On absence of embedded eigenvalues for schrÖdinger operators with perturbed periodic potentials

The problem of absence of eigenvalues imbedded into the continuous spectrum is considered for a Schrödinger operator with a periodic potential perturbed by a sufficiently fast decaying "impurity" potential. Results of this type have previously been known for the one-dimensional case only. Absence of embedded eigenvalues is shown in dimensions two and three if the corresponding Fermi surface is irreducible modulo natural symmetries. It is conjectured that all periodic potentials satisfy this condition. Separable periodic potentials satisfy it, and hence in dimensions two and three Schrödinger operator with a separable periodic potential perturbed by a sufficiently fast decaying "impurity" potential has no embedded eigenvalues     

Schrödinger operators with slowly decaying potentials

Several recent papers have obtained bounds on the distribution of eigenvalues of non-self-adjoint Schrödinger operators and resonances of self-adjoint operators. In this paper we describe two new methods of obtaining such bounds when the potential decays more slowly than previously permitted.     

An improved criterion for absence of eigenvalues of one-dimensional Schrödinger operators

We study one-dimensional Schrödinger operators with complex measures as potentials and present an improved criterion for absence of eigenvalues which involves a weak local periodicity condition. As an application we consider quasiperiodic measures as potentials. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the parametrix for a class of Schrödinger operators with potentials of quadratic growth

Summary Let E(t, h) be the Fourier Integral Operator representing the parametrix of the propagator for the Schrödinger equation Q=ih_t, Q=– 1/2h²_x + V(x), where V is a potential of quadratic growth inR ⁿ. It is proved that E(t, h) coincides exactly with the Schwarte kernel of the propagator if and only if V is a positive quadratic form, i.e. if the corresponding physical system is a n-dimensional harmonic oscillator.     

Spectral triangles of Schrödinger operators with complex potentials

Consider the Schrödinger operator with a complex-valued potential v of period Let and be the eigenvalues of L that are close to respectively, with periodic (for n even), antiperiodic (for n odd), and Dirichelet boundary conditions on [0,1], and let be the diameter of the spectral triangle with vertices We prove the following statement: If then v(x) is a Gevrey function, and moreover      

Schrödinger operators with singular potentials from the space of multiplicators

We study Schrödinger operatorsT+Q, whereT=?Δ is the Laplace operator andQ is the multiplication operator by a generalized function (distribution). We also consider generalizations for the case of the polyharmonic operatorT = (-δ) ⁿ      

On essential self-adjointness for Schrödinger operators with wildly oscillating potentials

A sufficient condition is given for the operator T₀: C^∞₀(R^m) → L²(R^m) given by

$\text{T}{}_0\text{∑}\text{K?1}\text{M}\text{(}\text{i?}\text{?x}{}_1\text{+b}{}_1\text{)a}{}_{\mathrm{1K}}\text{(}\text{i?}\text{?N}{}_{\mathrm{k}}\text{+b}{}_{\mathrm{k}}\text{)+q}$

to be essentially self-adjoint. This condition is sufficiently general to admit certain potentials q having unbounded oscillations in a neighborhood of ∞.     

Multidimensional Hardy inequalities and the absence of positive eigenvalues for the Schrödinger operator with a complex potential

The operator –(d/dx)²+ where is a complex function, |J(x)(1+|x|¹⁺|>, has no positive eigenvalues. This follows from the existence of a triangular Green function for the operator –(d/dx)^z–,gl>0, and from Hardy type inequalities. We give the multidimensional analogues of these inequalities and we prove the absence of positive eigenvalues for the operator –+.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 138, pp. 33–34, 1984.     

Schrödinger operators with highly singular oscillating potentials

We investigate the Feynman-Kac semigroupP _t ^V and its densityp ^V(t,.,.),t>0, associated with the Schrödinger operator ?1/2Δ+V on ?^d\{0}.V will be a highly singular, oscillating potential like $V\left( x \right) = k \cdot \left\| x \right\|^{ - 1} \cdot \sin \left( {\left\| x \right\|^{ - m} } \right)$ with arbitraryk, l, m>0. We derive conditions (onk,l,m) which are sufficientand necessary for the existence of constants α, β, γ, ∈ ? such that for allt, x, y p ^V(t, x, y)≤γ·p(βt, x, y)·e^at. On the other hand, also conditions are derived which imply thatp ^V (t, x, y)≡∞ for allt, x, y. The aim is to see to which extent quick oscillations can lead to annihilations of the singularities ofV. For this purpose, we analyse the above example in great detail. Note that forl≥2 the potential is so singular that none of the usual perturbation techniques applies.     

Monodromy-free Schrödinger operators with quadratically increasing potentials

We consider one-dimensional monodromy-free Schrödinger operators with quadratically increasing rational potentials. It is shown that all these operators can be obtained from the operator -?² + x² by finitely many rational Darboux transformations. An explicit expression is found for the corresponding potentials in terms of Hermite polynomials.     

