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  • 1. Benguria, Rafael D.
    et al.
    Frank, Rupert L.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Loss, Michael
    The sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half-space2008In: Mathematical Research Letters, ISSN 1073-2780, E-ISSN 1945-001X, Vol. 15, no 4, p. 613-622Article in journal (Refereed)
    Abstract [en]

    It is shown that the sharp constant in the Hardy-Sobolev-Maz'ya inequality on the upper half space H-3 subset of R-3 is given by the Sobolev constant. This is achieved by a duality argument relating the problem to a Hardy-Littlewood-Sobolev type inequality whose sharp constant is determined as well.

  • 2.
    Ekholm, Tomas
    et al.
    Department of Mathematics, Lund University.
    Frank, Rupert
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lieb-Thirring inequalities on the half-line with critical exponent2008In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 10, no 3, p. 739-755Article in journal (Refereed)
    Abstract [en]

    We consider the operator -d(2)/dr(2) - V in L-2(R+) with Dirichlet boundary condition at the origin. For the moments of its negative eigenvalues we prove the bound for any alpha is an element of [0, 1) and gamma >= (1 - alpha)/2. This includes a Lieb-Thirring inequality in the critical endpoint case.

  • 3.
    Ekholm, Tomas
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Frank, Rupert
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On Lieb-Thirring inequalities for Schrödinger operators with virtual level2006In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 264, no 3, p. 725-740Article in journal (Refereed)
    Abstract [en]

    We consider the operator H = - Delta- V in L-2(R-d), d >= 3. For the moments of its negative eigenvalues we prove the estimate

    tr H--(gamma) <= C-gamma,C-d integral(Rd) (V(x) - (d-2)(2)/4\x\(2))(gamma+d/2) dx, gamma > 0.

    Similar estimates hold for the one-dimensional operator with a Dirichlet condition at the origin and for the two-dimensional Aharonov-Bohm operator.

  • 4. Exner, Pavel
    et al.
    Frank, Rupert L.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Absolute continuity of the spectrum for periodically modulated leaky wires in R-32007In: Annales de l'Institute Henri Poincare. Physique theorique, ISSN 1424-0637, E-ISSN 1424-0661, Vol. 8, no 2, p. 241-263Article in journal (Refereed)
    Abstract [en]

    We consider a model of leaky quantum wires in three dimensions. The Hamiltonian is a singular perturbation of the Laplacian supported by a line with the coupling which is bounded and periodically modulated along the line. We demonstrate that such a system has a purely absolutely continuous spectrum and its negative part has band structure with an at most finite number of gaps. This result is extended also to the situation when there is an infinite number of the lines supporting the perturbations arranged periodically in one direction.

  • 5.
    Frank, Rupert L.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the tunneling effect for magnetic Schrödinger operators in antidot lattices2006In: Asymptotic Analysis, ISSN 0921-7134, E-ISSN 1875-8576, Vol. 48, no 1-2, p. 91-120Article in journal (Refereed)
    Abstract [en]

    We study the Schrodinger operator (hD-A)(2) with periodic magnetic field B=curl A in an antidot lattice Omega(infinity) = R-2\boolean OR(alpha is an element of Gamma)(U+alpha). Neumann boundary conditions lead to spectrum below hinf B. Under suitable assumptions on a "one-well problem" we prove that this spectrum is localized inside an exponentially small interval in the semi-classical limit h -> 0. For this purpose we construct a basis of the corresponding spectral subspace with natural localization and symmetry properties.

  • 6.
    Frank, Rupert L.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Hardy-Lieb-Thirring inequalities for eigenvalues of Schrödinger operators2007Doctoral thesis, comprehensive summary (Other scientific)
    Abstract [en]

    This thesis is devoted to quantitative questions about the discrete spectrum of Schrödinger-type operators.

    In Paper I we show that the Lieb-Thirring inequalities on moments of negative eigen¬values remain true, with possibly different constants, when the critical Hardy weight is subtracted from the Laplace operator.

