arXiv subjects
Publications and source records attributed to Ira Herbst.
We continue the search, begun by Kato, for all pairs of real, bounded, measurable functions $\{f,g\}$ that result in a positive commutator $[if(P),g(Q)]$. We prove a number of partial results including a connection with Loewner's celebrated theorem on matrix monotone functions.
We prove Griffiths inequalities for spins in n>1 dimensions with no interaction.
We prove the equivalence of being a Leray-Hopf weak solution to the Navier-Stokes equations in $\mathbb{R}^m, m \ge 3$, to satisfying a well known integral equation. We use this equation to derive some properties of these weak solutions.
We continue to discuss the following problem: For which bounded measurable real functions f and g is the commutator i[f(P),g(Q)] a non-negative operator on L^2(R)? In this work we concentrate on the situation where the commutator is a finite rank operator.
We discuss the resonances of Hamiltonians with constant electric field in one dimension in the limit of small field. These resonances occur near the real axis, near zeros of the analytic continuation of a reflection coefficient for potential scattering, and near the line arg z = -2π/3. We calculate their asymptotics. In conclusion we make some remarks about the higher dimensional problem.
We investigate the following problem: For what $f$ and $g$ is the commutator $i[f(P),g(Q)]$ positive when $f$ and $g$ are bounded measurable functions? This problem originated in work of James Howland and was pursued by Tosio Kato who suggested what might be the answer. So far there is no proof that Kato was correct but in this paper we discuss the problem and give some partial answers to the above question.
We consider the large $L$ limit of one dimensional Schrödinger operators $H_L=-d^2/dx^2 + V_1(x) + V_{2,L}(x)$ in two cases: when $V_{2,L}(x)=V_2(x-L)$ and when $V_{2,L}(x)=e^{-cL}δ(x-L)$. This is motivated by some recent work of Herbst and Mavi where $V_{2,L}$ is replaced by a Dirichlet boundary condition at $L$. The Hamiltonian $H_L$ converges to $H = -d^2/dx^2 + V_1(x)$ as $L\to \infty$ in the strong resolvent sense (and even in the norm resolvent sense for our second case). However, most of the resonances of $H_L$ do not converge to those of $H$. Instead, they crowd together and converge onto a horizontal line: the real axis in our first case and the line $\Im(k)=-c/2$ in our second case. In the region below the horizontal line resonances of $H_L$ converge to the reflectionless points of $H$ and to those of $-d^2/dx^2 + V_2(x)$. It is only in the region between the real axis and the horizontal line (empty in our first case) that resonances of $H_L$ converge to resonances of $H$. Although the resonances of $H$ may not be close to any resonance of $H_L$ we show that they still influence the time evolution under $H_L$ for a long time when $L$ is large.
We study exponential decay rates of eigenfunctions of self-adjoint higher order elliptic operators on R^n. We are interested in decay rates as a function of direction. We show that the possible decay rates are to a large extent determined algebraically.
We give a constructive proof for the existence of an $N$-dimensional Bloch basis which is both smooth (real analytic) and periodic with respect to its $d$-dimensional quasi-momenta, when $1\leq d\leq 2$ and $N\geq 1$. The constructed Bloch basis is conjugation symmetric when the underlying projection has this symmetry, hence the corresponding exponentially localized composite Wannier functions are real. In the second part of the paper we show that by adding a weak, globally bounded but not necessarily constant magnetic field, the existence of a localized basis is preserved.
We show that the shape resonances induced by a one dimensional well of delta functions disappear as soon as a small constant electric field is applied. In particular, in any compact subset below the positive real axis there are no resonances if the non-zero field is small enough. In contrast to the lack of convergence of the lifetimes computed from the widths of the resonances we show that the "experimental lifetimes" are continuous at zero field. The shape resonances are replaced by an infinite set of other resonances whose location and number we analyze.
We study exponential decay of eigenfunctions of self-adjoint higher order elliptic operators on $\R^d$. We show that the possible critical decay rates are determined algebraically. In addition we show absence of super-exponentially decaying eigenfunctions and a refined exponential upper bound.
We consider the bosonic Fock space over the Hilbert space of transversal vector fields in three dimensions. This space carries a canonical representation of the group of rotations. For a certain class of operators in Fock space we show that rotation invariance implies the absence of terms which either create or annihilate only a single particle. We outline an application of this result in an operator theoretic renormalization analysis of Hamilton operators, which occur in non-relativistic qed.
In the operator theoretic renormalization analysis introduced by Bach, Froehlich, and Sigal we prove uniqueness of the ground state.
We consider conditions under which an embedded eigenvalue of a self-adjoint operator remains embedded under small perturbations. In the case of a simple eigenvalue embedded in continuous spectrum of multiplicity m < \infty we show that in favorable situations the set of small perturbations of a suitable Banach space which do not remove the eigenvalue form a smooth submanifold of co-dimension m.
We discuss recent results concerning the ground state of non-relativistic quantum electrodynamics as a function of a magnetic coupling constant or the fine structure constant, obtained by the authors in [12,13,14].
We consider the ground state of an atom in the framework of non-relativistic qed. We show that the ground state as well as the ground state energy are analytic functions of the coupling constant which couples to the vector potential, under the assumption that the atomic Hamiltonian has a non-degenerate ground state. Moreover, we show that the corresponding expansion coefficients are precisely the coefficients of the associated Raleigh-Schroedinger series. As a corollary we obtain that in a scaling limit where the ultraviolet cutoff is of the order of the Rydberg energy the ground state and the ground state energy have convergent power series expansions in the fine structure constant $α$, with $α$ dependent coefficients which are finite for $α\geq 0$.
We consider the ground state of an atom in the framework of non-relativistic qed. We assume that the ultraviolet cutoff is of the order of the Rydberg energy and that the atomic Hamiltonian has a non-degenerate ground state. We show that the ground state energy and the ground state are k-times continuously differentiable functions of the fine structure constant and respectively the square root of the fine structure constant on some nonempty interval [0,c_k).
We prove that the Hamiltonian of the model describing a spin which is linearly coupled to a field of relativistic and massless bosons, also known as the spin-boson model, admits a ground state for small values of the coupling constant lambda. We show that the ground state energy is an analytic function of lambda and that the corresponding ground state can also be chosen to be an analytic function of lambda. No infrared regularization is imposed. Our proof is based on a modified version of the BFS operator theoretic renormalization analysis. Moreover, using a positivity argument we prove that the ground state of the spin-boson model is unique. We show that the expansion coefficients of the ground state and the ground state energy can be calculated using regular analytic perturbation theory.