On Absolute Continuity of Spectra of Periodic Elliptic Operators

Communications in Mathematical Physics, 1983

We discuss the absolutely continuous spectrum of H =-d 2 / d x 2 + V(x) with Valmost periodic and its discrete analog (hu)(n) = u(n + 1) + u (n-1) + V(n)u(n). Especial attention is paid to the set, A, of energies where the Lyaponov exponent vanishes. This set is known to be the essential support of the a.c. part of the spectral measure. We prove for a.e. Vin the hull and a.e. E in A, H and h have continuum eigenfunctions, u, with [u[ almost periodic. In the discrete case, we prove that IAI < 4 with equality only if V=const. If k is the integrated density of states, we prove that on A, 2kdk/dE>rc-2 in the continuum case and that 2nsinrckdk/dE> 1 in the discrete case. We also provide a new proof of the Pastur-Ishii theorem and that the multiplicity of the absolutely continuous spectrum is 2.

2010

We study Schrödinger operator H = −∆ + V (x) in dimension two, V (x) being a limit-periodic potential. We prove that the spectrum of H contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves e i k, x at the high energy region. Second, the isoenergetic curves in the space of momenta k corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). Third, the spectrum corresponding to the eigenfunctions (the semiaxis) is absolutely continuous. *. Considering that ϕ is close to ϕ b ± π/2, we readily obtain: b, ν * = o(b 0), b, µ * = ±b 0 (1 + o(1)). Using also estimates (2.70) for h 1 , we get T 1 = ±2b 0 k(1 + o(1)). By (2.63), ∇f 1 y (1) (ϕ) − ∇f 1 y (1) (ϕ) − b = O(b 0 k −2+36s 1 +24δ). Hence, T 2 = o (b 0 k). Adding the estimates for T 1 , T 2 , we get (2.132).

2003

In addition we assume that

2008

Schrödinger operator family H (g) := −d 2 x + V g (x) depending on the real parameter g, where V g (x) is a complex-valued but P T-symmetric periodic potential. An explicit condition on V is obtained which ensures that the spectrum of H (g) is purely real and band shaped; furthermore, a further condition is obtained which ensures that the spectrum contains at least a pair of complex analytic arcs.

arXiv (Cornell University), 2010

We study Schrödinger operator H = −∆ + V (x) in dimension two, V (x) being a limit-periodic potential. We prove that the spectrum of H contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves e i k, x at the high energy region. Second, the isoenergetic curves in the space of momenta k corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). Third, the spectrum corresponding to the eigenfunctions (the semiaxis) is absolutely continuous. *. Considering that ϕ is close to ϕ b ± π/2, we readily obtain: b, ν * = o(b 0), b, µ * = ±b 0 (1 + o(1)). Using also estimates (2.70) for h 1 , we get T 1 = ±2b 0 k(1 + o(1)). By (2.63), ∇f 1 y (1) (ϕ) − ∇f 1 y (1) (ϕ) − b = O(b 0 k −2+36s 1 +24δ). Hence, T 2 = o (b 0 k). Adding the estimates for T 1 , T 2 , we get (2.132).

Communications in Mathematical Physics, 2005

We consider Schrödinger operators H h = (ih d + A) * (ih d + A) with the periodic magnetic field B = dA on covering spaces of compact manifolds. Under some assumptions on B, we prove that there are arbitrarily large number of gaps in the spectrum of these operators in the semiclassical limit of strong magnetic field h → 0.

Transactions of the American Mathematical Society, 2008

We show that, under some very weak assumption of effective variation for the magnetic field, a periodic Schrödinger operator with magnetic wells on a noncompact Riemannian manifold M such that H 1 (M, R) = 0 equipped with a properly disconnected, cocompact action of a finitely generated, discrete group of isometries has an arbitrarily large number of spectral gaps in the semi-classical limit.

2021

We consider Schrödinger operators with periodic electric and magnetic potentials on periodic discrete graphs. The spectrum of such operators consists of an absolutely continuous (a.c.) part (a union of a finite number of non-degenerate bands) and a finite number of eigenvalues of infinite multiplicity. We prove the following results: 1) the a.c. spectrum of the magnetic Schrödinger operators is empty for specific graphs and magnetic fields; 2) we obtain necessary and sufficient conditions under which the a.c. spectrum of the magnetic Schrödinger operators is empty; 3) the spectrum of the magnetic Schrödinger operator with each magnetic potential tα, where t is a coupling constant, has an a.c. component for all except finitely many t from any bounded interval.

Mathematische Zeitschrift, 1999

We demonstrate that the structure of complex second-order strongly elliptic operators H on R d with coefficients invariant under translation by Z d can be analyzed through decomposition in terms of versions H z , z ∈ T d , of H with z-periodic boundary conditions acting on L 2 (I d ) where I = [0, 1 . If the semigroup S generated by H has a Hölder continuous integral kernel satisfying Gaussian bounds then the semigroups S z generated by the H z have kernels with similar properties and z → S z extends to a function on C d \{0} which is analytic with respect to the trace norm. The sequence of semigroups S (m),z obtained by rescaling the coefficients of H z by c(x) → c(mx) converges in trace norm to the semigroup S z generated by the homogenization H z of H z . These convergence properties allow asymptotic analysis of the spectrum of H.

Rendiconti Lincei - Matematica e Applicazioni, 2000

Consider in L 2 (R) the Schrödinger operator family H(g) := −d 2