Michael J Gruber
We study Schroedinger operators with periodic magnetic field in Euclidean 2-space, in the case of irrational magnetic flux. Positive measure Cantor spectrum is generically expected in the presence of an electric potential. We show that, even without electric potential, the spectrum has positive measure if the magnetic field is a perturbation of a constant one.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We study discrete Schroedinger operators with analytic potentials. In particular, we are interested in the connection between the absolutely continuous spectrum in the almost periodic case and the spectra in the periodic case. We prove a weak form of a precise conjecture relating the two. We also bound the measure of the spectrum in the periodic case in terms of the Lyapunov exponent in the almost periodic case. In the proofs, we use a partial generalization of Chambers' formula. As an additional application of this generalization, we provide a new proof of Herman's lower bound for the Lyapuno…
We develop direct and inverse scattering theory for one-dimensional Schroedinger operators with steplike potentials which are asymptotically close to different finite-gap periodic potentials on different half-axes. We give a complete characterization of the scattering data, which allow unique solvability of the inverse scattering problem in the class of perturbations with finite second moment.
We consider a periodic system of domains coupled by small windows. In such domain we study the band spectrum of a Schroedinger operator subject to Neumann condition. We show that near each isolated eigenvalue of the similar operator but in the periodicity cell, there are several non-intersecting bands of the spectrum for the perturbed operator. We also discuss the position of the points at which the band functions attain the edges of each band.
An explicit construction is provided for embedding n positive eigenvalues in the spectrum of a Schroedinger operator on the half-line with a Dirichlet boundary condition at the origin. The resulting potential is of von Neumann-Wigner type, but can be real valued as well as complex valued.
Fermi surfaces are basic objects in solid state physics and in the spectral theory of periodic operators. We define several measures connected to Fermi surfaces and study their measure theoretic properties. From this we get absence of singular continuous spectrum and of singular continuous components in the density of states for symmetric periodic elliptic differential operators acting on vector bundles. This includes Schroedinger operators with periodic magnetic field and rational flux, as well as the corresponding Pauli and Dirac-type operators.
The paper offers the method of discovering of some class of solutions for the nonlinear Schroedinger equation. An algorithm of constructive solving of the Cauchy periodic problem with a finite-gap initial condition was also obtained.
We study the Schrödinger operator on $L_2(\mathbb R^3)$ with periodic variable metric, and periodic electric and magnetic fields. It is assumed that the operator is reflection symmetric and the (appropriately defined) flux of the magnetic field is rational. Under these assumptions it is shown that the spectrum of the operator is absolutely continuous. Previously known results on absolute continuity for periodic operators were obtained for the zero magnetic flux.
The two-dimensional Schroedinger operator with a uniform magnetic field and a periodic zero-range potential is considered. For weak magnetic fields we reduce the spectral problem to the semiclassical analysis of one-dimensional Harper-like operators. This shows the existence of parts of Cantor structure in the spectrum for special values of the magnetic flux.
Resonant Solutions of the Non-linear Schroedinger Equation with Periodic Potential
本文研究二维周期非线性薛定谔方程在高能量下两个平面波非平凡组合附近的稳态解构造问题，并刻画了所有足够大能量对应的等能面。
In the large coupling constant limit, we obtain an asymptotic expansion in powers of $μ^{-\frac{1}δ}$ of the derivative of the spectral shift function corresponding to the pair $\big(P_μ=P_0+μW(x),P_0=-Δ+V(x)\big),$ where $W(x)$ is positive, $W(x)\sim w_0(\frac{x}{|x|})|x|^{-δ}$ near infinity for some $δ>n$ and $w_0\in {\mathcal C}^\infty(\mathbb S^{n-1};\,\mathbb R_+).$ Here $\mathbb S^{n-1}$ is the unite sphere of the space $\mathbb R^n$ and $μ$ is a large parameter. The potential $V$ is real-valued, smooth and periodic with respect to a lattice $Γ$ in ${\mathbb R}^n$.
We prove a localization theorem for continuous ergodic Schrödinger operators $ H_ω:= H_0 + V_ω$, where the random potential $ V_ω$ is a nonnegative Anderson-type perturbation of the periodic operator $ H_0$. We consider a lower spectral band edge of $ σ(H_0) $, say $ E= 0 $, at a gap which is preserved by the perturbation $ V_ω$. Assuming that all Floquet eigenvalues of $ H_0$, which reach the spectral edge 0 as a minimum, have there a positive definite Hessian, we conclude that there exists an interval $ I $ containing 0 such that $ H_ω$ has only pure point spectrum in $ I $ for almost all $ …
We consider the dynamics generated by the Schroedinger operator $H=-{1/2}Δ+ V(x) + W(\epsi x)$, where $V$ is a lattice periodic potential and $W$ an external potential which varies slowly on the scale set by the lattice spacing. We prove that in the limit $\epsi \to 0$ the time dependent position operator and, more generally, semiclassical observables converge strongly to a limit which is determined by the semiclassical dynamics.
