An infinite family of superintegrable systems from higher order ladder operators and supersymmetry

We will discuss how we can obtain new quantum superintegrable Hamiltonians allowing the separation of variables in Cartesian coordinates with higher order integrals of motion from ladder operators. We will discuss also how higher order supersymmetric quantum mechanics can be used to obtain systems with higher order ladder operators and their polynomial Heisenberg algebra. We will present a new family of superintegrable systems involving the fifth Painleve transcendent which possess fourth order ladder operators constructed from second order supersymmetric quantum mechanics. We present the polynomial algebra of this family of superintegrable systems.Comment: 8 pages, presented at ICGTMP 28, accepted for j.conf.serie

Addition theorems and the Drach superintegrable systems

We propose new construction of the polynomial integrals of motion related to the addition theorems. As an example we reconstruct Drach systems and get some new two-dimensional superintegrable Stackel systems with third, fifth and seventh order integrals of motion.Comment: 18 pages, the talk given on the conference "Superintegrable Systems in Classical and Quantum Mechanics", Prague 200

Third order superintegrable systems separating in polar coordinates

A complete classification is presented of quantum and classical superintegrable systems in $E_2$ that allow the separation of variables in polar coordinates and admit an additional integral of motion of order three in the momentum. New quantum superintegrable systems are discovered for which the potential is expressed in terms of the sixth Painlev\'e transcendent or in terms of the Weierstrass elliptic function

The Star Blended with the MOA-2008-BLG-310 Source Is Not the Exoplanet Host Star

High resolution Hubble Space Telescope (HST) image analysis of the MOA-2008-BLG-310 microlens system indicates that the excess flux at the location of the source found in the discovery paper cannot primarily be due to the lens star because it does not match the lens-source relative proper motion, $\mu_{\rm rel}$, predicted by the microlens models. This excess flux is most likely to be due to an unrelated star that happens to be located in close proximity to the source star. Two epochs of HST observations indicate proper motion for this blend star that is typical of a random bulge star, but is not consistent with a companion to the source or lens stars if the flux is dominated by only one star, aside from the lens. We consider models in which the excess flux is due to a combination of an unrelated star and the lens star, and this yields 95\% confidence level upper limit on the lens star brightness of $I_L > 22.44$ and $V_L >23.62$. A Bayesian analysis using a standard Galactic model and these magnitude limits yields a host star mass $M_h = 0.21 ^{+0.21}_{-0.09}~ M_\odot$, a planet mass of $m_p = 23.4 ^{+23.9}_{-9.9}~M_\oplus$ at a projected separation of $a_\perp = 1.12^{+0.16}_{-0.17},$AU. This result illustrates excess flux in a high resolution image of a microlens-source system need not be due to the lens. It is important to check that the lens-source relative proper motion is consistent with the microlensing prediction. The high resolution image analysis techniques developed in this paper can be used to verify the WFIRST exoplanet microlensing survey mass measurements.Comment: Submitted to AJ on March 18, 201

Families of superintegrable Hamiltonians constructed from exceptional polynomials

We introduce a family of exactly-solvable two-dimensional Hamiltonians whose wave functions are given in terms of Laguerre and exceptional Jacobi polynomials. The Hamiltonians contain purely quantum terms which vanish in the classical limit leaving only a previously known family of superintegrable systems. Additional, higher-order integrals of motion are constructed from ladder operators for the considered orthogonal polynomials proving the quantum system to be superintegrable

Classical ladder operators, polynomial Poisson algebras and classification of superintegrable systems

We recall results concerning one-dimensional classical and quantum systems with ladder operators. We obtain the most general one-dimensional classical systems respectively with a third and a fourth order ladder operators satisfying polynomial Heisenberg algebras. These systems are written in terms of the solutions of quartic and quintic equations. We use these results to present two new families of superintegrable systems and examples of trajectories that are deformed Lissajous's figures.Comment: 18 page

Two-dimensional superintegrable metrics with one linear and one cubic integral

We describe all local Riemannian metrics on surfaces whose geodesic flows are superintegrable with one integral linear in momenta and one integral cubic in momenta. We also show that some of these metrics can be extended to the 2-sphere. This gives us new examples of Hamiltonian systems on the sphere with integrals of degree three in momenta, and the first examples of superintegrable metrics of nonconstant curvature on a closed surfaceComment: 35 page

