Continuous spectrum for a class of smooth mixing Schrödinger operators

Continuous spectrum for a class of nonhomogeneous differential operators

We study the boundary value problem −div((|∇u|1 + |∇u|2)∇u) = λ|u|u in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in R with smooth boundary, λ is a positive real number, and the continuous functions p1, p2, and q satisfy 1 < p2(x) < q(x) < p1(x) < N and maxy∈Ω q(y) < Np2(x) N−p2(x) for any x ∈ Ω. The main result of this paper establishes the existence of two positive constants λ0 and λ1 with λ...

Generalized Continuous Frames for Operators

In this note, the notion of generalized continuous K- frame in a Hilbert space is defined. Examples have been given to exhibit the existence of generalized continuous $K$-frames. A necessary and sufficient condition for the existence of a generalized continuous $K$-frame in terms of its frame operator is obtained and a characterization of a generalized continuous $K$-frame for $ mathcal{H} $ wi...

On a class of nonlinear fractional Schrödinger-Poisson systems

In this paper, we are concerned with the following fractional Schrödinger-Poisson system:    (−∆s)u + V (x)u + φu = m(x)|u|q−2|u|+ f(x,u), x ∈ Ω, (−∆t)φ = u2, x ∈ Ω, u = φ = 0, x ∈ ∂Ω, where s,t ∈ (0,1], 2t + 4s > 3, 1 < q < 2 and Ω is a bounded smooth domain of R3, and f(x,u) is linearly bounded in u at infinity. Under some assumptions on m, V and f we obtain the existence of non-trivial so...

Locating the Spectrum for Magnetic Dirac and Schrödinger Operators

Imbedded 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...