Absence Of Eigenvalues Research Articles

Spectral Theory of Schrödinger Operators over Circle Diffeomorphisms

Abstract We initiate the study of Schrödinger operators with ergodic potentials defined over circle map dynamics, in particular over circle diffeomorphisms. For analytic circle diffeomorphisms and a set of rotation numbers satisfying Yoccoz’s ${{\mathcal{H}}}$ arithmetic condition, we discuss an extension of Avila’s global theory. We also give an abstract version and a short proof of a sharp Gordon-type theorem on the absence of eigenvalues for general potentials with repetitions. Coupled with the dynamical analysis, we obtain that, for every $C^{1+BV}$ circle diffeomorphism, with a super Liouville rotation number and an invariant measure $\mu $, and for $\mu $-almost all $x\in{{\mathbb{T}}}^1$, the corresponding Schrödinger operator has purely continuous spectrum for every Hölder continuous potential $V$.

Spectral stability of small amplitude solitary waves of the Dirac equation with the Soler-type nonlinearity

We study the point spectrum of the linearization at a solitary wave solution ϕω(x)e−iωt to the nonlinear Dirac equation in Rn, for all n≥1, with the nonlinear term given by f(ψ⁎βψ)βψ (known as the Soler model). We focus on the spectral stability, that is, the absence of eigenvalues with positive real part, in the non-relativistic limit ω→m−0, in the case when f∈C1(R∖{0}), f(τ)=|τ|κ+O(|τ|K) for τ→0, with 0<κ<K. For n≥1, we prove the spectral stability of small amplitude solitary waves (ω⪅m) for the charge-subcritical cases κ⪅2/n (in particular, 1<κ≤2 when n=1) and for the “charge-critical case” κ=2/n (with K>4/n).An important part of the stability analysis is the proof of the absence of bifurcation of nonzero-real-part eigenvalues from the embedded threshold points at ±2mi. Our approach is based on constructing a new family of exact bi-frequency solitary wave solutions in the Soler model and on the analysis of the behavior of “nonlinear eigenvalues” (characteristic roots of holomorphic operator-valued functions).

Sub-exponential decay of eigenfunctions for some discrete Schrödinger operators

Following the method of Froese and Herbst, we show for a class of potentials V that an embedded eigenfunction \psi with eigenvalue E of the multi-dimensional discrete Schrödinger operator H = \Delta + V on \mathbb Z^d decays sub-exponentially whenever the Mourre estimate holds at E . In the one-dimensional case we further show that this eigenfunction decays exponentially with a rate at least of cosh ^{-1}((E-2)/(\theta_E-2)) , where \theta_E is the nearest threshold of H located between E and 2 . A consequence of the latter result is the absence of eigenvalues between 2 and the nearest thresholds above and below this value. The method of Combes–Thomas is also reviewed for the discrete Schrödinger operators.

О спектре двумерного оператора Шрёдингера с однородным магнитным полем и периодическим электрическим потенциалом

