TY - JOUR AB - We investigate the frequency of positive squareful numbers x, y, z≤B for which x+y=z and present a conjecture concerning its asymptotic behavior. AU - Timothy Browning AU - Valckenborgh, K Van ID - 240 IS - 2 JF - Experimental Mathematics TI - Sums of three squareful numbers VL - 21 ER - TY - JOUR AB - If the polaron coupling constant α is large enough, bipolarons or multi-polarons will form. When passing through the critical α c from above, does the radius of the system simply get arbitrarily large or does it reach a maximum and then explode? We prove that it is always the latter. We also prove the analogous statement for the Pekar-Tomasevich (PT) approximation to the energy, in which case there is a solution to the PT equation at α c. Similarly, we show that the same phenomenon occurs for atoms, e. g., helium, at the critical value of the nuclear charge. Our proofs rely only on energy estimates, not on a detailed analysis of the Schrödinger equation, and are very general. They use the fact that the Coulomb repulsion decays like 1/r, while 'uncertainty principle' localization energies decay more rapidly, as 1/r 2. AU - Frank, Rupert L AU - Lieb, Élliott H AU - Robert Seiringer ID - 2400 IS - 2 JF - Communications in Mathematical Physics TI - Binding of polarons and atoms at threshold VL - 313 ER - TY - JOUR AB - We study the effects of random scatterers on the ground state of the one-dimensional Lieb-Liniger model of interacting bosons on the unit interval in the Gross-Pitaevskii regime. We prove that Bose-Einstein condensation survives even a strong random potential with a high density of scatterers. The character of the wavefunction of the condensate, however, depends in an essential way on the interplay between randomness and the strength of the two-body interaction. For low density of scatterers and strong interactions the wavefunction extends over the whole interval. A high density of scatterers and weak interactions, on the other hand, lead to localization of the wavefunction in a fragmented subset of the interval. AU - Robert Seiringer AU - Yngvason, Jakob AU - Zagrebnov, Valentin A ID - 2403 IS - 11 JF - Journal of Statistical Mechanics Theory and Experiment TI - Disordered Bose-Einstein condensates with interaction in one dimension VL - 2012 ER - TY - JOUR AB - We consider a model of quantum-mechanical particles interacting via point interactions of infinite scattering length. In the case of fermions we prove a Lieb-Thirring inequality for the energy, i.e., we show that the energy is bounded from below by a constant times the integral of the particle density to the power. AU - Frank, Rupert L AU - Robert Seiringer ID - 2402 IS - 9 JF - Journal of Mathematical Physics TI - Lieb-Thirring inequality for a model of particles with point interactions VL - 53 ER - TY - JOUR AB - We find further implications of the BMV conjecture, which states that for hermitian matrices B≥0 and A, the function λ {mapping} Tr exp(A - λB) is the Laplace transform of a positive measure supported on [0,∞]. AU - Lieb, Élliott H AU - Robert Seiringer ID - 2401 IS - 1 JF - Journal of Statistical Physics TI - Further implications of the Bessis-Moussa-Villani conjecture VL - 149 ER -