Threshold Eigenvalue Research Articles

Gevrey estimates of the resolvent and sub-exponential time-decay for the heat and Schrödinger semigroups. II

In this paper, we improve and generalize the results of [1] on large-time expansion of the heat and Schrödinger semigroups with sub-exponential time-decay estimates on the remainder for a class of non-selfadjoint Schrödinger operators. For the Schrödinger semigroup, the large-time expansion is established for distortion analytic potentials satisfying a damped virial condition outside some compact and the contributions from threshold eigenvalue and outgoing positive resonances are explicitly calculated.

Convergent Expansions of Eigenvalues of the Generalized Friedrichs Model with a Rank-One Perturbation

We study analytic behavior of eigenvalues of the generalized Friedrichs model \(H_\mu (p)\), with a rank-one perturbation, depending on parameters \(\mu >0\) and \(p\in {\mathbb {T}}^2\). Under certain conditions, the existence of a unique eigenvalue lying below the essential spectrum has been shown in Lakaev (Abstract Appl Anal 2012, 2012). Here, we obtain an absolutely convergent expansion for that eigenvalue at \(\mu (p)\), the coupling constant threshold. The expansion is dependent to a large extent on whether the lower bound of the essential spectrum is a threshold resonance, a threshold eigenvalue or neither of them.

The massless Dirac equation in two dimensions: zero-energy obstructions and dispersive estimates

We investigate $L^1\\to L^\\infty$ dispersive estimates for the massless two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies the natural $t^{-\\frac{1}{2}}$ decay rate, which may be improved to $t^{-\\frac{1}{2} - \\gamma}$ for any $0\\leq \\gamma<\\frac{3}{2}$ at the cost of spatial weights. We classify the structure of threshold obstructions as being composed of a two dimensional space of p-wave resonances and a finite dimensional space of eigenfunctions at zero energy. We show that, in the presence of a threshold resonance, the Dirac evolution satisfies the natural decay rate except for a finite-rank piece. While in the case of a threshold eigenvalue only, the natural decay rate is preserved. In both cases we show that the decay rate may be improved at the cost of spatial weights.

Principal Component Analysis for Yield in Blackgram(Vignamungo L. Hepper) under Organic and Inorganic Fertilizer Managements

The present investigation on thirty black gram diverse genotypes for 12 yield and yield attributing traits under organic and inorganic fertilizer managements was carried out to study genetic variation among traits and genotypes in the respective environments that would equip the selection criteria using principal Component Analysis. First four vectors with threshold Eigen value greater than one (>1) contributed to 77.56% and 70.74% variation under organic and inorganic fertilizer managements respectively. Characters viz., number of clusters per plant (0.395), number of seeds per pod (0.354), days to maturity (0.336), number of pods per plant (0.300), harvest index (0.244), plant height (0.073) and seed yield per plant (0.015), whereas under inorganic fertilizer management number of primary branches per plant (0.43), followed by number of pods per plant (0.43), seed yield per plant (0.31), number of pods per cluster (0.29), number of clusters per plant (0.29) explained the maximum variance in first principal component (PC1) under organic conditions. Based on comparison of trait contribution to total variability under PC1 under both the managements it can be concluded that the traits viz., number of clusters per plant, number of pods per plant and seed yield per plant were the potential traits that accounted for maximum share towards variability. These traits may be taken into consideration as selection criteria in breeding programmes aimed at developing high yielding varieties.

Spectral and threshold analysis of a small rank perturbation of the discrete Laplacian

We consider a family of the discrete Schrödinger operators Hλμ, depending on two parameters, in the d-dimensional lattice with a potential constructed via the delta function and the shift operator. The existence of eigenvalues, threshold eigenvalues and threshold resonances and their dependence on the parameters of the operator and dimension of the lattice are explicitly derived.

