Stefan Boettcher

Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry

The condition of self-adjointness ensures that the eigenvalues of a Hamiltonian are real and bounded below. Replacing this condition by the weaker condition of

PT-symmetric quantum mechanics

\mathrm{PT}

Nature's way of optimizing

symmetry, one obtains new infinite classes of complex Hamiltonians whose spectra are also real and positive. These

Extremal optimization for graph partitioning

\mathrm{PT}

Quasi-exactly solvable quartic potential

symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space. This paper describes the unusual classical and quantum properties of these theories.

Extremal Optimization: An Evolutionary Local-Search Algorithm

This paper proposes to broaden the canonical formulation of quantum mechanics. Ordinarily, one imposes the condition H†=H on the Hamiltonian, where † represents the mathematical operation of complex conjugation and matrix transposition. This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian H has a real spectrum. However, replacing this mathematical condition by the weaker and more physical requirement H‡=H, where ‡ represents combined parity reflection and time reversal PT, one obtains new classes of complex Hamiltonians whose spectra are still real and positive. This generalization of Hermiticity is investigated using a complex deformation H=p2+x2(ix)e of the harmonic oscillator Hamiltonian, where e is a real parameter. The system exhibits two phases: When e⩾0, the energy spectrum of H is real and positive as a consequence of PT symmetry. However, when −1<e<0, the spectrum contains an infinite number of complex eigenvalues and a finite number of real, positive eigenvalues b...

Extremal optimization for Sherrington-Kirkpatrick spin glasses

We propose a general-purpose method for finding high-quality solutions to hard optimization problems, inspired by self-organizing processes often found in nature. The method, called Extremal Optimization, successively eliminates extremely undesirable components of sub-optimal solutions. Drawing upon models used to simulate far-from-equilibrium dynamics, it complements approximation methods inspired by equilibrium statistical physics, such as Simulated Annealing. With only one adjustable parameter, its performance proves competitive with, and often superior to, more elaborate stochastic optimization procedures. We demonstrate it here on two classic hard optimization problems: graph partitioning and the traveling salesman problem.

Interoccurrence Times in the Bak-Tang-Wiesenfeld Sandpile Model: A Comparison with the Observed Statistics of Solar Flares

Extremal optimization is a new general-purpose method for approximating solutions to hard optimization problems. We study the method in detail by way of the computationally hard (NP-hard) graph partitioning problem. We discuss the scaling behavior of extremal optimization, focusing on the convergence of the average run as a function of run time and system size. The method has a single free parameter, which we determine numerically and justify using a simple argument. On random graphs, our numerical results demonstrate that extremal optimization maintains consistent accuracy for increasing system sizes, with an approximation error decreasing over run time roughly as a power law t(-0.4). On geometrically structured graphs, the scaling of results from the average run suggests that these are far from optimal with large fluctuations between individual trials. But when only the best runs are considered, results consistent with theoretical arguments are recovered.

Universality in Sandpiles, Interface Depinning, and Earthquake Models.

A new two-parameter family of quasi-exactly solvable quartic polynomial potentials is introduced. Heretofore, it was believed that the lowest-degree one-dimensional quasi-exactly solvable polynomial potential is sextic. This belief is based on the assumption that the Hamiltonian must be Hermitian. However, it has recently been discovered that there are huge classes of non-Hermitian, -symmetric Hamiltonians whose spectra are real, discrete, and bounded below. Replacing hermiticity by the weaker condition of symmetry allows for new kinds of quasi-exactly solvable theories. The spectra of this family of quartic potentials discussed here are also real, discrete and bounded below and the quasi-exact portion of the spectra consists of the lowest J eigenvalues. These eigenvalues are the roots of a Jth-degree polynomial.

Conjecture on the interlacing of zeros in complex Sturm–Liouville problems

A recently introduced general-purpose heuristic for finding high-quality solutions for many hard optimization problems is reviewed. The method is inspired by recent progress in understanding far-from-equilibrium phenomena in terms ofself-organized criticality, a concept introduced to describe emergent complexity in physical systems. This method, calledextremal optimization, successively replaces the value of extremely undesirable variables in a sub-optimal solution with new, random ones. Large, avalanche-like fluctuations in the cost function self-organize from this dynamics, effectively scaling barriers to explore local optima in distant neighborhoods of the configuration space while eliminating the need to tune parameters. Drawing upon models used to simulate the dynamics of granular media, evolution, or geology, extremal optimization complements approximation methods inspired by equilibrium statistical physics, such assimulated annealing. It may be but one example of applying new insights intonon-equilibrium phenomenasystematically to hard optimization problems. This method is widely applicable and so far has proved competitive with — and even superior to — more elaborate general-purpose heuristics on testbeds of constrained optimization problems with up to 105variables, such as bipartitioning, coloring, and satisfiability. Analysis of a suitable model predicts the only free parameter of the method in accordance with all experimental results.