ExELS: an exoplanet legacy science proposal for the ESA Euclid mission. II. Hot exoplanets and sub-stellar systems

The Exoplanet Euclid Legacy Survey (ExELS) proposes to determine the frequency of cold exoplanets down to Earth mass from host separations of ~1 AU out to the free-floating regime by detecting microlensing events in Galactic Bulge. We show that ExELS can also detect large numbers of hot, transiting exoplanets in the same population. The combined microlensing+transit survey would allow the first self-consistent estimate of the relative frequencies of hot and cold sub-stellar companions, reducing biases in comparing "near-field" radial velocity and transiting exoplanets with "far-field" microlensing exoplanets. The age of the Bulge and its spread in metallicity further allows ExELS to better constrain both the variation of companion frequency with metallicity and statistically explore the strength of star-planet tides. We conservatively estimate that ExELS will detect ~4100 sub-stellar objects, with sensitivity typically reaching down to Neptune-mass planets. Of these, ~600 will be detectable in both Euclid's VIS (optical) channel and NISP H-band imager, with ~90% of detections being hot Jupiters. Likely scenarios predict a range of 2900-7000 for VIS and 400-1600 for H-band. Twice as many can be expected in VIS if the cadence can be increased to match the 20-minute H-band cadence. The separation of planets from brown dwarfs via Doppler boosting or ellipsoidal variability will be possible in a handful of cases. Radial velocity confirmation should be possible in some cases, using 30-metre-class telescopes. We expect secondary eclipses, and reflection and emission from planets to be detectable in up to ~100 systems in both VIS and NISP-H. Transits of ~500 planetary-radius companions will be characterised with two-colour photometry and ~40 with four-colour photometry (VIS,YJH), and the albedo of (and emission from) a large sample of hot Jupiters in the H-band can be explored statistically.Comment: 18 pages, 16 figures, accepted MNRA

ExELS: an exoplanet legacy science proposal for the ESA Euclid mission. II. Hot exoplanets and sub-stellar systems

The Exoplanet Euclid Legacy Survey (ExELS) proposes to determine the frequency of cold exoplanets down to Earth mass from host separations of ~1 AU out to the free-floating regime by detecting microlensing events in Galactic Bulge. We show that ExELS can also detect large numbers of hot, transiting exoplanets in the same population. The combined microlensing+transit survey would allow the first self-consistent estimate of the relative frequencies of hot and cold sub-stellar companions, reducing biases in comparing "near-field" radial velocity and transiting exoplanets with "far-field" microlensing exoplanets. The age of the Bulge and its spread in metallicity further allows ExELS to better constrain both the variation of companion frequency with metallicity and statistically explore the strength of star-planet tides. We conservatively estimate that ExELS will detect ~4100 sub-stellar objects, with sensitivity typically reaching down to Neptune-mass planets. Of these, ~600 will be detectable in both Euclid's VIS (optical) channel and NISP H-band imager, with ~90% of detections being hot Jupiters. Likely scenarios predict a range of 2900-7000 for VIS and 400-1600 for H-band. Twice as many can be expected in VIS if the cadence can be increased to match the 20-minute H-band cadence. The separation of planets from brown dwarfs via Doppler boosting or ellipsoidal variability will be possible in a handful of cases. Radial velocity confirmation should be possible in some cases, using 30-metre-class telescopes. We expect secondary eclipses, and reflection and emission from planets to be detectable in up to ~100 systems in both VIS and NISP-H. Transits of ~500 planetary-radius companions will be characterised with two-colour photometry and ~40 with four-colour photometry (VIS,YJH), and the albedo of (and emission from) a large sample of hot Jupiters in the H-band can be explored statistically.Comment: 18 pages, 16 figures, accepted MNRA

Superintegrability and higher order polynomial algebras II

In an earlier article, we presented a method to obtain integrals of motion and polynomial algebras for a class of two-dimensional superintegrable systems from creation and annihilation operators. We discuss the general case and present its polynomial algebra. We will show how this polynomial algebra can be directly realized as a deformed oscillator algebra. This particular algebraic structure allows to find the unitary representations and the corresponding energy spectrum. We apply this construction to a family of caged anisotropic oscillators. The method can be used to generate new superintegrable systems with higher order integrals. We obtain new superintegrable systems involving the fourth Painleve transcendent and present their integrals of motion and polynomial algebras.Comment: 11 page