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We consider Schrodinger 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 Schrodinger 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 Schrodinger operators is empty; 3) the spectrum of the magnetic Schrodinger operator with each magnetic potential $talpha$, where $t$ is a coupling constant, has an a.c. component for all except finitely many $t$ from any bounded interval.
Continuous movement of discrete spectrum of the Schr{o}dinger operator $H(z)=-frac{d^2} {dx^2}+V_0+z V_1$, with $int_0^infty {x |V_j(x)| dx} < infty$, on the half-line is studied as $z$ moves along a continuous path in the complex plane. The analysis
We show that a generic quasi-periodic Schrodinger operator in $L^2(mathbb{R})$ has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling functions such that f
We consider the Schrodinger operator $H_{eta W} = -Delta + eta W$, self-adjoint in $L^2({mathbb R}^d)$, $d geq 1$. Here $eta$ is a non constant almost periodic function, while $W$ decays slowly and regularly at infinity. We study the asymptotic behav
Let $H_0$ be a purely absolutely continuous selfadjoint operator acting on some separable infinite-dimensional Hilbert space and $V$ be a compact non-selfadjoint perturbation. We relate the regularity properties of $V$ to various spectral properties
We show that the spectral flow of a one-parameter family of Schrodinger operators on a metric graph is equal to the Maslov index of a path of Lagrangian subspaces describing the vertex conditions. In addition, we derive an Hadamard-type formula for t