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We discuss features of coronal mass ejections (CMEs) that are specific to heliospheric observations at large elongation angles. Our analysis is focused on a series of two eruptions that occurred on 2007 January 24-25, which were tracked by the Heliospheric Imagers (HIs) onboard STEREO. |
The interactions between magnetic islands are considered to play an important role in electron acceleration during magnetic reconnection. In this paper, two-dimensional (2-D) particle-in-cell (PIC) simulations are performed to study electron acceleration during multiple X line reconnection with a guide field. |
This paper presents a novel encoding method for fine time data of a tapped delay line (TDL) time-to-digital Converter (TDC). It is based on divide-and-conquer strategy, and has the advantage of significantly reducing logic resource utilization while retaining low dead-time performance. |
To test the cosmic spatial isotropy, we use a rotatable torsion balance carrying a transversely spin-polarized ferrimagnetic Dy_{6}Fe_{23} mass. With a rotation period of one hour, the period of anisotropy signal is reduced from one sidereal day by about 24 times, and hence the 1/f noise is greatly reduced. |
We present a self-consistent Particle-in-Cell simulation of the resonant interactions between anisotropic energetic electrons and a population of whistler waves, with parameters relevant to the Earths radiation belt. By tracking PIC particles, and comparing with test-particle simulations we emphasize the importance of... |
This study investigates the modulation of particle fluxes at the Earths surface influenced by the intensity and orientation of the Interplanetary magnetic field (IMF) carried by the Coronal Mass Ejecta (ICME). We examine how IMF and its Bz component, opposing the magnetosphere, significantly enhance geomagnetic activi... |
Potentially impacting asteroids were analysed for their impact risk on the Earth. To this end, the Asteroid Risk Mitigation Optimization and Research (ARMOR) tool is currently being developed. |
The ambient solar wind conditions in interplanetary space and in the near-Earth environment are determined by activity on the Sun. Steady solar wind streams modulate the propagation behaviour of interplanetary coronal mass ejections and are themselves an important driver of recurrent geomagnetic storm activity. |
Volume estimation of onshore cargo piles is of economic importance for shipping and mining companies as well as public authorities for real-time planning of logistics, business intelligence, transport services by land or sea and governmental oversight. In remote sensing literature, the volume of pile is estimated by r... |
A weakly dispersive sub-range (WDR) of kinetic Alfvén turbulence is distinguished and investigated for the first time in the context of MHD/kinetic turbulence transition. We found perpendicular wavenumber spectra ~ k^{-3} and ~ k^{-4} formed in WDR by strong and weak turbulence of kinetic Alfvén waves (KAWs), respecti... |
Recent statistical studies of observational data unveil relevant correlations between whistler fluctuations and the anisotropic electron populations present in space plasmas, e.g., solar wind and planetary magnetospheres. Locally, whistlers can be excited by two sources of free energy associated with anisotropic elect... |
Small-scale magnetic flux ropes (SFRs), in the solar wind, have been studied for decades. Statistical analysis utilizing various in situ spacecraft measurements is the main observational approach which helps investigate the generation and evolution of these small-scale structures. |
Adiabatic and non-adiabatic electron dynamics have been proposed to explain electron heating across collisionless shocks. We analyze the evolution of the suprathermal electrons across 310 quasi-perpendicular shocks with $1.7<M_A<48$ using in-situ measurements. |
Shocks and turbulence are ubiquitous phenomena, responsible for particle acceleration to very high energies in a large collection of astrophysical systems. Using self-consistent, hybrid-kinetic simulations with and without pre-existing turbulence, we study the transition of a shock from ``laminar'' to turbulen... |
A self-consistent procedure for the ion-induced soft error rate calculation in space environment taking into account Error Correcting Codes is proposed. The method is based on the partitioning of the multiple cell events into groups with different multiplicities. |
A new variational technique determines the general condition of equilibrium of a rotating gravito-electromagnetic system and provides a modified dynamical equation of motion from where it emerges a so-far unforseen topological torsion current (TTC) [Mario J. Pinheiro (2013) 'A Variational Method in Out-of-Equilibri... |
