Herbert Koch, Daniel Tataru
Let L be a Schroedinger operator with potential W in L^{(n+1)/2}. We prove that there is no embedded eigenvalue. The main tool is an Lp Carleman type estimate, which builds on delicate dispersive estimates established in a previous paper. The arguments extend to variable coefficient operators with long range potentials and with gradient potentials.
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Translationnally invariant bidimensional magnetic Laplacians are considered. Using an improved version of the harmonic approximation, we establish the absence of point spectrum under various assumptions on the behavior of the magnetic field.
In the presence of the homogeneous electric field ${\bf E}$ and the homogeneous perpendicular magnetic field ${\bf B}$, the classical trajectory of a quantum particle on ${\mathbb R}^2$ moves with drift velocity $α$ which is perpendicular to the electric and magnetic fields. For such Hamiltonians the absence of the embedded eigenvalues of perturbed Hamiltonian has been conjectured. In this paper one proves this conjecture for the perturbations $V(x, y)$ which have sufficiently small support in direction of drift velocity.
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…
We show that, in a model where a non-relativistic particle is coupled to a quantized relativistic scalar Bose field, the embedded mass shell of the particle dissolves in the continuum when the interaction is turned on, provided the coupling constant is sufficiently small. More precisely, under the assumption that the fiber eigenvectors corresponding to the putative mass shell are differentiable as functions of the total momentum of the system, we show that a mass shell could exist only at a strictly positive distance from the unperturbed embedded mass shell near the boundary of the energy-mome…
Absence of embedded eigenvalues of Pauli and Dirac operators
本文研究 Pauli 算子嵌入连续谱的特征值，证明在依赖电磁场无穷远渐近行为的阈值之上不存在此类特征值，并说明衰减条件的精确性，结论亦适用于纯磁 Dirac 算子。
This paper is a continuation of the paper \cite{W} by the third author, which studied quantum walks with special long-range perturbations of the coin operator. In this paper, we consider general long-range perturbations of the coin operator and prove the non-existence of a singular continuous spectrum and embedded eigenvalues. The proof relies on the construction of generalized eigenfunctions (Jost solutions) which was studied in the short-range case in \cite{MSSSSdis}.
On absence of embedded eigenvalues and stability of BGK waves
该研究针对一维静电Vlasov-Poisson系统的BGK波平衡态，证明其线性化算子在本质谱内无嵌入本征值，推导了非定量朗道阻尼，提出基于能量的方法处理共振相互作用，方法可推广至含椭圆、双曲临界点的谱问题。
In this paper we study absence of embedded eigenvalues for Schrödinger operators on non-compact connected Riemannian manifolds. A principal example is given by a manifold with an end (possibly more than one) in which geodesic coordinates are naturally defined. In this case one of our geometric conditions is a positive lower bound of the second fundamental form of angular submanifolds at infinity inside the end. Another condition may be viewed (at least in a special case) as being a bound of the trace of this quantity, while similarly, a third one as being a bound of the derivative of this trac…
We construct the one-dimensional analogous of von-Neumann Wigner potential to the relativistic Klein-Gordon operator, in which is defined taking asymptotic mathematical rules in order to obtain existence conditions of eigenvalues embedded in the continuous spectrum. Using our constructed potential, we provide an explicit and analytical example of the Klein-Gordon operator with positive eigenvalues embedded in the so called relativistic "continuum region". Even so in this not standard example, we present the region of the "continuum" where those eigenvalues cannot occur. Besides, the absence of…
We generalize in this paper a theorem of Titchmarsh for the positivity of Fourier sine integrals. We apply then the theorem to derive simple conditions for the absence of positive energy bound states (bound states embedded in the continuum) for the radial Schrödinger equation with nonlocal potentials which are superposition of a local potential and separable potentials.
A remark on the absence of eigenvalues in continuous spectra for discrete Schrödinger operators on periodic lattices
本文在包含正方、三角、六角格点及其梯状结构的一类周期格点上，对具有指数衰减位势的离散薛定谔算子证明 Rellich-Vekua 型定理，并由此讨论唯一延拓性与连续谱中无嵌入特征值。
We study the Helmholtz equation with electromagnetic-type perturbations, in the exterior of a domain, in dimension $n\geq3$. We prove, by multiplier techniques in the sense of Morawetz, a family of a priori estimates from which the limiting absorption principle follows. Moreover, we give some standard applications to the absence of embedded eigenvalues and zero-resonances, under explicit conditions on the potentials.
On the point spectrum of electromagnetic Dirac operators
本文提出了一种用于电磁狄拉克算子的方法，以确定点谱和嵌入本征值的不存在条件。
We consider Schödinger operators on the half-line, both discrete and continuous, and show that the absence of bound states implies the absence of embedded singular spectrum. More precisely, in the discrete case we prove that if $Δ+ V$ has no spectrum outside of the interval $[-2,2]$, then it has purely absolutely continuous spectrum. In the continuum case we show that if both $-Δ+ V$ and $-Δ- V$ have no spectrum outside $[0,\infty)$, then both operators are purely absolutely continuous. These results extend to operators with finitely many bound states.
We study the nature of the essential spectrum of the Dirichlet Laplacian in tubes about infinite curves embedded in Euclidean spaces. Under suitable assumptions about the decay of curvatures at infinity, we prove the absence of singular continuous spectrum and state properties of possible embedded eigenvalues. The argument is based on Mourre conjugate operator method developed for acoustic multistratified domains by Benbernou and Dermenjian et al. As a technical preliminary, we carry out a spectral analysis for Schrodinger-type operators in straight Dirichlet tubes. We also apply the result to…
We derive bounds on the location of non-embedded eigenvalues of Dirac operators on the half-line with non-Hermitian $L^1$-potentials. The results are sharp in the non-relativistic or weak-coupling limit. In the massless case, the absence of discrete spectrum is proved under a smallness assumption.
The spectrum of discrete Schrödinger operator $L+V$ on the $d$-dimensional lattice is considered, where $L$ denotes the discrete Laplacian and $V$ a delta function with mass at a single point. Eigenvalues of $L+V$ are specified and the absence of singular continuous spectrum is proven. In particular it is shown that an embedded eigenvalue does appear for $d\geq5$ but does not for $1\leq d\leq 4$.
We obtain a representation formula for the derivative of the spectral shift function $ξ(λ; B, ε)$ related to the operators $H_0(B,ε) = (D_x - By)^2 + D_y^2 + εx$ and $H(B, ε) = H_0(B, ε) + V(x,y), \: B > 0, ε> 0$. We prove that the operator $H(B, ε)$ has at most a finite number of embedded eigenvalues on $\R$ which is a step to the proof of the conjecture of absence of embedded eigenvalues of $H$ in $\R.$ Applying the formula for $ξ'(λ, B, ε)$, we obtain a semiclassical asymptotics of the spectral shift function related to the operators $H_0(h) = (hD_x - By)^2 + h^2D_y^2 + εx$ and $H(h) = H_0(…
Sharp decay thresholds for eigenvalues of discrete Schrödinger operators on $\mathbb{Z}^2$
本文确定了 $\mathbb{Z}^2$ 上离散薛定谔算子在不同谱区域（内部、临界点、边缘）特征值存在的尖锐衰减阈值，并给出构造与不存在性证明。
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