    In Paper II we prove that the one-dimensional analog of this inequality holds even for the critical value of the moment parameter. In Paper III we establish Hardy-Lieb-Thirring inequalities for fractional powers of the Laplace operator and, in particular, relativistic Schrödinger operators. We do so by first establishing Hardy-Sobolev inequalities for such operators. We also allow for the inclu¬sion of magnetic fields.

    As an application, in Paper IV we give a proof of stability of relativistic matter with magnetic fields up to the critical value of the nuclear charge.

    In Paper V we derive inequalities for moments of the real part and the modulus of the eigen¬values of Schrödinger operators with complex-valued potentials.

  • 7.
    Frank, Rupert L.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the asymptotic number of edge states for magnetic Schrodinger operators2007In: Proceedings of the London Mathematical Society, ISSN 0024-6115, E-ISSN 1460-244X, Vol. 95, no 1, p. 1-19Article in journal (Refereed)
    Abstract [en]

    We consider a Schrodinger operator (hD - A)(2) with a positive magnetic field B = curl A in a domain Omega subset of R-2. The imposing of Neumann boundary conditions leads to the existence of some spectrum below h inf B. This is a boundary effect and it is related to the existence of edge states of the system. We show that the number of these eigenvalues, in the semi-classical limit h -> 0, is governed by a Weyl-type law and that it involves a symbol on partial derivative Omega. In the particular case of a constant magnetic field, the curvature plays a major role.

  • 8.
    Frank, Rupert L.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the Laplacian in the halfspace with a periodic boundary condition2006In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 44, no 2, p. 277-298Article in journal (Refereed)
    Abstract [en]

    We study spectral and scattering properties of the Laplacian H-(sigma)= - Delta in L-2 (R-+(d+1)) corresponding to the boundary condition (partial derivative u)/(partial derivative v) + partial derivative u = 0 with a periodic function sigma. For non-negative sigma we prove that H-(sigma) is unitarily equivalent to the Neumann Laplacian H-(0). In general, there appear additional channels of scattering due to surface states. We prove absolute continuity of the spectrum of H-(sigma) under mild assumptions on sigma.

  • 9.
    Frank, Rupert L.
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Hainzl, Christian
    Naboko, Serguei
    Seiringer, Robert
    The critical temperature for the BCS equation at weak coupling2007In: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002X, Vol. 17, no 4, p. 559-567Article in journal (Refereed)
    Abstract [en]

    For the BCS equation with local two-body interaction lambda V(x), we give a rigorous analysis of the asymptotic behavior of the critical temperature as lambda -> 0. We derive necessary and sufficient conditions on V(x)for the existence of a nontrivial solution for all values of lambda > 0.

  • 10.
    Frank, Rupert L.
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). Princeton University, United States.
    Laptev, Ari
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). Imperial College London, United Kingdom.
    Molchanov, Stanislav
    Eigenvalue estimates for magnetic Schrödinger operators in domains2008In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 136, no 12, p. 4245-4255Article in journal (Refereed)
    Abstract [en]

    Inequalities are derived for sums and quotients of eigenvalues of magnetic Schrodinger operators with non-negative electric potentials in domains. The bounds reflect the correct order of growth in the semi-classical limit.

  • 11.
    Frank, Rupert L.
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lieb, Elliott H.
    Department of Mathematics and Physics, Princeton University.
    Seiringer, Robert
    Department of Physics, Princeton University.
    Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators2008In: Journal of The American Mathematical Society, ISSN 0894-0347, E-ISSN 1088-6834, Vol. 21, no 4, p. 925-950Article in journal (Refereed)
  • 12.
    Frank, Rupert L.
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lieb, Elliott H.
    Seiringer, Robert
    Siedentop, Heinz
    Muller's exchange-correlation energy in density-matrix-functional theory2007In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, E-ISSN 1094-1622, Vol. 76, no 5, p. 052517-Article in journal (Refereed)
    Abstract [en]

    The increasing interest in the Muller density-matrix-functional theory has led us to a systematic mathematical investigation of its properties. This functional is similar to the Hartree-Fock (HF) functional, but with a modified exchange term in which the square of the density matrix gamma(x,x(')) is replaced by the square of gamma(1/2)(x,x(')). After an extensive introductory discussion of density-matrix-functional theory we show, among other things, that this functional is convex (unlike the HF functional) and that energy minimizing gamma's have unique densities rho(r), which is a physically desirable property often absent in HF theory. We show that minimizers exist if N <= Z, and derive various properties of the minimal energy and the corresponding minimizers. We also give a precise statement about the equation for the orbitals of gamma, which is more complex than for HF theory. We state some open mathematical questions about the theory together with conjectured solutions.

