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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.