Roger T. Lewis

The Effect of Variable Change on Oscillation and Disconjugacy Criteria with Applications to Spectral Theory and Asymptotic Theory

This study of the effect of variable change on differential operators was motivated by recent papers of Tipler [ 131 and Hinton and Lewis [37] which appeared in the same issue of the Journal of D@‘erential Equations. Tipler gave a geometric proof [ 13, p. 1671 of the following result of Hawking and Penrose (see [29, 301). Let F(t) be a real valued continuous, nonnegative, function on (-W, co) such that F(b) > 0 for some point b. Then there exists a pair of conjugate points on (-co, co) for the differential equation

On the number of negative eigenvalues of Schrödinger operators with an Aharanov-Bohm magnetic field

It is proved that for V+=max(V,0) in the subspace L1(R+ ; L∞(S1); r dr) of L1(R2), there is a Cwikel–Lieb–Rosenblum–type inequality for the number of negative eigen2 R values of the operator ((1/i)∇ + A)2 — V in L2(R2) when A is an Aharonov-Bohm magnetic potential with non-integer flux. It is shown that the L1(R+, L∞ (S1),r dr)-norm cannot be replaced by the L1(R2)-norm in the inequality.

Geometric spectral properties of N-body Schrödinger operators. II

Conditions for the finiteness and for the infiniteness of bound states of N-body Schrödinger operators are presented. These bound states correspond to eigenvalues below the essential spectrum of the operator. Previous work of the authors which extended geometric methods and localization techniques of Agmon are used to establish these conditions. An application to a diatomic system having N electrons and two nuclei is given.

Sobolev, Hardy and CLR inequalities associated with Pauli operators in ℝ3

In a previous article, the first two authors have proved that the existence of zero modes of Pauli operators is a rare phenomenon; inter alia, it is proved that for |B|L3/2(3), the set of magnetic fields B which do not yield zero modes contains an open dense subset of [L3/2(3)]3. Here the analysis is taken further, and it is shown that Sobolev, Hardy and Cwikel-Lieb-Rosenbljum (CLR) inequalities hold for Pauli operators for all magnetic fields in the aforementioned open dense set.

A Friedrichs inequality and an application

An inequality whose origins date to the work of G. H. Hardy is presented. This Hardy-type inequality applies to derivatives of arbitrary order of functions whose domain is a subset of ℝ n . The Friedrichs inequality is a corollary. The result is then used to establish lower bounds on the essential spectra of even-order elliptic partial differential operators on unbounded domains.

The Agmon spectral function for molecular hamiltonians with symmetry restrictions

In this paper we introduce symmetry considerations into our earlier work, which was concerned with geometric spectral properties of Schrödinger operators including the N-body operators of quantum mechanics. The point of emphasis is a function introduced by Shmuel Agmon which we have named the Agmon spectral function. We show that this function is symmetric for an N-body Schrödinger operator restricted to a subspace of prescribed symmetry. We then show how it can be used to obtain criteria for the finiteness and infiniteness of bound states of polyatomic systems.

Hardy, Sobolev, and CLR Inequalities

The Hardy and Sobolev inequalities are of fundamental importance in many branches of mathematical analysis and mathematical physics, and have been intensively studied since their discovery. A rich theory has been developed with the original inequalities on \((0,\infty )\) extended and refined in many ways, and an extensive literature on them now exists. We shall be focusing throughout the book on versions of the inequalities in L p spaces, with \(1 < p < \infty \). In this chapter we shall be mainly concerned with the inequalities in \((0,\infty )\) or \(\mathbb{R}^{n},\ n \geq 1\). Later in the chapter we shall also discuss the CLR (Cwikel, Lieb, Rosenbljum) inequality, which gives an upper bound to the number of negative eigenvalues of a lower semi-bounded Schrodinger operator in \(L^{2}(\mathbb{R}^{n})\). This has a natural place with the Hardy and Sobolev inequalities as the three inequalities are intimately related, as we shall show. Where proofs are omitted, e.g., of the Sobolev inequality, precise references are given, but in all cases we have striven to include enough background analysis to enable a reader to understand and appreciate the result.

Boundary Curvatures and the Distance Function

Let \(\Omega \) be an open subset of \(\mathbb{R}^{n},\ n \geq 2,\) with non-empty boundary, and set

Inequalities and Operators Involving Magnetic Fields

\displaystyle{\delta (\mathbf{x}):=\inf \{ \vert \mathbf{x} -\mathbf{y}\vert: \mathbf{y} \in \mathbb{R}^{n}\setminus \Omega \}}