Hot flow anomalies (HFAs) were first discovered in the early 1980s at the bow shock of the Earth. In the 1990s these features were studied, observed and simulated very intensively and many new missions focused the attention to this phenomenon again. |
We study the interaction of upstream ultra-low frequency (ULF) waves with collisionless shocks by analyzing the outputs of eleven 2D local hybrid simulation runs. Our simulated shocks have Alfvénic Mach numbers between 4.29-7.42 and their $\theta_{BN}$ angles are 15$^\circ$, 30$^\circ$, 45$^\circ$ and 50$^\circ$. |
Models for collisionless magnetic reconnection in near-Earth space are distinctly characterized as 2D or 3D. In 2D kinetic models, the frozen-in law for the electron fluid is usually broken by laminar dynamics involving structures set by the electron orbit size, while in 3D models the width of the electron diffusion r... |
Using a refined setup process, we simulated the propagation of six observed Coronal Mass Ejections (CMEs) with the 2012 Block-Adaptive-Tree-Solarwind-Roe-Upwind-Scheme (BATS-R-US) code from the Sun to the Earth or STEREO A and compared the outputs with observations. A linear relation between the average CME speed belo... |
In this paper we examine suprathermal He ions measured by the SIT (Suprathermal Ion Telescope) instrument associated with tilted corotating interaction regions (CIRs). We use observations of the two STEREO spacecraft (s/c) for the first 2.7 years of the mission, along with ground-based measurements of the solar magnet... |
Simultaneous radio and extreme ultraviolet (EUV)/white-light imaging data are examined for a solar type II radio burst occurring on 2010 March 18 to deduce its source location. Using a bow-shock model, we reconstruct the 3-dimensional EUV wave front (presumably the type-II emitting shock) based on the imaging data of ... |
We examine the range of applicability of Taylor's hypothesis used in observations of magnetic turbulence in the solar wind. We do not refer to turbulence theory. |
The treatment of the electron-nucleus interaction based on the Mott differential cross section was extended to account for effects due to screened Coulomb potentials, finite sizes and finite rest masses of nuclei for electrons above 200 keV and up to ultra high energies. This treatment allows one to determine both the... |
The increasing number of objects orbiting the Earth justifies the great attention and interest in the observation, spacecraft protection and collision avoidance. These studies involve different disturbances and resonances in the orbital motions of these objects distributed by the distinct altitudes. |
Lower atmospheric global dust storms affect the small- and large-scale weather and variability of the whole Martian atmosphere. Analysis of the CO$_2$ density data from the Neutral Gas and Ion Mass Spectrometer instrument (NGIMS) on board NASA's Mars Atmosphere Volatile EvolutioN (MAVEN) spacecraft show a remarkab... |
After the disappearance of lower energy heliospheric particles at Voyager 1 starting on August 25th, 2012, spectra of H, He and C/O nuclei were revealed that resembled those to be expected for galactic cosmic rays. These spectra had intensity peaks in the range of 30-60 MeV, decreasing at both lower energies down to a... |
Ultra-low frequency (ULF) waves can contribute to the transport, acceleration and loss of electrons in the radiation belts through inward and outward diffusion. However, the most appropriate parameters to use to specify the ULF wave diffusion rates are unknown. |
Transit observations of HD 209458b in the stellar Lyman-$\alpha$ (Ly$\alpha$) line revealed strong absorption in both blue and red wings of the line interpreted as hydrogen atoms escaping from the planet's exosphere at high velocities. The following sources for the absorption were suggested: acceleration by the st... |
Utilizing new Voyager measurements at lower energies and higher energy spacecraft measurements near the Earth, the interstellar spectra of primary cosmic ray nuclei from H to Fe have now been determined from ~10 MeV/nuc to > 100 GeV/nuc. These measurements are compared with the predictions from a Leaky Box propagat... |
Phase space holes, double layers and other solitary electric field structures, referred to as time domain structures (TDSs), often occur around dipolarization fronts in the Earth's inner magnetosphere. They are considered to be important because of their role in the dissipation of the injection energy and their po... |
In this paper we define the a posteriori probability W. This quantity is introduced with the aim of suggesting a reliable interpretation of the actual threat posed by a newly discovered Near Earth Asteroid (NEA), for which impacting orbital solutions (now commonly known as Virtual Impactors or VIs) can not be excluded ... |