Threshold resonances and virtual levels in the spectrum of cylindrical and periodic waveguides

We describe and classify the thresholds of the continuous spectrum and the resulting resonances for general formally self-adjoint elliptic systems of second-order differential equations with Dirichlet or Neumann boundary conditions in domains with cylindrical and periodic outlets to infinity (in waveguides). These resonances arise because there are “almost standing” waves, that is, non-trivial solutions of the homogeneous problem which do not transmit energy. We consider quantum, acoustic, and elastic waveguides as examples. Our main focus is on degenerate thresholds which are characterized by the presence of standing waves with polynomial growth at infinity and produce effects lacking for ordinary thresholds. In particular, we describe the effect of lifting an eigenvalue from the degenerate zero threshold of the spectrum. This effect occurs for elastic waveguides of a vector nature and is absent from the scalar problems for cylindrical acoustic and quantum waveguides. Using the technique of self-adjoint extensions of differential operators in weighted spaces, we interpret the almost standing waves as eigenvectors of certain operators and the threshold as the corresponding eigenvalue. Here the threshold eigenvalues and the corresponding vector-valued functions not decaying at infinity can be obtained by approaching the threshold (the virtual level) either from below or from above. Hence their properties differ essentially from the customary ones. We state some open problems.

Threshold Analysis of the Three Dimensional Lattice Schrödinger Operator with Non-Local Potential

We consider a family of the discrete Schrodinger operators $$H_{\lambda\mu}$$ , depending on parameters, in the $$3$$ -dimensional lattice, $$\mathbb{Z}^{3}$$ with a non-local potential constructed via the Dirac delta function and the shift operator. The existence of eigenvalues outside the essential spectrum and their dependence on the parameters of the operator are explicitly derived. The threshold eigenvalue is proven to be absorbed into the essential spectrum and it turns into an embedded eigenvalue at the left intercept of a particular parabola, and the threshold resonance at the other points of the parabola.

Threshold of discrete Schrödinger operators with delta potentials on n-dimensional lattice

ABSTRACT Eigenvalue behaviours of Schrödinger operator defined on n-dimensional lattice with n + 1 delta potentials are studied. It can be shown that lower threshold eigenvalue and lower threshold resonance appear for , and lower super-threshold resonance appears for n = 1.

Gevrey estimates of the resolvent and sub-exponential time-decay for the heat and Schrödinger semigroups

In this article, we prove Gevrey estimates of the resolvent near threshold zero for a class of second order elliptic operators satisfying a weighted coercive condition. As application, we obtain large time expansions with sub-exponential time-decay estimates on the remainder for the heat and Schrödinger semigroups generated by non-selfadjoint Schrödinger operators. Our results cover the cases of threshold eigenvalue and positive resonances.

THRESHOLD EIGENVALUES AND RESONANCES OF A FRIEDRICHS MODEL WITH RANK TWO PERTURBATION

In this paper in the Hilbert space a bounded self-adjoint Friedrichs model with rank two perturbation is considered. Number and location of the eigenvalues of are studied. An existence conditions of these eigenvalues are found. Under some conditions we prove that the lower (upper) bound of the essential spectrum of is either threshold eigenvalue or virtual level of .

Erratum for “The Dirac Operator with Mass $m_0 \geq 0$: Non-Existence of Zero Modes and of Threshold Eigenvalues”

We give a proof of Theorem 2.1 in [ H. Kalf et al., Doc. Math. 20, 37–64 (2015; Zbl 1333.35227)], namely of the following assertion. Let Q \colon \mathbb{R}^n \rightarrow \mathbb{C}^{N\times N} be measurable with \sup_{x \in \mathbb{R}^n} |x||Q(x)| \leq C \ \ \text{ for some}\ \ 0<C<\frac{n-1}{2}. Then any solution u \in H_{\text{loc}}^1(\mathbb{R}^n)^N \cap L^2(\mathbb{R}^n, r^{-1}dx)^N of (\alpha\cdot p +Q)u=0 is identically zero.

