Bloch electrons in a magnetic field: Rigorous justification of the Peierls-Onsager effective Hamiltonian

We present a rigorous justification of the Peierls-Onsager one-band effective Hamiltonian, starting from the one-particle Hamiltonian for Bloch electrons in a homogeneous magnetic field.     

A Tensor Product of Representations of Cuntz Algebras

We introduce a nonsymmetric and associative tensor product among representations of Cuntz algebras by using embeddings. Since the tensor product of permutative representations is also a permutative representation, the decomposition of such a tensor product is unique up to unitary equivalence. We show the decomposition formulae explicitly. As an application, we show properties of concrete endomorphisms.      

A lower bound on the size of molecules

We give lower bounds on the size of a molecule in the Born-Oppenheimer approximation. For diatomic molecules, we give a new lower bound on the number of electrons in terms of the nuclear charges.     

Geometric quantization of an OSp(1/2) coadjoint orbit

In this Letter, we show how the complete geometric quantization extends to specific supersymplectic supermanifolds. More precisely, we extend this procedure to OSp(1/2)-coadjoint orbits, which are graded extensions of elliptic Sp(2, )-coadjoint orbits. Our approach exploits results obtained in a previous work, where the notion of a super-Kähler supermanifold was defined, and the former orbits were shown to be nontrivial examples of such a notion. As their underlying Kähler manifolds, these supermanifolds carry a natural (super-Kähler) polarization, a crucial notion that was so far lacking. Geometric quantization leads here to a nontrivial representation of osp(1/2), which is realized in a space of square integrable holomorphic sections of a super-Hermitian complex line bundle sheaf-with-connection over the homogenous space OSp(1/2)/U(1).     

Limits on stability of positive molecular ions

It is shown that a positive diatomic molecule consisting of N electrons and two nuclei with charges Z ₁ and Z ₂ is unstable with respect to breakup into two atomic subsystems if the nuclear charges are sufficiently large. Bounds on the critical charge are obtained in the limit as N .Research supported in part by NSF grants DMS-8709805 and DMS-8808112.     

Some remarks concerning unique continuation for the Dirac operator

We show that the strong unique continuation property for the Dirac operator holds if the potential belongs toL ^p withp = max(d,(3d – 4)/2).     

Adiabatic dynamics for a class of quantum Hamiltonians

We establish the existence of a special class of unitary transformations that act on the parameter space of a broad class of physical Hamiltonians (including externally imposed electromagnetic fields). For this class, we calculate the quantum amplitudes for adiabatically induced transitions. Processes described include transitions between bound states and transitions from a bound state to the continuum. The leading terms of the adiabatic limit are evaluated in closed form.     

Selfadjointness and positivity of a certain partial differential operator associated with the Kerr metric in general relativity

A second-order partial differential operator singular on the boundary, is shown to be positive selfadjoint without zero eigenvalue. The operator may serve as a kind of the Hamiltonian operator for the Klein-Gordon equation in the Kerr metric of general relativity.     

Geometrical construction of a generalized Skyrme Lagrangian

A geometrical scheme is proposed that leads to a straightforward generalization of a Skyrme Lagrangian, comprising higher-order terms up to the eighth power in pion fields.Boursière DGRST.     

Ground state of a spin 1/2 charged particle in an even-dimensional magnetic field

We investigate the ground state structure of the Schrödinger operator (Pauli Hamiltonian)H with a magnetic fieldb for a spin 1/2 charged particle in ^{2d 2d d} . We consider the case whereb is given by the complex exterior derivative of a functionW on ^d of the form W. We find that dim kerH is related to the asymptotic behavior ofW at infinity. More precisely, if there exists a constantC such that there exists the nonzero limit lim_|z|e ^w(z) /|z|^–C , then dim kerH is equal to the number of all monomialsf ind variables such that the degree off is smaller than |C| -d. In the case whereC , under a weaker assumption this conclusion holds. Moreover, we clarify the structure of kerH.     

On modular inclusion and spectrum condition

For a family of von Neumann algebras, we investigate the relations between spectrum condition, half-sided modular inclusion, and reflection symmetry. It is shown that the geometric interpretation of modular actions puts strong restrictions on the family.     

Spéculation sur la cosmologie quantique primordiale

An old-fashioned axiomatic formalism for the quantization of the gravitational field is adapted to a recently introduced notion of cosmological gravitational wave. Dynamical equations for the quantum field are proposed. An outline for the primordial cosmological scenario is suggested.

     

A relativistic method for the nonrelativistic gap problem

We examine several criteria concerning the problem of gaps in the spectrum of Hamiltonians for one-dimensional alloys. In particular, a criterion which uses a theorem from the special theory of relativity is compared with criteria already known.     

A note on essential duality

By considering some simple models, it is shown that the essential duality condition for local nets of von Neumann algebras associated with Wightman fields need not be fulfilled if Lorentz covariance is dropped. These models illustrate a point made by Borchers in the proof of his two-dimensional CPT theorem for local nets: The Lorentz covariant net constructed from the wedge algebras of a given two-dimensional net may not be unique. It is also shown that in higher dimensions, the Lorentz boosts constructed by means of the modular groups of wedge algebras may act nonlocally in the directions parallel to the edge of the wedge.     

A stochastic posterior Schrödinger equation for counting nondemolition measurement

A stochastic model of the continuous counting nondemolition measurement of photons emitted by an excited atom in a quantum electromagnetic field is given. It is proved that the posterior dynamics of such an atom is described by a new nonlinear stochastic wave equation showing the relaxation of the atom to its ground state due to the photon emission.     

Cosmological principle and primordial spinor fields

An exact solution of Dirac's equation in an open Robertson-Walker universe is constructed such that the corresponding energy-momentum densityT _µv obeys the cosmological principle. However, in anopen universe, this principlemust always be violated by the other physical densities of the cosmological solution (e.g. pseudoscalar, (axial-) current and polarization). If the current densityj _µ = isrequired to obey the cosmological principle, then the universe must beflat.     

C *-algebraic quantization and the origin of topological quantum effects

Concepts from the theory of abstract operator algebras are used to solve the problem of quantizing a particle moving on an arbitrary locally compact homogeneous space. Inequivalent quantizations are identified with inequivalent irreducible representations of the corresponding C ^*-algebra. Topological terms in the action (or Hamiltonian) are found to be representation-dependent, and are automatically induced by the quantization procedure. Known charge quantization conditions turn out to be identically satisfied. Several examples are considered, among them the Dirac monopole and the Aharonov-Bohm effect.     

Vertex operator representations of the quantum affine algebra U q(B r (1) )

We construct the level one vertex operator representations of the q-deformation U _q(B _r ⁽¹⁾ ) of the affine Kac-Moody algebra B _r ⁽¹⁾ . Beside the q-deformed vertex operators introduced by Frenkel and Jing, this construction involves a q-deformation of free fermionic fields.     

The vacuum and lightcone quantization of interaction Hamiltonians

Let denote the conformally invariant neutral free scalar field on ×S ⁿ. The naive lightcone Hamiltonian for a ^p interaction is given by c^p, where C denotes a lightcone in ×S ⁿ, and the Wick power is relative to the free vacuum. We show that this sesquilinear form annihilates the free vacuum if n3 is odd, p>2, and p(n–1)0 mod 4.