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Register a new userOn the spectral properties of non-selfadjoint discrete Schrodinger operators
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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 of the perturbed operator $H_0+V$. The structure of the discrete spectrum and the embedded eigenvalues are analysed jointly with the existence of limiting absorption principles in a unified framework. Our results are based on a suitable combination of complex scaling techniques, resonance theory and positive commutators methods. Various results scattered throughout the literature are recovered and extended. For illustrative purposes, the case of the one-dimensional discrete Laplacian is emphasized.
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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 behaviour of the discrete spectrum of $H_{eta W}$ near the origin, and due to the irregular decay of $eta W$, we encounter some non semiclassical phenomena. In particular, $H_{eta W}$ has less eigenvalues than suggested by the semiclassical intuition.
Let $H_0 = -Delta + V_0(x)$ be a Schroedinger operator on $L_2(mathbb{R}^ u),$ $ u=1,2,$ or 3, where $V_0(x)$ is a bounded measurable real-valued function on $mathbb{R}^ u.$ Let $V$ be an operator of multiplication by a bounded integrable real-valued function $V(x)$ and put $H_r = H_0+rV$ for real $r.$ We show that the associated spectral shift function (SSF) $xi$ admits a natural decomposition into the sum of absolutely continuous $xi^{(a)}$ and singular $xi^{(s)}$ SSFs. This is a special case of an analogous result for resolvent comparable pairs of self-adjoint operators, which generalises the known case of a trace class perturbation while also simplifying its proof. We present two proofs -- one short and one long -- which we consider to have value of their own. The long proof along the way reframes some classical results from the perturbation theory of self-adjoint operators, including the existence and completeness of the wave operators and the Birman-Krein formula relating the scattering matrix and the SSF. The two proofs demonstrate the equality of the singular SSF with two a priori different but intrinsically integer-valued functions: the total resonance index and the singular $mu$-invariant.
Let $T$ be the generator of a $C_0$-semigroup $e^{-Tt}$ which is of finite trace for all $t>0$ (a Gibbs semigroup). Let $A$ be another closed operator, $T$-bounded with $T$-bound equal to zero. In general $T+A$ might not be the generator of a Gibbs semigroup. In the first half of this paper we give sufficient conditions on $A$ so that $T+A$ is the generator of a Gibbs semigroup. We determine these conditions in terms of the convergence of the Dyson-Phillips expansion corresponding to the perturbed semigroup in suitable Schatten-von Neumann norms. In the second half of the paper we consider $T=H_vartheta=-e^{-ivartheta}partial_x^2+e^{ivartheta}x^2$, the non-selfadjoint harmonic oscillator, on $L^2(mathbb{R})$ and $A=V$, a locally integrable potential growing like $|x|^{alpha}$ for $0leq alpha<2$ at infinity. We establish that the Dyson-Phillips expansion converges in this case in an $r$ Schatten-von Neumann norm for $r>frac{4}{2-alpha}$ and show that $H_vartheta+V$ is the generator of a Gibbs semigroup $mathrm{e}^{-(H_vartheta+V)tau}$ for $|arg{tau}|leq frac{pi}{2}-|vartheta|$. From this we determine asymptotics for the eigenvalues and for the resolvent norm of $H_vartheta+V$.
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 for each of these sampling functions, the Schrodinger operator with the resulting potential has empty absolutely continuous spectrum.
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 provides information regarding the members of the discrete spectrum of the non-selfadjoint operator that are evolved from the discrete spectrum of the corresponding selfadjoint operator.
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