1,721,073 research outputs found
Functional Analysis and Operator Theory for Quantum Physics. The Pavel Exner Anniversary Volume
No full textHardy inequalities for p-Laplacians with Robin boundary conditions
Full text linkIn this paper we study the best constant in a Hardy inequality for the p-Laplace operator on convex domains with Robin boundary conditions. We show, in particular, that the best constant equals ((p−1)/p)p whenever Dirichlet boundary conditions are imposed on a subset of the boundary of non-zero measure. We also discuss some generalizations to non-convex domains
A sharp bound on eigenvalues of Schröedinger operators on the half-line with complex-valued potentials
Full text linkWe derive a sharp bound on the location of non-positive eigenvalues of Schröedinger operators on the halfline with complex-valued potentials
Estimates for the counting function of the laplace operator on domains with rough boundaries
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Sharp Fractional Hardy Inequalities in Half-Spaces
Full text linkWe determine the sharp constant in the Hardy inequality for fractional Sobolev spaces on half-spaces. Our proof relies on a nonlinear and nonlocal version of the ground state representation
Spectral bounds for infinite dimensional polydiagonal symmetric matrix operators on discrete spaces
Full text linkIn this thesis, we prove a variety of discrete Agmon Kolmogorov inequalities and apply them to prove Lieb Thirring inequalities for discrete Schrodinger operators on ℓ[superscript 2](ℤ). We generalise these results in two ways: Firstly, to higher order difference operators, leading to spectral bounds for Tri-, Penta- and Polydiagonal Jacobi-type matrix operators. Secondly, to ℓ[superscript 2]-spaces on higher dimensional domains, specifically on ℓ[superscript 2](ℤ[superscript 2]), ℓ[superscript 2](ℤ[superscript 3]) and finally ℓ[superscript 2](ℤ[superscript d]).
In the Introduction we discuss previous work on Landau Kolmogorov inequalities on a variety of Banach Spaces, Lieb Thirring inequalities in ℓ[superscript 2](ℝ[superscript d]), and the use of Jacobi Matrices in relation to the discrete Schrodinger Operator. We additionally give our main results with some introduction to the notation at hand.
Chapters 2, 3 and 4 follow a similar structure.
We first introduce the relevant difference operators and examine their properties. We then move on to prove the Agmon Kolmogorov and Generalised Sobolev inequalities over ℤ of order 1, 2 and σ respectively. Furthermore, we prove the Lieb Thirring inequality for the respective discrete Schrodinger-type operators, which we subsequently lift to arbitrary moments. Finally we apply this inequality to obtain spectral bounds for tri-, penta- and polydiagonal matrices.
In Chapter 5, we prove a variety of Agmon Kolmogorov inequalities on ℓ[superscript 2](ℤ[superscript 2]) and ℓ[superscript 2](ℤ[superscript 3]). We use these intuitive ideas to obtain 2[superscript d-1] Agmon Kolmogorov inequalities on ℓ[superscript 2](ℤ[superscript d]). We continue from here in the same manner as before and prove the discrete Generalised Sobolev and Lieb Thirring inequalities for a variety of exponent combinations on ℓ[superscript 2](ℤ[superscript d]).Open Acces
Optimisation of Hardy - type inequalities with applications in infinite dimensional Geometry
Full text linkThe following thesis concerns the generalisation of Hardy's integral inequality to multiple dimensions. The structure involves three main papers, two published in journals and one paper in preparation.
In the first article, we make use of the classical Hardy inequality away from a point and a multi-dimensional Hardy inequality associated to the Laplacian operator. The inequality's domain is extended to an in finite case and the behaviour of systems is studied when multiple particles are introduced as a concept. The result we obtain resembles the classical Hardy inequality, but, as an extension, we introduce new vector fields and boundary approximations on a many-particle quantum system.
The second article presents the interest in the re finement of Hardy inequalities using Fourier expansions applied to a new Hopf system of coordinates. As a starting point, we con figured the coordinates in four dimensions for
block - radial functions such that our argument could be easier understood. In comparison to previous research, our result behaves promising in the sense that the inequality discovered is true for functions on the whole space.
Moreover, by introducing a new system of coordinates, we are able to apply our result to the extension of the Hopf fi bration to dimension eight.
Last but not least, in article three we look at discrete Hardy Inequalities and their link to the continuous Hardy inequalities discovered so far. Considering that Hardy inequalities are more difficult to analyse for discrete operators, we make use of two important results in order to establish whether for the discrete Hardy inequality discovered there is a sharp constant on a closed disk in Z^2. The results involve a specifi c type of Aharonov-Bohm magnetic potential which facilitates the analysis of the constant's behaviour, which is present in the new discrete Hardy inequality.Open Acces
Spectral bounds for Schrödinger operators in dimensions one and two
Full text linkThis thesis investigates Lieb-Thirring and Cwikel-Lieb-Rozenblum (CLR) type inequalities for Schrödinger operators in dimensions one and two, with a focus on overcoming the weak coupling problem associated with the failure of the CLR inequality in these dimensions. To this end we start by investigating three two-dimensional Schrödinger operators with extra repulsive factors in the form of: a repulsive Hardy potential, a restriction to antisymmetric functions, and an Aharonov-Bohm magnetic field.
Under the assumption that the electric potential is radially non-increasing we establish semi-classical bounds on the number of negative eigenvalues for each of these operators. Our proof lies in generalising a one-dimensional bound of Calogero and Cohn to operator-valued potentials.
For the same operators we then derive a family of weighted CLR type inequalities by applying the Birman-Schwinger principle. An interpolation argument leads us to ascertain weak forms, which we show to be saturated in the strong coupling limit by a class of long-range potentials. In the Aharonov-Bohm case this enables us to deduce the optimal dependence of the constants on the flux of the field.
Finally in one dimension, we examine Schrödinger operators with an additional fixed attractive potential. Following the method of factorisation we derive Lieb-Thirring type bounds which measure the distance between the eigenvalues of the original and the perturbed operator. Applying this to the Pölsch-Teller potential leads to an improvement of the Lieb-Thirring inequality for partial sums of eigenvalues.Open Acces
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