for the ground state solutions to a fractional magnetic critical system.

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

Abstract 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.

Year	2017
Authors	Z. Guo, M. Melgaard, W. Zou
Title	Schrödinger equations with magnetic fields and Hardy-Sobolev critical exponents
Reference	Elec. J. Diff. Eqns. Vol 201, 1--18.

Abstract This article is motivated by problems in astrophysics. We consider nonlinear Schrödinger equations and related systems with magnetic fields and Hardy-Sobolev critical exponents. Under proper conditions, existence of ground state solutions to these equations and systems are established.

Year	2017
Authors	J. Kungsman and M. Melgaard
Title	Poisson wave trace formula for perturbed Dirac operators
Reference	J. Operator Theory 77, 133--147.

Abstract We consider self-adjoint Dirac operators $\ham{D}=\ham{D}_0 + V(x)$, where $\ham{D}_0$ is the free three-dimensional Dirac operator and $V(x)$ is a smooth compactly supported Hermitian matrix. We define resonances of $\ham{D}$ as poles of the meromorphic continuation of its cut-off resolvent. An upper bound on the number of resonances in disks, an estimate on the scattering determinant and the Lifshits-Krein trace formula then leads to a global Poisson wave trace formula for resonances of $\ham{D}$.

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

Abstract We study the open-shell, spin-polarized Kohn-Sham models for non-relativistic and quasi-relativistic $N$-electron Coulomb systems, that is, systems where the kinetic energy of the electrons is given by either the non-relativistic operator $-\D_{x_{n}}$ or the quasi-relativistic operator $\sqrt{ -\a^{-2} \D_{x_{n}} + \a^{-4}} -\a^{-2}$. For standard and extended Kohn-Sham models in the local density approximation, we prove existence of a ground state (or minimizer) provided that the total charge $Z_{\rm tot}$ of $K$ nuclei is greater than $N-1$. For the quasi-relativistic setting we also need that $Z_{\rm tot}$ is smaller than a critical charge $Z_{\rm c}=2 \a^{-1} \pi^{-1}$.

Year	2014d
Authors	J. Kungsman and M. Melgaard
Title	Existence of Dirac resonances in the semi-classical limit
Reference	Dynamics of P.D.E. 11, no 4, 381--395.

Abstract We study the existence of quantum resonances of the three-dimensional semiclassical Dirac operator perturbed by smooth, bounded and real-valued scalar potentials $V$ decaying like $\langle x \rangle^{-\epsilon}$ at infinity for some $\epsilon > 0$. By studying analytic singularities of a certain distribution related to $V$ and by combining two trace formulas, we prove that the perturbed Dirac operators possess resonances near $\sup V + 1$ and $\inf V − 1$. We also provide a lower bound for the number of resonances near these points expressed in terms of the semiclassical parameter.

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

Abstract The Complex Absorbing Potential (CAP) method is widely used to compute resonances in Quantum Chemistry, both for nonrelativistic and relativistic Hamiltonians. In the semiclassical limit $\hbar \to 0$ we consider resonances near the real axis and we establish the CAP method rigorously for the perturbed Dirac operator by proving that individual resonances are perturbed eigenvalues of the nonselfadjoint CAP Hamiltonian, and vice versa. The proofs are based on pseudodifferential operator theory and microlocal analysis.

Year	2014a
Authors	J. Kungsman and M. Melgaard
Title	Complex absorbing potential method for Dirac operators. Clusters of resonances
Reference	Journal of Operator Theory 71 (2014), issue 1, 259--283.

Abstract For both nonrelativistic and relativistic Hamiltonians, the Complex Absorbing Potential (CAP) method has been applied extensively to calculate resonances in Physics and Chemistry. We study clusters of resonances for the perturbed Dirac operator near the real axis and, in the semiclassical limit, we establish the CAP method rigorously by showing that resonances are perturbed eigenvalues of the nonselfadjoint CAP Hamiltonian, and vice versa.