Two generic properties of the Neumann--Poincaré operator are investigated. We prove that non-zero eigenvalues of the Neumann--Poincaré operator on smooth boundaries in three dimensions and higher are generically simple in the sense of Baire category. |
We consider discrete Schrödinger operators with pattern Sturmian potentials. This class of potentials strictly contains the class of Sturmian potentials, for which the spectral properties of the associated Schrödinger operators are well understood. |
We obtain the asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators with general regular boundary conditions. Using these formulas, we find sufficient conditions on the potential q such that the root functions of these operators do not form a Riesz basis. |
The norm of an integral operator occurring in the partial wave decomposition of an operator B introduced by Brown and Ravenhall in a model for relativistic one-electron atoms is determined. The result implies that B is non-negative and has no eigenvalue at 0 when the nuclear charge does not exceed a specified critical... |
Canonical Polyadic Decomposition (CPD) of a third-order tensor is a minimal decomposition into a sum of rank-$1$ tensors. We find new mild deterministic conditions for the uniqueness of individual rank-$1$ tensors in CPD and present an algorithm to recover them. |
We study Schrödinger operators on star metric graphs with potentials of the form $\alpha\varepsilon^{-2}Q(\varepsilon^{-1}x)$. In dimension 1 such potentials, with additional assumptions on $Q$, approximate in the sense of distributions as $\varepsilon\to0$ the first derivative of the Dirac delta-function. |
Boundary value problems for Sturm-Liouville operators with potentials from the class $W_2^{-1}$ on a star-shaped graph are considered. We assume that the potentials are known on all the edges of the graph except two, and show that the potentials on the remaining edges can be constructed by fractional parts of two spec... |
The $H^\infty$-functional calculus is a two-step procedure, introduced by A. McIntosh, that allows the definition of functions of sectorial operators in Banach spaces. It plays a crucial role in the spectral theory of differential operators, as well as in their applications to evolution equations and various other fie... |
The purpose of this note is to review some recent results concerning the pseudospectra and the eigenvalues asymptotics of non-selfadjoint semiclassical pseudo-differential operators subject to small random perturbations. |
Let $(\lambda_-,\lambda_+)$ be a spectral gap of a periodic Schrödinger operator $A$ on the lattice ${\mathbb Z}^d$. Consider the operator $A(\alpha)=A-\alpha V$ where $V$ is a decaying positive potential on ${\mathbb Z}^d$. |
We study the properties of eigenvalues and corresponding eigenfunctions generated by a defect in the gaps of the spectrum of a high-contrast random operator. We consider a family of elliptic operators $\mathcal{A}^\varepsilon$ in divergence form whose coefficients are random, possess double porosity type scaling, and ... |
We study spectral approximations of Schrödinger operators $T=-\Delta+Q$ with complex potentials on $\Omega=\mathbb{R}^d$, or exterior domains $\Omega\subset \mathbb{R}^d$, by domain truncation. Our weak assumptions cover wide classes of potentials $Q$ for which $T$ has discrete spectrum, of approximating domains $\Ome... |
Bounds are obtained for the efficiency or mean to peak ratio $E(\Omega)$ for the first Dirichlet eigenfunction (positive) for open, connected sets $\Omega$ with finite measure in Euclidean space $\R^m$. It is shown that |
We consider the combinatorial Laplacian on a sequence of discrete tori which approximate the m-dimensional torus. In the special case m=1, Friedli and Karlsson derived an asymptotic expansion of the corresponding spectral zeta function in the critical strip, as the approximation parameter goes to infinity. |
The work is devoted to the study of the similarity of a correct restriction to some self-adjoint operator in the case when the minimal operator is symmetric. The resulting theorem was applied to the Sturm-Liouville operator and the Laplace operator. |
In this paper we study the Riesz transform on complete and connected Riemannian manifolds $M$ with a certain spectral gap in the $L^2$ spectrum of the Laplacian. We show that on such manifolds the Riesz transform is $L^p$ bounded for all $p \in (1,\infty)$. |
Product graphs have been gainfully used in literature to generate mathematical models of complex networks which inherit properties of real networks. Realizing the duplication phenomena imbibed in the definition of corona product of two graphs, we define Corona graphs. |