Dispersive estimates for higher dimensional Schrödinger operators with threshold eigenvalues. II: The even dimensional case

We investigate L^1(\mathbb R^n)\to L^\infty(\mathbb R^n) dispersive estimates for the Schrödinger operator H=-\Delta+V when there is an eigenvalue at zero energy in even dimensions n\geq 6 . In particular, we show that if there is an eigenvalue at zero energy then there is a time dependent, rank one operator F_t satisfying \|F_t\|_{L^1\to L^\infty} \lesssim |t|^{2-\frac {n}{2}} for |t|>1 such that \|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim |t|^{1-\frac {n}{2}},\,\,\,\,\,\text{ for } |t|>1. With stronger decay conditions on the potential it is possible to generate an operator-valued expansion for the evolution, taking the form e^{itH} P_{ac}(H)=|t|^{2-\frac {n}{2}}A_{-2}+ |t|^{1-\frac {n}{2}} A_{-1}+|t|^{-\frac {n}{2}}A_0, with A_{-2} and A_{-1} mapping L^1(\mathbb R^n) to L^\infty(\mathbb R^n) while A_0 maps weighted L^1 spaces to weighted L^\infty spaces. The leading-order terms A_{-2} and A_{-1} are both finite rank, and vanish when certain orthogonality conditions between the potential V and the zero energy eigenfunctions are satisfied. We show that under the same orthogonality conditions, the remaining |t|^{-\frac {n}{2}}A_0 term also exists as a map from L^1(\mathbb R^n) to L^\infty(\mathbb R^n) , hence e^{itH}P_{ac}(H) satisfies the same dispersive bounds as the free evolution despite the eigenvalue at zero.

Dispersive estimates for higher dimensional Schrödinger operators with threshold eigenvalues I: The odd dimensional case

We investigate $L^1(\mathbb R^n)\to L^\infty(\mathbb R^n)$ dispersive estimates for the Schr\"odinger operator $H=-\Delta+V$ when there is an eigenvalue at zero energy and $n\geq 5$ is odd. In particular, we show that if there is an eigenvalue at zero energy then there is a time dependent, rank one operator $F_t$ satisfying $\|F_t\|_{L^1\to L^\infty} \lesssim |t|^{2-\frac{n}{2}}$ for $|t|>1$ such that $$\|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim |t|^{1-\frac{n}{2}},\qquad\textrm{ for } |t|>1.$$ With stronger decay conditions on the potential it is possible to generate an operator-valued expansion for the evolution, taking the form $$ e^{itH} P_{ac}(H)=|t|^{2-\frac{n}{2}}A_{-2}+ |t|^{1-\frac{n}{2}} A_{-1}+|t|^{-\frac{n}{2}}A_0, $$ with $A_{-2}$ and $A_{-1}$ finite rank operators mapping $L^1(\mathbb R^n)$ to $L^\infty(\mathbb R^n)$ while $A_0$ maps weighted $L^1$ spaces to weighted $L^\infty$ spaces. The leading order terms $A_{-2}$ and $A_{-1}$ vanish when certain orthogonality conditions between the potential $V$ and the zero energy eigenfunctions are satisfied. We show that under the same orthogonality conditions, the remaining $|t|^{-\frac{n}{2}}A_0$ term also exists as a map from $L^1(\mathbb R^n)$ to $L^\infty(\mathbb R^n)$, hence $e^{itH}P_{ac}(H)$ satisfies the same dispersive bounds as the free evolution despite the eigenvalue at zero.

The Dirac operator with mass $m_0 \ge 0$: non-existence of zero modes and of threshold eigenvalues

A simple global condition on the potential is given which excludes zero modes of the massless Dirac operator. As far as local conditions at infinity are concerned, it is shown that at energy zero the Dirac equation without mass term has no non-trivial L^2 -solutions at infinity for potentials which are either very slowly varying or decaying at most like r^{-s} with s \in (0,1) . When a mass term is present, it is proved that at the thresholds there are again no such solutions when the potential decays at most like r^{-s} with s \in (0,2) . In both situations the decay rate is optimal.

