Note on Lieb-Thirring Type Inequalities for a Complex Perturbation of Fractional Laplacian

For s > 0, let H0 = (-∆)s be the fractional Laplacian. In this paper, we obtain Lieb-Thirring type inequalities for the fractional Schrödinger operator defined as H = H0 + V , where V ∈ Lp(ℝd), p ≥ 1, d ≥ 1, is a complex-valued potential. Our methods are based on the results of articles by Borichev-Golinskii-Kupin [BGK09] and Hansmann [Han11]Для s > 0 пусть H0 = (-∆)s будет дробным лапласианом. В данной статье мы получаем неравенства типа Либа-Тирринга для дробного оператора Шредингера, который определяется как H = H0 + V, где V ∈ Lp(ℝd), p ≥ 1, d ≥ 1, - комплексный потенциал. Наши методы основываются на результатах работ Borichev-Golinskii-Kupin (Bull. Lond. Math. Soc. 41 (2009), No. 1, 117-123) и Hansmann (Lett. Math. Phys. 98 (2011), No. 1, 79-95)

Finite-size scaling of helix-coil transitions in poly-alanine studied by multicanonical simulations

We report results from multicanonical simulations of poly-alanine. Homopolymers of up to 30 amino acids were considered and various thermodynamic quantities as a function of temperature calculated. We study the nature of the observed helix-coil transition and present estimates for critical exponents.Comment: to appear in J.Chem.Phys. (Jan 99

Side-chain and backbone ordering in Homopolymers

In order to study the relation between backbone and side chain ordering in proteins, we have performed multicanonical simulations of deka-peptide chains with various side groups. Glu10, Gln10, Asp10, Asn10, and Lys10 were selected to cover a wide variety of possible interactions between the side chains of the monomers. All homopolymers undergo helix-coil transitions. We found that peptides with long side chains that are capable of hydrogen bonding, i.e. Glu10, and Gln10, exhibit a second transition at lower temperatures connected with side chain ordering. This occurs in gas phase as well as in solvent, although the character of the side chain structure is different in each case. However, in polymers with short side chains capable of hydrogen bonding, i.e. Asp10 and Asn10, side chain ordering takes place over a wide temperature range and exhibits no phase transition like character. Moreover, non-backbone hydrogen bonds show enhanced formation and fluctuations already at the helix-coil transition temperature, indicating competition between side chain and backbone hydrogen bond formation. Again, these results are qualitatively independent of the environment. Side chain ordering in Lys10, whose side groups are long and polar, also takes place over a wide temperature range and exhibits no phase transition like character in both environments. Reasons for the observed chain length threshold and consequences from these results for protein folding are discussed.Comment: 12 pages,11 figure