Mathematical Physics and Harmonic Analysis Seminar

The following topics will be covered:

1. Complexity of Bipartite Entanglement: It is known [Gurvits, 2003] that to check entanglement for a bipartite (n,m) mixed state, with n>=m>=n^a, is NP-hard for any a>0. I will sketch a reduction of this problem to checking whether a particular well-defined real polynomial in m^2 variables of degree n is positive on R^{m^2}. For fixed m this gives a polynomial-time algorithm.
2. Largest Separable Balls: I will review results on the so-called largest separable ball. The main focus will be on exact, (1/\sqrt{6})^n asymptotics for the radius of the largest separable ball in the case of n qubits.
3. ``Quantum'' van der Waerden Conjecture: I will state and prove sufficient conditions for Bipartite Entanglement, expressed in terms of the 4-dimensional Pascal Hyperdeterminant.

For a fixed natural number n, we consider a family of rank n unitary perturbations of a completely non-unitary contraction with deficiency indices (n, n) on a separable Hilbert space. We relate the unitary dilation of such a contraction to its rank n unitary perturbations. Based on this construction, we prove that the spectra of the perturbed operators are purely singular if and only if the operator-valued characteristic function corresponding to the unperturbed operator is inner. In the case where n = 1, the latter statement reduces to a well-known result in the theory of rank one perturbations. However, our method of proof via the theory of dilations extends to the case of arbitrary n.

In the talk we discuss Barry Simon's subshift conjecture, which states that the (OPUC or Schr\"odinger) spectrum associated with a minimal aperiodic subshift has zero Lebesgue measure. We use approximation with subshifts of finite type along with an analysis of the spectra associated with periodic orbits in them to produce a counterexample. This is joint work with Artur Avila and Zhenghe Zhang.

The vacuum (Casimir) energy in quantum field theory is a problem relevant both to new nanotechnology devices and to dark energy in cosmology. The crucial question is the dependence of the energy on the system geometry under study. Despite much progress since the first prediction of the Casimir effect in 1948 and its subsequent experimental verification in simple geometries, even the sign of the force in nontrivial situations is still a matter of controversy. Mathematically, vacuum energy fits squarely into the spectral theory of second-order self-adjoint elliptic linear differential operators. Specifically, one promising approach is based on the small-$t$ asymptotics of the cylinder kernel $e^{-t\sqrt{H}}$, where $H$ is the self-adjoint operator under study. In contrast with the well-studied heat kernel $e^{-tH}$, the cylinder kernel depends in a non-local way on the geometry of the problem. We discuss analytical and numerical results by the Louisiana-Oklahoma-Texas collaboration on vacuum energy in model systems, including quantum graphs and two- and three-dimensional cavities. The results may shed light on general questions, including the relationship between vacuum energy and periodic or closed classical orbits, and the contribution to vacuum energy of boundaries, edges, and corners.

Hertz potentials are "second-generation" potentials for the electromagnetic field: The ordinary 4-vector potential is formed from first derivatives of the Hertz potential. For the field inside a cavity with conducting walls, the existence of the Hertz potential and even of the vector potential itself depends on the topology of the cavity. Nontrivial second (co)homology gives rise to field modes with no vector potential, which are "nonpropagating" modes effectively determined by electric or magnetic charges inside the "holes". Similarly, nontrivial first homology gives rise to modes with no Hertz potential, linked to currents inside excluded "tubes"; but these are "propagating" modes that are part of the independent dynamics of the field. In systems with cylindrical symmetry (translation along one axis), the Hertz potential can be reduced to two scalar potentials, corresponding to TM and TE modes; the remaining, topological modes are called TEM modes. Correct commutation relations for the electric and magnetic fields are be obtained by quantizing the scalar Hertz potentials with slightly noncanonical commutation relations. The verification of this fact requires subtle reasoning with the Green functions of the Dirichlet and Neumann problems on the 2-dimensional cross section, which we have completed only for the special cases of circular, annular, and rectangular cross sections. We have verified that the TEM modes do need to be included and quantized to achieve the right commutators. This result is significant because others have noted that existence of TEM modes (createable by the mere insertion of longitudinal wires in the cavity) should produce a Casimir attraction between the end plates of longer range than the usual one. Our work indicates that this counterintuitive result is not an error. More generally, the facts reported by Jeff Bouas and me appear to be fundamental to electromagnetism, classical and quantum, but are far from clear in textbook treatments of the subject.
Abstract

Ultracold atoms make very versatile systems for studying the dynamics of isolated quantum systems because of their low damping rates and high levels of tunability. In this talk we discuss the dynamics of a bosonic Josephson junction made from an atomic Bose-Einstein condensate trapped in a double well potential. In particular, we consider the dynamics that ensue following a sudden change in the tunneling barrier, and find that this generates rainbows in Fock space. In fact, elementary catastrophe theory shows that such structures are generic in this type of dynamics. In the Gross-Pitaevskii mean-field theory these rainbows are singular caustics, but in the second-quantized theory they are described by well behaved Airy functions. Using the properties of Airy functions we are able to discuss the analyticity of the macroscopic limit and also show that the long-time dynamics are ergodic. Our results are relevant to the question posed by Berry [M.V.Berry, Nonlinearity 21, T19 (2008)]: are there circumstances when it is necessary to second-quantize wave theory in order to avoid singularities?

See arXiv:1202.1341 for a preprint of this work.