Urban land cover classification using hyperspectral data

Abstract. Urban land cover classification using remote sensing data is quite challenging due to spectrally and spatially complex urban features. The present study describes the potential use of hyperspectral data for urban land cover classification and its comparison with multispectral data. EO-1 Hyperion data of October 05, 2012 covering parts of Bengaluru city was analyzed for land cover classification. The hyperspectral data was initially corrected for atmospheric effects using MODTRAN based FLAASH module and Minimum Noise Fraction (MNF) transformation was applied to reduce data dimensionality. The threshold Eigen value of 1.76 in VNIR region and 1.68 in the SWIR region was used for selection of 145 stable bands. Advanced per pixel classifiers viz., Spectral Angle Mapper (SAM) and Support Vector Machine (SVM) were used for general urban land cover classification. Accuracy assessment of the classified data revealed that SVM was quite superior (82.4 per cent) for urban land cover classification as compared to SAM (67.1 per cent). Selecting training samples using end members significantly improved the classification accuracy by 20.1 per cent in SVM. The land cover classification using multispectral LISS-III data using SVM showed lower accuracy mainly due to limitation of spectral resolution. The study indicated the requirement of additional narrow bands for achieving reasonable classification accuracy of urban land cover. Future research is focused on generating hyperspectral library for different urban features.

On Threshold Eigenvalues and Resonances for the Linearized NLS Equation

We prove the instability of threshold resonances and eigenvalues of the linearized NLS operator. We compute the asymptotic approximations of the eigenvalues appearing from the endpoint singularities in terms of the perturbations applied to the original NLS equation. Our method involves such techniques as the Birman-Schwinger principle and the Feshbach map.

Nonexponential decay laws in perturbation theory of near threshold eigenvalues

We consider a two channel model of the form Hε=[Hop00E0]+ε[0W12W210] on H=Hop⊕C. The operator Hop is assumed to have the properties of a Schrödinger operator in odd dimensions, with a threshold at zero. As the energy parameter E0 is tuned past the threshold, we consider the survival probability |⟨Ψ0,e−itHεΨ0⟩|2, where Ψ0 is the eigenfunction corresponding to eigenvalue E0 for ε=0. We find nonexponential decay laws for ε small and E0 close to zero provided that the resolvent of Hop is not at least Lipschitz continuous at the threshold zero.

On the number of eigenvalues of a model operator associated to a system of three-particles on lattices

A model operator H associated to a system of three particles on the threedimensional lattice ℤ3 that interact via nonlocal pair potentials is studied. The following results are established. (i) The operator H has infinitely many eigenvalues lying below the bottom of the essential spectrum and accumulating at this point if both the Friedrichs model operators $$h_{\mu _\alpha } $$ (0), α = 1, 2, have threshold resonances. (ii) The operator H has finitely many eigenvalues lying outside the essential spectrum if at least one of the operators $$h_{\mu _\alpha } $$ (0), α = 1, 2, has a threshold eigenvalue.

On the Spectrum of an Hamiltonian in Fock Space. Discrete Spectrum Asymptotics

A model operator H associated with the energy operator of a system describing three particles in interaction, without conservation of the number of particles, is considered. The location of the essential spectrum of H is described. The existence of infinitely many eigenvalues (resp. the finiteness of eigenvalues) below the bottom τess(H) of the essential spectrum of H is proved for the case where the associated Friedrichs model has a threshold energy resonance (resp. a threshold eigenvalue). For the number N(z) of eigenvalues of H lying below z < τess(H) the following asymptotics is found $$\lim\limits_{z \to \tau_{\rm ess}(H)-0}\frac{N(z)}{|\log |z-\tau_{\rm ess}(H)||}={U}_0 (0<{U}_0 <\infty).$$

The threshold effects for a family of Friedrichs models under rank one perturbations

A family of Friedrichs models under rank one perturbations h μ ( p ) , p ∈ ( − π , π ] 3 , μ > 0 , associated to a system of two particles on the three-dimensional lattice Z 3 is considered. We prove the existence of a unique eigenvalue below the bottom of the essential spectrum of h μ ( p ) for all non-trivial values of p under the assumption that h μ ( 0 ) has either a threshold energy resonance (virtual level) or a threshold eigenvalue. The threshold energy expansion for the Fredholm determinant associated to a family of Friedrichs models is also obtained.