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

Abstract We consider a system of nonlinear coupled equations involving magnetic Schrödinger operators and general potentials. We provide a criteria for the existence of multiple solutions to these equations. As special cases we get the classical results on existence of infinitely many distinct solutions within Hartree and Hartree-Fock theory of atoms and molecules subject to an external magnetic fields. We also extend recent results within this theory, including Coulomb system with a constant magnetic field, a decreasing magnetic field and a ``physically measurable" magnetic field.

Year	2012d
Authors	E. Chiumiento and M. Melgaard
Title	Stiefel and Grassmann manifolds in quantum chemistry
Reference	Journal of Geometry and Physics 62, issue 8, 1866-1881. doi: 10.1016/j.geomphys.2012.04.005

Abstract We establish geometric properties of Stiefel and Grassmann manifolds which arise in relation to Slater type variational spaces in many-particle Hartree-Fock theory and beyond. In particular, we prove that they are analytic homogeneous spaces and submanifolds of the space of bounded operators on the single-particle Hilbert space. As a by-product we obtain that they are complete Finsler manifolds. These geometric properties underpin state-of-the-art results on the existence of solutions to Hartree-Fock type equations.

Year	2012c
Authors	M. Melgaard, F. D. Y. Zongo
Title	Multiple solutions of the quasi relativistic Choquard equation
Reference	Journal of Mathematical Physics 53, 033709 (12 pp) doi: 10.1063/1.3695991

Abstract We prove existence of multiple solutions to the quasirelativistic Choquard equations with a scalar potential using critical point theory.

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

Abstract We study the standard and extended Kohn-Sham models for quasi-relativistic N-electron Coulomb systems; that is, systems where the kinetic energy of the electrons is given by the quasi-relativistic operator (see article) . For spin-unpolarized systems in the local density approximation, we prove existence of a ground state (or minimizer) provided that the total charge Z of K nuclei is greater than N-1 and that Z is smaller than a critical charge (see article).

Year	2012a
Authors	C. Argaez and M. Melgaard
Title	Solutions to quasi-relativistic multi-configurative Hartree-Fock equations in quantum chemistry
Reference	Nonlinear Analysis: theory, methods and applications 75, 384--404. doi: 10.1016/j.na.2011.08.038

Abstract We establish existence of infinitely many distinct solutions to the multi-configurative Hartree-Fock type equations for $N$-electron Coulomb systems with quasi-relativistic kinetic energy $\sqrt{ -\a^{-2} \D_{x_{n}} + \a^{-4}} -\a^{-2}$ for the $n^{\rm th}$ electron. Finitely many of the solutions are interpreted as excited states of the molecule. Moreover, we prove existence of a ground state. The results are valid under the hypotheses that the total charge $Z_{\rm tot}$ of $K$ nuclei is greater than $N-1$ and that $Z_{\rm tot}$ is smaller than a critical charge $Z_{\rm c}$. The proofs are based on a new application of the Lions-Fang-Ghoussoub critical point approach to nonminimal solutions on a complete analytic Hilbert-Riemann manifold.

Year	2010
Authors	J. Kungsman and M. Melgaard
Title	Complex absorbing potential method for systems
Reference	Dissertationes Mathematicae 469 (58 pages), Polish Academy of Sciences, doi:10.4064/dm469-0-1

Abstract The Complex Absorbing Potential (CAP) method is widely used to compute resonances in Quantum Chemistry, both for scalar valued and matrix valued Hamiltonians. In the semiclassical limit $\hbar \to 0$ we consider resonances near the real axis and we establish the CAP method rigorously in an abstract matrix valued setting by proving that resonances are perturbed eigenvalues of the nonselfadjoint CAP Hamiltonian, and vice versa. The proof is based on pseudodifferential operator theory and microlocal analysis.

