Singular Continuous Spectrum for Palindromic Schrödinger Operators
نویسندگان
چکیده
We give new examples of discrete Schrödinger operators with potentials taking finitely many values that have purely singular continuous spectrum. If the hull X of the potential is strictly ergodic, then the existence of just one potential x in X for which the operator has no eigenvalues implies that there is a generic set in X for which the operator has purely singular continuous spectrum. A sufficient condition for the existence of such an x is that there is a z ∈ X that contains arbitrarily long palindromes. Thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset in X . The class includes well-known substitutions like Fibonacci, ThueMorse, Period Doubling, binary non-Pisot and ternary non-Pisot. We also show that the operator has no absolutely continuous spectrum for all x ∈ X if X derives from a primitive substitution. For potentials defined by circle maps, xn = 1J (θ0 + nα), we show that the operator has purely singular continuous spectrum for a generic subset in X for all irrational α and every half-open interval J .
منابع مشابه
Uniform Cantor Singular Continuous Spectrum for Nonprimitive Schrödinger Operators
It is shown that some Schrödinger operators, with nonprimitive substitution potentials, have pure singular continuous Cantor spectrum with null Lebesgue measure for all elements in the respective hulls.
متن کاملOperators with Singular Continuous Spectrum: Iii. Almost Periodic Schrödinger Operators
We prove that one-dimensional Schrödinger operators with even almost periodic potential have no point spectrum for a dense Gδ in the hull. This implies purely singular continuous spectrum for the almost Mathieu equation for coupling larger than 2 and a dense Gδ in θ even if the frequency is an irrational with good Diophantine properties. §
متن کاملSingular Continuous Spectrum for Certain Quasicrystal Schrödinger Operators
We give a short introduction into the theory of one-dimensional discrete Schrödinger operators associated to quasicrystals. We then report on recent results, obtained in jont work with D. Damanik, concerning a special class of these operators viz Quasi-Sturmian operators. These results show, in particular, uniform singular continuous spectrum of Lebesgue measure zero.
متن کاملImbedded Singular Continuous Spectrum for Schrödinger Operators
We construct examples of potentials V (x) satisfying |V (x)| ≤ h(x) 1+x , where the function h(x) is growing arbitrarily slowly, such that the corresponding Schrödinger operator has imbedded singular continuous spectrum. This solves one of the fifteen “twenty-first century” problems for Schrödinger operators posed by Barry Simon in [22]. The construction also provides the first example of a Sch...
متن کاملSingular Continuous Spectrum of Half-line Schrödinger Operators with Point Interactions on a Sparse Set
We say that a discrete set X = {xn}n∈N0 on the half-line 0 = x0 < x1 < x2 < x3 < · · · < xn < · · · < +∞ is sparse if the distances ∆xn = xn+1−xn between neighbouring points satisfy the condition ∆xn ∆xn−1 → +∞. In this paper half-line Schrödinger operators with point δand δ′-interactions on a sparse set are considered. Assuming that strengths of point interactions tend to ∞ we give simple suff...
متن کامل