Hölder Regularity of Integrated Density of States for the Almost Mathieu Operator in a Perturbative Regime

Consider the Almost Mathieu operator H = cos 2(k +)+ on the lattice. It is shown that for large , the integrated density of states is Hölder continuous of exponent < . This result gives a precise version in the perturbative regime of recent work by M. Goldstein and W. Schlag on Hölder regularity of the integrated density of states for 1D quasi-periodic lattice Schrödinger operators, assuming positivity of the Lyapunov exponent (and proven by different means). Our approach provides also a new way to control Green's functions, in the spirit of the author's work in KAM theory. It is by no means restricted to the cosine-potential and extends to band operators.     

Hermitian Operators Conjugate to Two-Mode Number-Difference Operator Studied in Entangled State Representation

In similar to the derivation of phase angle operator conjugate to the number operator by Arroyo Carrasco-Moya Cessay we deduce the Hermitian phase operators that are conjugate to the two-mode number-difference operator and the three-mode number combination operator. It is shown that these operators are on the same footing in the entangled state representation as the one of Turski in the coherent state representation.     

Consistency check on the fundamental and alternative flux operators in loop quantum gravity

There are different constructions of the flux of triad in loop quantum gravity, namely the fundamental and alternative flux operators. In parallel to the consistency check on the two versions of operator by the algebraic calculus in the literature, we check their consistency by the graphical calculus. Our calculation based on the original Brink graphical method is obviously simpler than the algebraic calculation. It turns out that our consistency check fixes the regulating factor κreg of the Ashtekar-Lewandowski volume operator as1/2, which corrects its previous value in the literature.     

Maximal operators of Fejér means of double Walsh-Fourier series

The main aim of this paper is to prove that the maximal operator of the Fejér mean of the double Walsh-Fourier series is not bounded from the Hardy space H _1/2 to the space weak-L _1/2. This paper was written during the visit of the author at the College of Nyíregyháza in Hungary.     

Hard-sphere binary-collision operators

The time displacement operator is described for a system of hard-sphere particles. We show how to avoid needing a representation for this operator in unphysical regions of phase space, and how to construct a useful representation in terms of binary collision operators in the physical region. The various binary collision operators used for hard-sphere systems are derived for the case of a system of two spheres, and the results are generalized toN-particle systems.Dedicated to Prof. E. G. D. Cohen on the occasion of his 65th birthday.     

Deficiency Indices of a Symmetric Ordinary Differential Operator with Infinitely Many Degeneration Points

Let H be the minimal symmetric operator in L ²() generated by the differential expression (-1) ⁿ (c(x)f ⁽ⁿ⁾)⁽ⁿ⁾, n 1, with a real coefficient c(x) that has countably many zeros without finite accumulation points and is infinitely smooth at all points x with c(x) 0. We study the value Def H of the deficiency indices of H. It is shown that DefH=+ if infinitely many zeros of c(x) have multiplicities p satisfying the inequality n – 1/2 < p < 2n – 1/2. Our second result pertains to the case in which the set of zeros of c(x) is bounded neither above nor below. Under this condition, Def H = 0 provided that the multiplicity of each zero is greater than or equal to 2n – 1/2. The multiplicities of zeros of c(x) are understood in the paper in a broader sense than in the standard definition.     

The well-posedness of the Korteweg-de Vries-Benjamin-Ono equation

The well-posedness of the Cauchy problem to the generalized Korteweg-de Vries-Benjamin-Ono equation is considered. Local results for data in (s?−1/8) and the global well-posedness for data in are obtained if l=2. Moreover, for l=3, the problem is locally well-posed for data in H^s (s?1/4). The main idea is to use the Fourier restriction norm method.     

Singular Sturm-Liouville problems whose coefficients depend rationally on the eigenvalue parameter

Let −Dω(·,z)D+q be a differential operator in L²(0,∞) whose leading coefficient contains the eigenvalue parameter z. For the case that ω(·,z) has the particular form