Year	2009
Authors	M. Enstedt and M. Melgaard
Title	Existence of infinitely many distinct solutions to the quasi-relativistic Hartree-Fock equations
Reference	International Journal of Mathematics and Mathematical Sciences, vol. 2009, Article ID 651871, 20 pages, 2009. doi:10.1155/2009/651871

Abstract We establish existence of infinitely many distinct solutions to the semilinear elliptic Hartree-Fock equations for $N$ electron Coulomb systems with quasi-relativistic kinetic energy $\sqrt{ -\a^{-2} \D_{x_{n}} + \a^{-4}} -\a^{-2}$ for the $n^{\rm th}$ electron. Moreover, we prove existence of a ground state. The results are valid under the hypotheses that the total charge $Z_{\rm tot}$ of $K$ nuclei is greater than $N-1$ and that $Z_{\rm tot}$ is smaller than some critical charge $Z_{\rm c}$. The proofs are based on a new application of the Fang-Ghoussoub critical point approach to multiple solutions on a noncompact Riemannian manifold, in combination with density operator techniques. In particular, compact Sobolev imbeddings are not available.

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

Abstract In the presence of an external magnetic field, we prove absence of a ground state within the Hartree-Fock theory of atoms and molecules. The result is established for a wide class of magnetic fields when the number of electrons is greater than or equal to 2Z+K , where Z is the total charge of K nuclei. Positivity properties are instrumental in the proof of this bound for the maximal ionization.

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

Abstract In the presence of an external magnetic field, we prove existence of a ground state within the Hartree-Fock theory of atoms and molecules. The ground state exists provided the magnetic field decreases at infinity and the total charge Z of K nuclei exceeds N-1, where N is the number of electrons. In the opposite direction, no ground state exists if N > 2Z+K

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

Abstract The motion of a nanoparticle in a narrow, bend channel is used to illustrate features of scattering in systems with semi-open geometries. Under certain general constraints on the geometry, results on the scattering process are established.

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

Abstract Strong asymptotic completeness is shown for a pair of Schrödinger type operators on a cylindrical Lipschitz domain. A key ingredient is a limiting absorption principle valid in a scale of weighted (local) Sobolev spaces with respect to the uniform topology. The results are based on a refined version of Mourre's method within the context of pseudo-selfadjoint operators.

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

Abstract The problem of a quantum particle moving in a narrow, bend two-dimensional channel is used to illustrate features of collisions in systems with semi-constrainted structures. The collision theory of such systems is developed and, under certain general restrictions on the geometry, the following properties are rigorously established: (1) The (localized) M{\o}ller wave operator exist and the $S$ operator is unitary in the scattering channels. (2) The singular spectrum is discrete in intervals away from the continuum thresholds. The discussion is based on a time-dependent approach.

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

Abstract We present some results on the perturbation of eigenvalues embedded at a threshold for a matrixvalued Hamiltonian with three-dimensional dilation analytic Schrödinger operators as entries and with a small off-diagonal perturbation. The main result describes how a threshold eigenvalue generates resonances (that is, poles of the meromorphic continuation of the perturbed Hamiltonian).

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

Abstract We show that the Aharonov-Bohm Hamiltonian considered on a disc has a four-parameter family of self-adjoint extensions. Among the infinitely many self-adjoint extensions, we determine to which parameters the Friedrichs extension $H^{F}$ corresponds and its lowest eigenvalue is found. Moreover, we note that the diamagnetic inequality holds for $H^{F}$.

See below for entry 2005d

Year	2005c
Authors	M. Melgaard
Title	Optimal limiting absorption principle for a Schrödinger type operator on a Lipschitz cylinder
Reference	Manuscripta Mathematica 118 (2005), no. 2, 253-270

Abstract Spectral properties of a Schrödinger type operator on a cylinder is established by a version of Mourre's conjugate operator method. In particular, the limiting absorption principle is proved in an optimal scale of Besov spaces.

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

Abstract Within the Born-Oppenheimer approximation Lieb proved that the number of non-relativistic, spin-$\frac{1}{2}$ particles that can be bound to an atom of nuclear charge $Z$ in the presence of an external magnetic field satisfies $N_{\max} < 2Z+1$, provided the magnetic field tends to zero at infinity and the coupling between the magnetic field and the spin is ignored. Assuming that the magnetic field is generic, we prove an upper bound which holds when the spin-field coupling is included; the set of generic magnetic fields contains an open, dense subset of $[L^{3/2}(\R^{3})]^{3}$.