We consider the two-dimensional Schrodinger operator $\widehat H_B+V$ with a uniform magnetic field $B$ and a periodic electric potential $V$. The absence of eigenvalues (of infinite multiplicity) in the spectrum of the operator $\widehat H_B+V$ is proved if the electric potential $V$ is a nonconstant trigonometric polynomial and the condition $(2\pi )^{-1}\, Bv(K)=Q^{-1}$ for the magnetic flux is fulfilled where $Q\in \mathbb{N}$ and the $v(K)$ is the area of the elementary cell $K$ of the period lattice $\Lambda \subset \mathbb{R}^2$ of the potential $V$. In this case the singular component of the spectrum is absent so the spectrum is absolutely continuous. In this paper, we use the magnetic Bloch theory. Instead of the lattice $\Lambda $ we choose the lattice $\Lambda _{\, Q}=\{ N_1QE^1+N_2E^2:N_j\in \mathbb{Z} , j=1,2\} $ where $E^1$ and $E^2$ are basis vectors of the lattice $\Lambda $. The operator $\widehat H_B+V$ is unitarily equivalent to the direct integral of the operators $\widehat H_B(k)+V$ with $k\in 2\pi K_{\, Q}^*$ acting on the space of magnetic Bloch functions where $K_{\, Q}^*$ is an elementary cell of the reciprocal lattice $\Lambda _{\, Q}^*\subset \mathbb{R}^2$. The proof of the absence of eigenvalues in the spectrum of the operator $\widehat H_B+V$ is based on the following assertion: if $\lambda $ is an eigenvalue of the operator $\widehat H_B+V$, then the $\lambda $ is an eigenvalue of the operators $\widehat H_B(k+i\varkappa )+V$ for all $k,\, \varkappa \in \mathbb{R}^2$ and, moreover, (under the assumed conditions on the $V$ and the $B$) there is a vector $k_0\in \mathbb{C}^2\, \backslash \, \{0\}$ such that the eigenfunctions of the operators $\widehat H_B(k+\zeta k_0)+V$ with $\zeta \in \mathbb{C}$ are trigonometric polynomials $\sum \zeta ^j\Phi _j$ in $\zeta $.

On the discrete spectrum of the Dirac operator on bent chain quantum graph

We study Dirac operators on an infinite quantum graph of a bent chain form which consists of identical rings connected at the touching points by δ-couplings with a parameter α ∈ ℝ. We are interested in the discrete spectrum of the corresponding Hamiltonian. It can be non-empty due to a local (geometrical perturbation of the corresponding infinite chain of rings. The quantum graph of analogous geometry with the Schrodinger operator on the edges was considered by Duclos, Exner and Turek in 2008. They showed that the absence of δ-couplings at vertices (i.e. the Kirchhoff condition at the vertices) lead to the absence of eigenvalues. We consider the relativistic particle (the Dirac operator instead of the Schrodinger one) but the result is analogous. Quantum graphs of such type are suitable for description of grapheme-based nanostructures. It is established that the negativity of α is the necessary and sufficient condition for the existence of eigenvalues of the Dirac operator (i.e. the discrete spectrum of the Hamiltonian in this case is not empty). The continuous spectrum of the Hamiltonian for bent chain graph coincides with that for the corresponding straight infinite chain. Conditions for appearance of more than one eigenvalue are obtained. It is related to the bending angle. The investigation is based on the transfer-matrix approach. It allows one to reduce the problem to an algebraic task. δ-couplings was introduced by the operator extensions theory method.

Criteria for the absence of eigenvalues of Jacobi matrices with matrix entries

Criteria for the absence of eigenvalues of Jacobi matrices with matrix entries

New Expressions for the Wave Operators of Schrödinger Operators in $${\mathbb{R}^3}$$ R 3

We prove new and explicit formulas for the wave operators of Schroedinger operators in R^3. These formulas put into light the very special role played by the generator of dilations and validate the topological approach of Levinson's theorem introduced in a previous publication. Our results hold for general (not spherically symmetric) potentials decaying fast enough at infinity, without any assumption on the absence of eigenvalue or resonance at 0-energy.

Absence of eigenvalues for the periodic Schrödinger operator with singular potential in a rectangular cylinder

We consider the periodic Schrodinger operator on a d-dimensional cylinder with rectangular section. The electric potential may contain a singular component of the form σ(x, y)δΣ(x,y), where Σ is a periodic system of hypersurfaces. We establish that there are no eigenvalues in the spectrum of this operator, provided that Σ is sufficiently smooth and σ ∈ Lp,loc(Σ), p > d − 1.

Absence of eigenvalue at the bottom of the continuous spectrum on asymptotically hyperbolic manifolds

For a class of asymptotically hyperbolic manifolds, we show that the bottom of the continuous spectrum of the Laplace–Beltrami operator is not an eigenvalue. Our approach only uses properties of the operator near infinity and, in particular, does not require any global assumptions on the topology or the curvature, unlike previous papers on the same topic.