We study Schroedinger operators with Robin boundary conditions on exterior domains in $\R^d$. We prove sharp point-wise estimates for the associated semi-groups which show, in particular, how the boundary conditions affect the time decay of the heat kernel in dimensions one and two. |
Karlin's (1982) Theorem 5.2 shows that linear systems alternating between growth and mixing phases have lower asymptotic growth with greater mixing. Here this result is extended to linear differential equations that combine site-specific growth or decay rates, and mixing between sites, showing that the spectral ab... |
I present an example of a discrete Schr"odinger operator that shows that it is possible to have embedded singular spectrum and, at the same time, discrete eigenvalues that approach the edges of the essential spectrum (much) faster than exponentially. This settles a conjecture of Simon (in the negative). |
We prove several new results on the absolutely continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely continuous spectrum. |
We prove sufficient conditions for Hausdorff convergence of the spectra of sequences of closed operators defined on varying Hilbert spaces and converging in norm-resolvent sense, i.e. $\|J_\varepsilon(1+A_\varepsilon)^{-1} - (1+A)^{-1}J_\varepsilon\|\to 0$ as $\varepsilon\to 0$, where $J_\varepsilon$ is a suitable iden... |
Let (M^n, g) be a closed smooth Riemannian spin manifold and denote by D its Atiyah-Singer-Dirac operator. We study the variation of Riemannian metrics for the zeta function and functional determinant of D^2, and prove finiteness of the Morse index at stationary metrics, and local extremality at such metrics under gen... |
Let $A$ and $B$ designate $n\times n$ matrices with coefficients in a field $F$. In this paper, we completely answer the following question: For $A$ fixed, what are the possible characteristic polynomials of $A+B$, where $B$ ranges over matrices of rank $\le m$? |
We present an explicit two-parameter family of finite-band Jacobi elliptic potentials for a non-self-adjoint Dirac operator which connects two previously known limiting cases in which the elliptic parameter is zero or one. A full characterization of the spectrum is obtained by relating the periodic and antiperiodic ei... |
We consider two-dimensional Schrödinger operators in bounded domains. We analyze relations between nodal domains of eigenfunctions, spectral minimal partitions and spectral properties of the corresponding operator. |
The second eigenvalue of the Robin Laplacian is shown to be maximal for a spherical cap among simply connected Jordan domains on the 2-sphere, for substantial intervals of positive and negative Robin parameters and areas. Geodesic disks in the hyperbolic plane similarly maximize the eigenvalue on a natural interval of... |
We will study the spectral problem related to the Laplace operator in a singularly perturbed periodic waveguide. The waveguide is a quasi-cylinder with contains periodic arrangement of inclusions. |
It is an open problem in general to prove that there exists a sequence of $\Delta_g$-eigenfunctions $\phi_{j_k}$ on a Riemannian manifold $(M, g)$ for which the number $N(\phi_{j_k}) $ of nodal domains tends to infinity with the eigenvalue. Our main result is that $N(\phi_{j_k}) \to \infty$ along a subsequence of eige... |
This paper develops a scattering theory for the asymmetric transport observed at interfaces separating two-dimensional topological insulators. Starting from the spectral decomposition of an unperturbed interface Hamiltonian, we present a limiting absorption principle and construct a generalized eigenfunction expansion... |
We extend the multiplicative submodularity of the principal determinants of a nonnegative definite hermitian matrix to other spectral functions. We show that if $f$ is the primitive of a function that is operator monotone on an interval containing the spectrum of a hermitian matrix $A$, then the function $I\mapsto {\r... |
We provide an elementary proof of the equivalence of various notions of uniform hyperbolicity for a class of $\mathrm{GL}(2,\mathbb{C})$ cocycles and establish a Johnson-type theorem for extended CMV matrices, relating the spectrum to the points on the unit circle for which the associated Szegő cocycle is not uniformly... |
We determine the density of eigenvalues of the scattering matrix of the Schrodinger operator with a short range potential in the high energy asymptotic regime. We give an explicit formula for this density in terms of the X-ray transform of the potential. |
