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Register a new userOn non-selfadjoint operators with finite discrete spectrum
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We consider some compact non-selfadjoint perturbations of fibered one-dimensional discrete Schrodinger operators. We show that the perturbed operator exhibits finite discrete spectrum under suitable- regularity conditions.
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We study the spectrum and dynamics of a one-dimensional discrete Dirac operator in a random potential obtained by damping an i.i.d. environment with an envelope of type $n^{-alpha}$ for $alpha>0$. We recover all the spectral regimes previously obtained for the analogue Anderson model in a random decaying potential, namely: absolutely continuous spectrum in the super-critical region $alpha>frac12$; a transition from pure point to singular continuous spectrum in the critical region $alpha=frac12$; and pure point spectrum in the sub-critical region $alpha<frac12$. From the dynamical point of view, delocalization in the super-critical region follows from the RAGE theorem. In the critical region, we exhibit a simple argument based on lower bounds on eigenfunctions showing that no dynamical localization can occur even in the presence of point spectrum. Finally, we show dynamical localization in the sub-critical region by means of the fractional moments method and provide control on the eigenfunctions.
We construct concrete examples of time operators for both continuous and discrete-time homogeneous quantum walks, and we determine their deficiency indices and spectra. For a discrete-time quantum walk, the time operator can be self-adjoint if the time evolution operator has a non-zero winding number. In this case, its spectrum becomes a discrete set of real numbers.
Using the general formalism of [12], a study of index theory for non-Fredholm operators was initiated in [9]. Natural examples arise from $(1+1)$-dimensional differential operators using the model operator $D_A$ in $L^2(mathbb{R}^2; dt dx)$ of the type $D_A = (d/dt) + A$, where $A = int^{oplus}_{mathbb{R}} dt , A(t)$, and the family of self-adjoint operators $A(t)$ in $L^2(mathbb{R}; dx)$ is explicitly given by $A(t) = - i (d/dx) + theta(t) phi(cdot)$, $t in mathbb{R}$. Here $phi: mathbb{R} to mathbb{R}$ has to be integrable on $mathbb{R}$ and $theta: mathbb{R} to mathbb{R}$ tends to zero as $t to - infty$ and to $1$ as $t to + infty$. In particular, $A(t)$ has asymptotes in the norm resolvent sense $A_- = - i (d/dx)$, $A_+ = - i (d/dx) + phi(cdot)$ as $t to mp infty$. Since $D_A$ violates the relative trace class condition introduced in [9], we now employ a new approach based on an approximation technique. The approximants do fit the framework of [9] and lead to the following results: Introducing $H_1 = {D_A}^* D_A$, $H_2 = D_A {D_A}^*$, we recall that the resolvent regularized Witten index of $D_A$, denoted by $W_r(D_A)$, is defined by $$ W_r(D_A) = lim_{lambda to 0} (- lambda) {rm tr}_{L^2(mathbb{R}^2; dtdx)}((H_1 - lambda I)^{-1} - (H_2 - lambda I)^{-1}). $$ In the concrete example at hand, we prove $$ W_r(D_A) = xi(0_+; H_2, H_1) = xi(0; A_+, A_-) = 1/(2 pi) int_{mathbb{R}} dx , phi(x). $$ Here $xi(, cdot , ; S_2, S_1)$, denotes the spectral shift operator for the pair $(S_2,S_1)$, and we employ the normalization, $xi(lambda; H_2, H_1) = 0$, $lambda < 0$.
We study the Bloch variety of discrete Schrodinger operators associated with a complex periodic potential and a general finite-range interaction, showing that the Bloch variety is irreducible for a wide class of lattice geometries in arbitrary dimension. Examples include the triangular lattice and the extended Harper lattice.
We study the spectrum of the linear operator $L = - partial_{theta} - epsilon partial_{theta} (sin theta partial_{theta})$ subject to the periodic boundary conditions on $theta in [-pi,pi]$. We prove that the operator is closed in $L^2([-pi,pi])$ with the domain in $H^1_{rm per}([-pi,pi])$ for $|epsilon| < 2$, its spectrum consists of an infinite sequence of isolated eigenvalues and the set of corresponding eigenfunctions is complete. By using numerical approximations of eigenvalues and eigenfunctions, we show that all eigenvalues are simple, located on the imaginary axis and the angle between two subsequent eigenfunctions tends to zero for larger eigenvalues. As a result, the complete set of linearly independent eigenfunctions does not form a basis in $H^1_{rm per}([-pi,pi])$.
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