Circular trichotomy of the spectrum of regular matrix pencils

We introduce a method for approximating the right and left deflating subspaces of a regular matrix pencil corresponding to the eigenvalues inside, on and outside the unit circle. The method extends the iteration used in the context of spectral dichotomy, where the assumption on the absence of eigenvalues on the unit circle is removed. It constructs two matrix sequences whose null spaces and the null space of their sum lead to approximations of the deflating subspaces corresponding to the eigenvalues of modulus less than or equal to 1, equal to 1 and larger than or equal to 1. An orthogonalization process is then used to extract the desired delating subspaces. The resulting algorithm is an inverse free, easy to implement, and sufficiently fast. The derived convergence estimates reveal the key parameters, which determine the rate of convergence. The method is tested on several numerical examples.

Zero energy asymptotics of the resolvent for a class of slowly decaying potentials

We prove a limiting absorption principle at zero energy for two-body Schrödinger operators with long-range potentials having a positive virial at infinity. More precisely, we establish a complete asymptotic expansion of the resolvent in weighted spaces when the spectral parameter tends to zero in cones which are adjacent to the positive real axis. The principal tools are absence of eigenvalue at zero, singular Mourre theory and microlocal estimates.

The Asymptotics of the Number of Eigenvalues of a Three-Particle Lattice Schrödinger Operator

The Hamiltonian of a system of three quantum-mechanical particles moving on the three-dimensional lattice \(\mathbb{Z}^3 \) and interacting via zero-range attractive potentials is considered. The location of the essential and discrete spectra of the three-particle discrete Schrodinger operator H(K), where K is the three-particle quasimomentum, is studied. The absence of eigenvalues below the bottom of the essential spectrum of H(K) for all sufficiently small values of the zero-range attractive potentials is established.

The Spectrum of the Two-Dimensional Periodic Schrödinger Operator

The absence of eigenvalues (of infinite multiplicity) for the two-dimensional periodic Schrodinger operator with a variable metric is proved. The method of proof does not use the change of variables reducing the metric to a scalar form.

Schrödinger operators on manifolds, essential self-adjointness, and absence of eigenvalues

Suppose thatM n is a complete, noncompact, Riemannian manifold. If Δ denotes the Laplace operator ofM, one has associated Schrodinger operators − Δ +V. Conditions onV are formulated, which ensures the essential self-adjointness of − Δ +V. In particular, ifV ∈ Qα,loc (M n), the local Stummel class, andV ≥ − c outside of a compact set, then − Δ +V is essentially self-adjoint on C 0 ∞ (M n). In addition, essential self-adjointness is proved for potentials which are strongly singular at a point. The absence of eigenvalues of −Δ +V is also studied. This relies upon Rellich-type identities. The results on strongly singular potentials make use of a generalization of the classical uncertainty principle, inR n, to Riemannian manifolds with a pole.

Resonances and standing waves

AbstractRecent investigations on aperiodic waves, which are generated by time‐harmonic forces in waveguides, indicate a close relationship between resonances and certain time‐harmonic solutions of homogeneous boundary value problems for the wave equation (“standing waves”). This paper is motivated by the observation that, in all known cases, standing waves are connected with resonances if and only if they are subject to suitable asymptotic restrictions as |x| → ∞. These asymptotic properties are used to introduce a class of “admissible” standing waves. We prove that admissible standing waves do not exist in a certain class of local perturbations Ω of the n‐dimensional domain Ω0 bounded by the hyperplanes xn = 0 and xn = π. This extends a result of M. Faulhaber on the absence of eigenvalues. As we shall show in a subsequent paper, absence of admissible standing waves implies absence of resonances.

On the spectra and eigenfunctions of the Schrödinger and Maxwell operators

The new sufficient conditions of the exponential decay of eigenfunctions and the absence of eigenvalues are obtained for the second-order elliptic operator in L 2( r n ), generalizing the Schrödinger operator, and for the Maxwell operator with the matrix medium characteristics (the crystal optics operator) in L 2( r n ).