We explore connections between Krein's spectral shift function $\xi(\lambda,H_0,H)$ associated with the pair of self-adjoint operators $(H_0,H)$, $H=H_0+V$ in a Hilbert space $\calH$ and the recently introduced concept of a spectral shift operator $\Xi(J+K^*(H_0-\lambda-i0)^{-1}K)$ associated with the operator-valu... |
We study partition problems based on two ostensibly different kinds of energy functionals defined on $k$-partitions of metric graphs: Cheeger-type functionals whose minimisers are the $k$-Cheeger cuts of the graph, and the corresponding values are the $k$-Cheeger constants of the graph; and functionals built using the ... |
We describe some aspects of spectral theory that involve algebraic considerations but need no analysis. Some of the important applications of the results are to the algebra of $n\times n$ matrices with entries that are polynomials or more general analytic functions. |
For a class of singular Sturm-Liouville equations on the unit interval with explicit singularity $a(a + 1)/x^2, a \in \mathbb{N}$, we consider an inverse spectral problem. Our goal is the global parametrization of potentials by spectral data noted by $\lambda^a$, and some norming constants noted by $\kappa^a$. |
The Neumann-Poincaré (NP) operator naturally appears in the context of metamaterials as it may be used to represent the solutions of elliptic transmission problems via potentiel theory. In particular, its spectral properties are closely related to the well-posedness of these PDE's, in the typical case where one co... |
In this work, we use regularized determinant approach to study the discrete spectrum generated by relatively compact non-self-adjoint perturbations of the magnetic Schrödinger operator $(-i\nabla - \textbf{\textup{A}})^{2} - b$ in dimension $3$ with constant magnetic field of strength $b>0$. The situation near the ... |
We establish Anderson localization for Schrödinger operators with even analytic potentials on the first supercritical stratum for Liouville frequencies in the sharp regime $\{E: L(\omega,E)>\beta(\omega)>0, \kappa(\omega,E)=1\}$, with $\kappa(\omega,E)$ being Avila's acceleration. This paper builds on the la... |
A generalized variant of the Calderón problem from electrical impedance tomography with partial data for anisotropic Lipschitz conductivities is considered in an arbitrary space dimension $n \geq 2$. The following two results are shown: |
We solve a spectral and an inverse spectral problem arising in the computation of peakon solutions to the two-component PDE derived by Geng and Xue as a generalization of the Novikov and Degasperis-Procesi equations. Like the spectral problems for those equations, this one is of a 'discrete cubic string' type ... |
In this note we study inverse spectral problems for canonical Hamiltonian systems, which encompass a broad class of second order differential equations on a half-line. Our goal is to extend the classical resultss developed in the work of Marchenko, Gelfand-Levitan, and Krein to broader classes of canonical systems and... |
A result of Borg--Hochstadt in the theory of periodic Jacobi matrices states that such a matrix has constant diagonals as long as all gaps in its spectrum are closed (have zero length). We suggest a quantitative version of this result by proving the two-sided bounds between oscillations of the matrix entries along the... |
In various aspects of the spectral analysis of random Schroedinger operators monotonicity with respect to the randomness plays a key role. In particular, both the continuity properties and the low energy behaviour of the integrated density of states (IDS) are much better understood if such a monotonicity is present in... |
We construct efficient approximations for the eigenfunctions of non-self-adjoint Schroedinger operators in one dimension. The same ideas also apply to the study of resonances of self-adjoint Schroedinger operators which have dilation analytic potentials. |
In this article we construct a family of domains $\Omega \subset \mathbb{R}^2$ with infinite volume such that the Dirichlet Laplacian $\Delta^D$ has purely discrete spectrum and give precise spectral asymptotics for the eigenvalue counting function in terms of the geometry of $\Omega$. This generalizes the well-known ... |
We consider the Stark operator perturbed by a compactly supported potential (of a certain class) on the real line. We prove the following results: |
We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. |
We prove a lower bound for the $k$-th Steklov eigenvalues in terms of an isoperimetric constant called the $k$-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the St... |
We discuss the spectral properties of three-dimensional Dirac operators with critical combinations of electrostatic and Lorentz scalar shell interactions supported by a compact smooth surface. It turns out that the criticality of the interaction may result in a new interval of essential spectrum. |
