Examples are given of non-Hermitian Hamiltonian operators which have a real spectrum. Some of the investigated operators are expressed in terms of the generators of the Weil-Heisenberg algebra. It is argued that the existence of an involutive operator Ĵ which renders the Hamiltonian Ĵ-Hermitian leads to the unambiguous definition of an associated positive definite norm allowing for the standard...

In appropriate units, the Brown-Ravenhall Hamiltonian for a system of 1 electron relativistic molecules with K fixed nuclei having charge and position Zk, Rk, k = 1, 2, . . . ,K, is of the form B1,K = Λ+ ( D0 + αVc ) Λ+, where Λ+ is the projection onto the positive spectral subspace of the free Dirac operator D0 and Vc = − ∑K k=1 αZk |x−Rk| + ∑K k<l, k,l=1 αZkZl |Rk−Rl| , with α Sommerfeld’s fi...

It is shown that, in Dirac theory, there is a spatial velocity of a free electron which commutes with the Hamiltonian, so it is a conserved quantity of the motion. Furthermore, there is a spatial orbital angular momentum which also commutes with the Hamiltonian and is a constant of the motion.

We study the Schrödinger operator −∆ − αδ(x − Γ) in L2(R3) with a δ interaction supported by an infinite non-planar surface Γ which is smooth, admits a global normal parameterization with a uniformly elliptic metric. We show that if Γ is asymptotically planar in a suitable sense and α > 0 is sufficiently large this operator has a non-empty discrete spectrum and derive an asymptotic expansion of...

We prove an analogue of a classical asymptotic stability result of standing waves of the Schrödinger equation originating in work by Soffer and Weinstein. Specifically, our result is a transposition on the lattice Z of a result by Mizumachi [M1] and it involves a discrete Schrödinger operator H = −∆+q. The decay rates on the potential are less stringent than in [M1], since we require q ∈ l1,τ w...

We show that for a Hamiltonian action of a compact torus G on a compact, connected symplectic manifold M , the Gequivariant cohomology is determined by the residual S action on the submanifolds of M fixed by codimension-1 tori. This theorem allows us to compute the equivariant cohomology of certain manifolds, which have pieces that are four-dimensional or smaller. We give several examples of th...

An action of a torus T on a compact connected symplectic manifold M is Hamiltonian if it admits a momentum map, which is a map on M taking values in the dual of the Lie algebra of T. In 1982, Atiyah and GuilleminSternberg proved that the image of the momentum map is a convex polytope, now called the momentum polytope. In 1988, Delzant showed that if the dimension of T is half of the dimension o...

We study the properties of a discrete Schrödinger operator. We prove that if its spectrum is discrete in the complement of [−2d, 2d], then it contains [−2d, 2d].

1. In this short paper we extend the method of Laptev-Naboko-Safronov [15]. New estimates for the discrete spectrum obtained in this paper allow one to prove stronger results compared to [15]. The main technical tool of the paper [15] is the so called trace inequality, which relates properties of negative eigenvalues to the properties of the a.c. spectrum. Based on this relation, the technique ...

The idea that spacetime points are to be identified by a fleet of clock–carrying particles can be traced to the earliest days of general relativity. Such a fleet of clocks can be described phenomenologically as a reference fluid. One approach to the problem of time consists in coupling the metric to a reference fluid and solving the super–Hamiltonian constraint for the momentum conjugate to the...