Year	2005a
Authors	M. Melgaard
Title	Thresholds properties for matrix-valued Schrödinger operators
Reference	Journal of Mathematical Physics 46 (2005), p 83507 (10 pages).

Abstract We present some results on the perturbation of eigenvalues embedded at a threshold for a two-channel Hamiltonian with three-dimensional Schrödinger operators as entries and with a small off-diagonal perturbation. In particular, we show how the threshold eigenvalue gives rise to discrete eigenvalues below the threshold and, moreover, we establish a criterion on existence of half-bound states associated with embedded pseudo eigenvalues.

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

Abstract Existence of bound states for the three-dimensional Aharonov-Bohm quantum wire, suggested by Dunne and Jaffe, is established provided the wire is thin enough. Recent results on the Aharonov-Bohm Schrödinger operator on a disk are used.

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

Abstract The diamagnetic inequality is established for the Schrödinger operator $H_{0}^{(d)}$ in $L^{2}(\R^{d})$, $d=2,3$, describing a particle moving in a magnetic field generated by finitely or infinitely many Aharonov-Bohm solenoids located at the points of a discrete set in $\Rl^{2}$, e.g. lattice. This fact is used to prove the Lieb-Thirring inequality as well as CLR-type eigenvalue estimates for the perturbed Schrödinger operator $H_{0}^{(d)}-V$, using new Hardy type inequalities. Large coupling constant eigenvalue asymptotic formulas for the perturbed operators are also proved.

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

Abstract The even-dimensional Dirac and Schrödinger operators with a constant magnetic field of full rank have purely essential spectrum consisting of isolated eigenvalues, so-called Landau levels. For a sign-definite electric potential V which tends to zero at infinity, not too fast, it is known for the Schrödinger operator that the number of eigenvalues near each Landau level is infinite and their leading (quasiclassical) asymptotics depends on the rate of decay for V. We show, both for Schrödinger and Dirac operators, that, for any sign-definite, bounded V which tends to zero at infinity, there still are an infinite number of eigenvalues near each Landau level. For compactly supported V, we establish the non-classical formula, not depending on V, describing how the eigenvalues converge to the Landau levels asymptotically.

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

Abstract For fixed magnetic quantum number $m$ results on spectral properties and scattering theory are given for the three-dimensional Schrödinger operator with a constant magnetic field and an axisymmetrical electric potential $V$. In various, mostly fairly singular settings asymptotic expansions for the resolvent of the Hamiltonian $H_{m}=H_{om}+V$ are deduced as the spectral parameter tends to the lowest Landau threshold. Furthermore, scattering theory for the pair $(H_{m}, H_{om})$ is established and asymptotic expansions of the scattering matrix are derived as the energy parameter tends to the lowest Landau threshold.

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

Abstract We establish the Birman-Schwinger relation for a class of Schrödinger operators $-d^{2}/dx^{2} \otimes 1_{\CH}+V$ on $L^{2}(\Rl,{\CH})$, where $\CH$ is an auxiliary Hilbert space and $V$ is a operator-valued potential. As an application we give an asymptotic formula for the bound states which may arise for a weakly coupled Schrödinger operator with a matrix potential (having one or more thresholds). In addition, for a two-channel system with eigenvalues embedded in the continuous spectrum we show that, under a small perturbation, such eigenvalues turn into resonances.

Year	2002c
Authors	M. Melgaard
Title	New approach to quantum scattering near the lowest Landau threshold for a Schrödinger operator with a constant magnetic field
Reference	Few-Body Systems (Formerly Acta Physica Austriaca) 32 (2002), 1-22.

Abstract For fixed magnetic quantum number $m$ results on spectral properties and scattering theory are given for the three-dimensional Schr\"{o}dinger operator with a constant magnetic field and an axisymmetrical electric potential $V$. Asymptotic expansions for the resolvent of the Hamiltonian $H_{m}=H_{om}+V$ are deduced as the spectral parameter tends to the lowest Landau threshold $E_{0}$. In particular it is shown that $E_{0}$ can be an eigenvalue of $H_{m}$. Furthermore, asymptotic expansions of the scattering matrix associated with the pair $(H_{m}, H_{om})$ are derived as the energy parameter tends to $E_{0}$.