  • 13.
    Frank, Rupert L.
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Safronov, Oleg
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Absolutely continuous spectrum of a class of random nonergodic Schrodinger operators2005In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, no 42, p. 2559-2577Article in journal (Refereed)
  • 14.
    Frank, Rupert L.
    et al.
    KTH, Superseded Departments, Mathematics.
    Shterenberg, R. G.
    On the scattering theory of the Laplacian with a periodic boundary condition. II. Additional channels of scattering2004In: Documenta Mathematica, ISSN 1431-0635, E-ISSN 1431-0643, Vol. 9, no 1, p. 57-77Article in journal (Refereed)
    Abstract [en]

    We study spectral and scattering properties of the Laplacian H (σ) = -Δ in L2(ℝ+2) corresponding to the boundary condition ∂u/∂ν + σu = 0 for a wide class of periodic functions σ. For non-negative σ we prove that H(σ) is unitarily equivalent to the Neumann Laplacian H(0). In general, there appear additional channels of scattering which are analyzed in detail.

  • 15.
    Frank, Rupert L.
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Shterenberg, Roman G.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    On the spectrum of partially periodic operators2007In: Operator Theory, Analysis and Mathematical Physics / [ed] Janas, J; Kurasov, P; Laptev, A; Naboko, S; Stolz, G, BASEL: BIRKHAUSER VERLAG AG , 2007, Vol. 174, p. 35-50Conference paper (Refereed)
    Abstract [en]

    We consider Schrodinger operators H = -Delta + V in L-2 (Omega) where the domain Omega subset of R-+(d+1) and the potential V = V (x, y) are periodic with respect to the variable x is an element of R-d. We assume that Omega is unbounded with respect to the variable y is an element of R and that V decays with respect to this variable. V may contain a singular term supported on the boundary. We develop a scattering theory for H and present an approach to prove absence of singular continuous spectrum. Moreover, we show that certain repulsivity conditions on the potential and the boundary of Omega exclude the existence of surface states. In this case, the spectrum of His purely absolutely continuous and the scattering is complete.

  • 16.
    Frank, Rupert L.
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Siedentop, Heinz
    Warzel, Simone
    The ground state energy of heavy atoms: Relativistic lowering of the leading energy correction2008In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 278, no 2, p. 549-566Article in journal (Refereed)
    Abstract [en]

    We describe atoms by a pseudo-relativistic model that has its origin in the work of Chandrasekhar. We prove that the leading energy correction for heavy atoms, the Scott correction, exists. It turns out to be lower than in the non-relativistic description of atoms. Our proof is valid up to and including the critical coupling constant. It is based on a renormalization of the energy whose zero level we adjust to be the ground-state energy of the corresponding non-relativistic problem. This allows us to roll the proof back to results for the Schrodinger operator.

  • 17.
    Frank, Rupert
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Laptev, Ari
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lieb, Elliott H.
    Department of Physics, Princeton University.
    Seiringer, Robert
    Department of Physics, Princeton University.
    Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials2006In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 77, no 3, p. 309-316Article in journal (Refereed)
    Abstract [en]

    Inequalities are derived for power sums of the real part and the modulus of the eigenvalues of a Schrodinger operator with a complex-valued potential.

  • 18.
    Frank, Rupert
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lieb, Elliott H.
    Departments of Mathematics and Physics, Princeton University.
    Seiringer, Robert
    Department of Physics, Princeton University.
    Stability of relativistic matter with magnetic fields for nuclear charges up to the critical value2007In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 275, no 2, p. 479-489Article in journal (Refereed)
    Abstract [en]

    We give a proof of stability of relativistic matter with magnetic fields all the way up to the critical value of the nuclear charge Z alpha = 2/pi.

1 - 18 of 18
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