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 Schrödinger operators ? Δ +V. Conditions onV are formulated, which ensures the essential self-adjointness of ? Δ +V. In particular, ifV ∈ Q_α,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.     

On the spectral properties of discrete Schrödinger operators

We use the method of the conjugate operator to prove a limiting absorption principle and the absence of the singular continuous spectrum for discrete Schrödinger operators. We also obtain local decay estimates. Our results apply to a large class of perturbating potentials V decaying arbitrarily slowly to zero at infinity.     

Some estimates for the Schrödinger type operators with nonnegative potentials

In this paper we consider the Schrödinger operator ?Δ + V on \({\mathbb R^d}\), where the nonnegative potential V belongs to the reverse Hölder class \({B_{q_{_1}}}\) for some \({q_{_1}\geq \frac{d}{2}}\) with d ≥ 3. Let \({H^1_L(\mathbb R^d)}\) denote the Hardy space related to the Schrödinger operator L = ?Δ + V and \({BMO_L(\mathbb R^d)}\) be the dual space of \({H^1_L(\mathbb R^d)}\). We show that the Schrödinger type operator \({\nabla(-\Delta +V)^{-\beta}}\) is bounded from \({H^1_L(\mathbb R^d)}\) into \({L^p(\mathbb R^d)}\) for \({p=\frac{d}{d-(2\beta-1)}}\) with \({ \frac{1}{2}<\beta<\frac{3}{2} }\) and that it is also bounded from \({L^p(\mathbb R^d)}\) into \({BMO_L(\mathbb R^d)}\) for \({p=\frac{d}{2\beta-1}}\) with \({ \frac{1}{2}<\beta< 2}\).     

Spectral representation for Schrödinger operators with long-range potentials

A spectral representation for the self-adjoint Schrödinger operator H = ?Δ + V(x), x? R³, is obtained, where V(x) is a long-range potential: $\text{V(x) = O(¦ x ¦}{}^{\mathrm{<mtext>?</mtext><mtext>(1</mtext><mtext>2)</mtext><mtext>?\delta </mtext>}}\text{)}$, grad $\text{V(x) = O(¦ x ¦}{}^{\mathrm{<mtext>?</mtext><mtext>(3</mtext><mtext>2)</mtext><mtext>?\delta </mtext>}}\text{)}$, $\text{ΛV(x) = O(¦ x}\text{s}\text{?}{}^{\mathrm{?\delta }}\text{) (δ > 0), Λ}$ being the Laplace-Beltrami operator on the unit sphere Ω. Namely, we shall construct a unitary operator $\text{F}$ from PL₂(R³) onto $\text{L}{}_2\text{((0, ∞); L}{}_2\text{(Ω)), P}$ being the orthogonal projection onto the absolutely continuous subspace for H, such that for any Borel function α(λ),

$\text{(α(H)(Pf,g)=}\text{∫}\text{0}\text{∞}\text{(α(λ)(Ff)(λ),(Fg)(λ))L}{}_2\text{(ω) dλ}$

.     

A characterization of the eigenvalues of Schrödinger operators with Dirichlet and Neumann boundary conditions

The eigenvalues of the self-adjoint Schrödinger operators on a bounded domain with Dirichlet and Neumann boundary conditions are characterized by the singularities of an associated Dirichlet-to-Neumann map and its inverse, respectively. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Lp boundedness of wave operators for two-dimensional Schrödinger operators with threshold obstructions

Let $H=?\mathrm{\Delta }+V$ be a Schrödinger operator on $L^2({\mathbb{R}}^2)$ with real-valued potential V, and let $H_0=?\mathrm{\Delta }$. If V has sufficient pointwise decay, the wave operators $W_{\pm }=s?{\lim }_{t\to \pm \infty }?e^{itH}{e}^{?it{H}_0}$ are known to be bounded on $L^p({\mathbb{R}}^2)$ for all $1<p<\infty$ if zero is not an eigenvalue or resonance. We show that if there is an s-wave resonance or an eigenvalue only at zero, then the wave operators are bounded on $L^p({\mathbb{R}}^2)$ for $1<p<\infty$. This result stands in contrast to results in higher dimensions, where the presence of zero energy obstructions is known to shrink the range of valid exponents p.     

On the absolutely continuous spectrum of magnetic Schrödinger operators

We prove that [0, ∞) is an essential support for the absolutely continuous part of the spectral measure associated with the magnetic Schr?dinger operator (i∇ + A)² in , given certain conditions on the decay of A. Bibliography: 8 titles. Illustrations: 1 figure. Translated from Problemy Matematicheskogo Analiza, No. 38, 2008, pp. 121–143.     

On absolute continuity of the spectrum of periodic Schrödinger operators

In this paper we find a new condition on a real periodic potential for which the self-adjoint Schrödinger operator may be defined by a quadratic form and the spectrum of the operator is purely absolutely continuous. This is based on resolvent estimates and spectral projection estimates in weighted \(L^2\) spaces on the torus, and an oscillatory integral theorem.