130,780 research outputs found
Orthogonal polynomials associated to almost periodic Schrödinger operators. A trend towards random orthogonal polynomials
No full textAbstractWe introduce a special class of Schrödinger type H-operators in l2 as (φ,Hψ) = ∑∞n=0 φ∗Rn+1ψn+1 +Rnψn−1,Rn being a nonnegative real number. H satisfies the renormalization equation HD = D(H2 - λ), with λ real, λ ⩾ 2. D is the decimation operator defined by (φ,Dψ) = ∑∞n=0φ∗nψ2n. A consequence of the renormalization equation is that the Rn fulfil the recursion relation R0 = 0, R2nR2n−1 = Rn, R2n + R2n+1 = λ. From the above relations, it can be shown that the Rn are quasi-periodic functions of their index n.The components of the eigenfunctions of H corresponding to the eigenvalue x are the orthonormalized polynomials Pn (x) satisfying Rn+1Pn+1(x) + RnPn−1(x) = (x)Pn(x). The spectrum of H is the support of the measure associated to the polynomials. In the present case it is a compact perfect set of Lebesque measure zero (Cantor set). It is therefore purely singular continuous.We are led to study classes of orthogonal polynomials whose three-terms recursive relations are quasi periodic functions of their index. We will present several results, conjectures and open questions which may have relevant physical applications. We study the randomness of the eigenfunctions, and we discuss their algorithmic complexity
Construction of multifractal measures in dynamical systems from their invariance properties
No full textOn D. Hägele’s approach to the Bessis–Moussa–Villani conjecture
No full textAbstractThe reformulation of the Bessis–Moussa–Villani (BMV) conjecture given by Lieb and Seiringer asserts that the coefficient αm,k(A,B) of tk in the polynomial Tr(A+tB)m, with A,B positive semidefinite matrices, is nonnegative for all m,k. We propose a natural extension of a method of attack on this problem due to Hägele, and investigate for what values of m,k the method is successful, obtaining a complete determination when either m or k is odd
Advances on the Bessis–Moussa–Villani trace conjecture
No full textAbstractA long-standing conjecture asserts that the polynomialp(t)=Tr[(A+tB)m]has nonnegative coefficients whenever m is a positive integer and A and B are any two n×n positive semidefinite Hermitian matrices. The conjecture arises from a question raised by Bessis et al. [D. Bessis, P. Moussa, M. Villani, Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics, J. Math. Phys. 16 (1975) 2318–2325] in connection with a problem in theoretical physics. Their conjecture, as shown recently by Lieb and Seiringer, is equivalent to the trace positivity statement above. In this paper, we derive a fundamental set of equations satisfied by A and B that minimize or maximize a coefficient of p(t). Applied to the Bessis–Moussa–Villani (BMV) conjecture, these equations provide several reductions. In particular, we prove that it is enough to show that (1) it is true for infinitely many m, (2) a nonzero (matrix) coefficient of (A+tB)m always has at least one positive eigenvalue, or (3) the result holds for singular positive semidefinite matrices. Moreover, we prove that if the conjecture is false for some m, then it is false for all larger m. Finally, we outline a general program to settle the BMV conjecture that has had some recent success
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