Year	2002b
Authors	M. Melgaard
Title	Spectral properties in the low-energy limit of one-dimensional Schrödinger operators $-d^{2}/dx^{2}+V$. The case $\langle 1, V1 \rangle \neq 0$
Reference	Mathematische Nachrichten 238 (2002), 113-143.

Abstract In this paper we consider the Schrödinger operator $H = ?d^{2}/dx^{2} +V$ in $L^{2}(\R)$, where $V$ satisfies an abstract shortÐrange condition and the (solvability) condition
$\langle 1, V 1\rangle = 0$. Spectral properties of H in the lowÐenergy limit are analyzed. Asymptotic expansions for $R(\zeta) = (H ? \zeta)^{-1}$ and the SÐmatrix $S(\lambda)$ are deduced for $\zeta \to 0$ and $\lambda \downarrow 0$, respectively. Depending on the zeroÐenergy properties of $H$, the expansions of $R(\zeta)$ take different forms. Generically, the expansions of $R(\zeta)$ do not contain negative powers; the appearance of negative powers in $\zeta^{1/2}$ is due to the possible presence of zeroÐenergy resonances (halfÐbound states) or the eigenvalue zero of $H$ (bound state), or both. It is found that there are at most two zero resonances modulo $L^2$Ðfunctions.

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

Abstract Results are obtained on perturbation of eigenvalues and half-bound states (zero-resonances) embedded at a threshold. The results are obtained in a two-channel framework for small off-diagonal perturbations. The results are based on given asymptotic expansions of the component Hamiltonians.

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

Abstract Spectral properties and scattering theory in the low-energy limit are investigated for two-channel Hamiltonians with Schrödinger operators as component Hamiltonians. In various, mostly fairly ÔÔsingularÕÕ settings asymptotic expansions of the resolvent are deduced as the spectral parameter tends to the threshold zero. Furthermore scattering theory for pairs of two-channel Hamiltonians is established. As an application of the expansions of the resolvent, asymptotic expansions of the scattering matrix are derived as the energy parameter tends to the threshold zero.

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

Abstract Spectral properties at thresholds are investigated for two-channel Hamiltonians in various, mostly fairly ÔÔsingularÕÕ settings. In an abstract framework we deduce asymptotic expansions of the resolvent as the spectral parameter tends to a threshold. The results are based on given asymptotic expansions of the component Hamiltonians. Applications to scattering theory are given in a companion paper.

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

Abstract The well-known CLR-estimate for the number of negative eigenvalues of the Schrödinger operator $-\Delta+V$ is carried over to a class of second differential operators, generalizing the magnetic Schrödinger operator. The coefficients in the magnetic operators are variable, they may be nonsmooth, unbounded and some degeneration is allowed.

Refereed contributions to books

Year	2005d
Authors	G. Rozenblum and M. Melgaard
Title	Schrödinger operators with singular potentials
Reference	In: Stationary Partial Differential Equations, Vol. II, 407--517. Handbook of Differential Equations, Elsevier/North-Holland, Amsterdam.

Abstract We describe classical and recent results on the spectral theory of Schrödinger and Pauli operators with singular electric and magnetic potentials

Refereed articles in conference proceedings

Year	2007
Authors	M. Melgaard
Title	Confinement effects on scattering for a nanoparticle
Reference	pp 225-233 in Proceedings of the International Conference APLIMAT 2007, February 6 - 9, 2007, Bratislava, Slovak Republic. ISBN: 978-80-969562-5-8.

Year	2007
Authors	M. Melgaard
Title	Strong asymptotic completeness of a quantum wire
Reference	pp 178-187 in Proceedings of the International Conference Days on Diffraction 2006, May 30 - jun 2, 2006, St. Petersburg, Russia. ISBN: 5-9651-0226-7