Donald Babbitt

Ground state representation of the infinite one-dimensional Heisenberg ferromagnet. II. An explicit Plancherel formula

In its ground state representation, the infinite, spin 1/2 Heisenberg chain provides a model for spin wave scattering, which entails many features of the quantum mechanicalN-body problem. Here, we give a complete eigenfunction expansion for the Hamiltonian of the chain in this representation, forall numbers of spin waves. Our results resolve the questions of completeness and orthogonality of the eigenfunctions given by Bethe for finite chains, in the infinite volume limit.

Algebraic Difficulties of Preserving Dynamical Relations When Forming Quantum‐Mechanical Operators

Proofs are presented showing impossibility of assigning differential operators (quantum observables) to classical mechanical observables in such a way as to preserve the usual bracket formalism. Difficulty is shown to arise even if we limit ourselves to preserving brackets between the Hamiltonian and a rather limited set of observables. Some other algebraic difficulties inherent in the operator assignment problem are also discussed.

Ground state representation of the infinite one-dimensional Heisenberg ferromagnet

In its ground state representation, the infinite, spin 1/2 Heisenberg chain provides a model for spin wave scattering, which entails many features of the quantum mechanicalN-body problem. Here, we give a complete eigenfunction expansion for the Hamiltonian of the chain in this representation, forall numbers of spin waves. Our results resolve the questions of completeness and orthogonality of the eigenfunctions given by Bethe for finite chains, in the infinite volume limit.

Local distortion techniques and unitarity of the S-matrix for the 2-body problem

Abstract The two-body S-matrix for an interaction with exponential decay at infinity is defined in a time-independent way and its unitarity is proved directly by local distortion techniques. Complete sets of incoming and outgoing states, or delicate resolvent estimates are not needed for the proof.

The plancherel formula for the infinite XXZ Heisenberg spin chain

In this Letter we discuss the explicit Plancherel formula for the Bethe Ansatz eigenstates for the Hamiltonian of the infinite XXZ Heisenberg-Ising spin chain in the N-magnon sectors: N=2, 3,... In particular, we shall point out that a striking spectral phenomenon occurs when the coupling constant c is such that 0<|c|<1.

Ground state representation of the infinite one-dimensional Heisenberg ferromagnet. IV. A completely integrable quantum system

Abstract The purpose of this article is to show that the ground state representation of the infinite one-dimensional spin 1 2 Heisenberg chain, with isotropic nearest neighbor interactions, provides an example of a completely integrable quantum system.

Dilation-analyticity and decay properties of interactions

LetH=H0+V be a Schrödinger operator onL2(ℝn). We show that the more dilation analyticV is, the slower it must decay at infinity.

Ground state representation of the infinite one‐dimensional Heisenberg ferromagnet. III. Scattering theory

This article gives a complete description of the scattering for the spin 1/2 Heisenberg ferromagnetic chain in its ground state representation.

Probabilistic Interpretation of the Classical Scattering Cross Section

The probabilistic interpretation of the classical scattering cross section is discussed in a mathematically rigorous framework. In particular, extensive use is made of the notion of a Poisson process based on a plane and on a sphere. The case of Rutherford scattering is given as a detailed illustration.

Probabilistic Interpretation of the Quantum Scattering Cross Section

The probabilistic interpretation of the quantum scattering cross section in the case of potential scattering is discussed in terms of Poisson random measures on the impact parameter plane and the sphere of outgoing directions.