In this paper, the geometry properties of Hankel form are studied, including their positive semi-definite (PSD) cone and sum-of-squares (SOS) cone. We denote them by $HPSD(m,n)$ and $HSOS(m,n)$, respectively. |
We study some spectral properties of random walks on infinite countable amenable groups with an emphasis on locally finite groups, e.g. the infinite symmetric group. On locally finite groups, the random walks under consideration are driven by infinite divisible distributions. |
A one-dimensional quantum harmonic oscillator perturbed by a smooth compactly supported potential is considered. For the corresponding eigenvalues $\lambda_n$, a complete asymptotic expansion for large $n$ is obtained, and the coefficients of this expansion are expressed in terms of the heat invariants. |
The spectral gap for Laplace operators on metric graphs is investigated in relation to graph's connectivity, in particular what happens if an edge is added to (or deleted from) a graph. It is shown that in contrast to discrete graphs connection between the connectivity and the spectral gap is not one-to-one. |
We consider a spectral stability estimate by Burenkov and Lamberti concerning the variation of the eigenvalues of second order uniformly elliptic operators on variable open sets in the N-dimensional euclidean space, and we prove that it is sharp for any dimension N. This is done by studying the eigenvalue problem for t... |
We consider the wave equation on non-compact star graphs, subject to a distributional damping defined through a Robin-type vertex condition with complex coupling. It is shown that the non-self-adjoint generator of the evolution problem admits an abrupt change in its spectral properties for a special coupling related t... |
For the the Schrödinger operator $H=-\Delta+ V(x)\cdot$, acting in the space L_2(\R^d)\,(d\ge 3), with V(x)\ge 0 and V(\cdot)\in L_{1,loc}(\R^d), we obtain some constructive conditions for discreteness of its spectrum. Basing on the Mazya-Shubin criterion for discreteness of the spectrum of $H$ and using the isocapaci... |
We prove that ellipses are infinitesimally spectrally rigid among $C^{\infty}$ domains with the symmetries of the ellipse. |
In this paper we consider the spectrum of the self-adjoint differential operator L generated by the differential expression of order n with the m by m periodic matrix coefficients, where n and m are respectively odd and even integers and n>1. We prove that the number of gaps in the spectrum of L is finite and find ... |
We introduce a notion of weak isospectrality for continuous deformations. Consider the Laplace-Beltrami operator on a compact Riemannian manifold with boundary with Robin boundary conditions. |
We investigate the relation between the spectrum of a non-normal matrix and the norm of its resolvent. We provide spectral estimates for the resolvent of matrices whose largest singular value is bounded by $1$ (so-called Hilbert space contractions) and for power-bounded matrices. |
We prove a partial result concerning the long-standing problem on limit periodicity of the Jacobi matrix associated with the balanced measure on the Julia set of an expending polynomial. Besides this, connections of the problem with the Faybusovich--Gekhtman flow and many other objects (the Hilbert transform, the Schw... |
This article is concerned with the semi-classical limits of matrix elements $<F \phi_j, \phi_j>$ of eigenfunctions of the Laplacian $\Delta_g$ of a compact Riemannian manifold $(M, g)$ with respect to a Fourier integral operator $F$ on $L^2(M)$. Many results exist for the case where $F$ is a pseudo-differential ... |
Denote by $L_D$ the Sturm-Liouville operator $Ly=-y" +q(x)y$ on the finite interval $[0,\pi]$ with Dirichlet boundary conditions $y(0)=y(\pi)=0$. Let $\{\lambda_k\}_1^\infty$ and $\{\alpha_k\}_1^\infty$ be the sequences of the eigenvalues and norming constants of this operator. |
The principal objective in this paper is a new inverse approach to general Dirac-type systems modeled after B. Simon's 1999 inverse approach to half-line Schrödinger operators. In particular, we derive the so-called A-equation associated to Dirac-type systems and, given a fundamental positivity condition, we prove... |
We prove that the exponential moments of the position operator stay bounded for the supercritical almost Mathieu operator with Diophantine frequency. |
Let $H_0 = -\Delta + V_0(x)$ be a Schroedinger operator on $L_2(\mathbb{R}^\nu),$ $\nu=1,2,$ or 3, where $V_0(x)$ is a bounded measurable real-valued function on $\mathbb{R}^\nu. $ 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. |
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