We consider a perturbed Hill’s equation of the form φ̈ + (p0(t) + εp1(t))φ = 0, where p0 is real analytic and periodic, p1 is real analytic and quasi-periodic and ε ∈ R is “small”. Assuming Diophantine conditions on the frequencies of the decoupled system, i.e. the frequencies of the external potentials p0 and p1 and the proper frequency of the unperturbed (ε = 0) Hill’s equation, but without ma...

Calculating the spectral function of two dimensional systems is arguably one most pressing challenges in modern computational condensed matter physics. While efficient techniques are available lower dimensions, present insurmountable hurdles, ranging from sign problem quantum Monte Carlo (MC), to entanglement area law tensor network based methods. We hereby a variational approach on Chebyshev e...

In a recent Letter [1] Öğüt, Chelikowsky, and Louie (OCL) calculated the optical gap of Si nanocrystals as ε g,OCL = ε qp g,LDA − E e−h Coul, where ε qp g,LDA is the quasi-particle gap in the local-density approximation (LDA) and E Coul is the electron-hole Coulomb energy. They argued that their method produces different results from conventional approaches (e.g. pseudopotential [2]). We show i...

We introduce two kinds of quasi-inner functions. Since every rationally invariant subspace for a shift operator SK on a vector-valued Hardy space H(Ω, K) is generated by a quasi-inner function, we also provide relationships of quasi-inner functions by comparing rationally invariant subspaces generated by them. Furthermore, we discuss fundamental properties of quasi-inner functions, and quasi-in...

We introduce two kinds of quasi-inner functions. Since every rationally invariant subspace for a shift operator SK on a vector-valued Hardy space H(Ω, K) is generated by a quasi-inner function, we also provide relationships of quasi-inner functions by comparing rationally invariant subspaces generated by them. Furthermore, we discuss fundamental properties of quasi-inner functions, and quasi-in...

The question whether every operator on H has an hyperinvariant subspace is one of the most difficult problems in operator theory. The purpose of this paper is to make a beginning on the hyperinvariant subspace problems for another class of operators closely related to the normal operators namely, the class of k -quasi-class A operators. A necessary and sufficient condition for the hypercyclicit...

In this paper, a two-dimensional multi-wavelet is constructed in terms of Chebyshev polynomials. The constructed multi-wavelet is an orthonormal basis for space. By discretizing two-dimensional Fredholm integral equation reduce to a algebraic system. The obtained system is solved by the Galerkin method in the subspace of by using two-dimensional multi-wavelet bases. Because the bases of subs...

In this paper, we derive an explicit formula for the bivariate Lagrange basis polynomials of a general set checkerboard nodes. This generalizes existing results at Padua nodes, Chebyshev Morrow-Patterson and Geronimus We also construct subspace spanned by linearly independent vanishing that vanish nodes prove uniqueness in quotient space defined as with certain degree over polynomials.

The electron spin resonance (ESR) of nanoscale molecular magnet V15 is studied. Since the Hamiltonian of V15 has a large Hilbert space and numerical calculations of the ESR signal evaluating the Kubo formula with exact diagonalization method is difficult, we implement the formula with the help of the random vector technique and the Chebyshev polynominal expansion, which we name the double Cheby...

and Applied Analysis 3 Proof. The proof follows from Theorem 2.1. The following theorem shows that on a slowly oscillating compact subset A of R, slowly oscillating continuity implies uniform continuity. Theorem 2.3. Let A be a slowly oscillating compact subset of R and let f : A → R be slowly oscillating continuous on A. Then f is uniformly continuous on A. Proof. Assume that f is not uniforml...