In this paper we study the asymptotic expansion of the spectral shift function for the slowly varying perturbations of periodic Schrödinger operators. We give a weak and pointwise asymptotics expansions in powers of $h$ of the derivative of the spectral shift function corresponding to the pair $\big(P(h)=P_0+ϕ(hx),P_0=-Δ+V(x)\big),$ where $ϕ(x)\in {\mathcal C}^\infty(\mathbb R^n,\mathbb R)$ is a decreasing function, ${\mathcal O}(|x|^{-δ})$ for some $δ>n$ and $h$ is a small positive parameter. Here the potential $V$ is real, smooth and periodic with respect to a lattice $Γ$ in ${\mathbb R}^n$.…
We consider radial tree extensions of one-dimensional quasi-periodic Schroedinger operators and establish the stability of their absolutely continuous spectra under weak but extensive perturbations by a random potential. The sufficiency criterion for that is the existence of Bloch-Floquet states for the one dimensional operator corresponding to the radial problem.
Consider in $L^2(\R)$ the \Sc operator family $H(g):=-d^2_x+V_g(x)$ depending on the real parameter $g$, where $V_g(x)$ is a complex-valued but $PT$ symmetric periodic potential. An explicit condition on $V$ is obtained which ensures that the spectrum of $H(g)$ is purely real and band shaped; furthermore, a further condition is obtained which ensures that the spectrum contains at least a pair of complex analytic arcs.
We consider a cubic nonlinear Schroedinger equation with periodic potential. In a semiclassical scaling the nonlinear interaction of modulated pulses concentrated in one or several Bloch bands is studied. The notion of closed mode systems is introduced which allows for the rigorous derivation of a finite system of amplitude equations describing the macroscopic dynamics of these pulses.
In this paper we consider the discrete one-dimensional Schroedinger operator with quasi-periodic potential v_n = λv (x + n ω). We assume that the frequency ωsatisfies a strong Diophantine condition and that the function v belongs to a Gevrey class, and it satisfies a transversality condition. Under these assumptions we prove - in the perturbative regime - that for large disorder λand for most frequencies ωthe operator satisfies Anderson localization. Moreover, we show that the associated Lyapunov exponent is positive for all energies, and that the Lyapunov exponent and the integrated density o…
We consider the Dirichlet Laplacian in a straight planar strip perturbed by a bounded periodic symmetric operator. We prove the classical Bethe-Sommerfeld conjecture for this operator, namely, that this operator has finitely many gaps in its spectrum provided a certain special function written as a series satisfies some lower bound. We show that this is indeed the case if the ratio of the period and the width of strip is less than a certain explicit number, which is approximately equal to 0.10121. We also find explicitly the point in the spectrum, above which there is no internal gaps. We then…
We prove Strichartz estimates for the Schroedinger operator $H = -Δ+ V(t,x)$ with time-periodic complex potentials $V$ belonging to the scaling-critical space $L^{n/2}_x L^\infty_t$ in dimensions $n \ge 3$. This is done directly from estimates on the resolvent rather than using dispersive bounds, as the latter generally require a stronger regularity condition than what is stated above. In typical fashion, we project onto the continuous spectrum of the operator and must assume an absence of resonances. Eigenvalues are permissible at any location in the spectrum, including at threshold energies,…
We derive a sharp bound on the location of non-positive eigenvalues of Schroedinger operators on the halfline with complex-valued potentials.
In this paper we study the operator $H_β=-Δ-βδ(\cdot-Γ)$ in $L^{2}(\mathbb{R}^{2})$, where $Γ$ is a smooth periodic curve in $\mathbb{R}^{2}$. We obtain the asymptotic form of the band spectrum of $H_β$ as $β$ tends to infinity. Furthermore, we prove the existence of the band gap of $σ(H_β)$ for sufficiently large $β>0$. Finally, we also derive the spectral behaviour for $β\to\infty$ in the case when $Γ$ is non-periodic and asymptotically straight.
Let $V_Γ$ be a lattice periodic potential and $A$ and $ϕ$ external electromagnetic potentials which vary slowly on the scale set by the lattice spacing. It is shown that the Wigner function of a solution of the Schroedinger equation with Hamiltonian operator $H = {1/2} (-\I\nabla_x - A(εx))^2 + V_Γ(x) + ϕ(εx)$ propagates along the flow of the semiclassical model of solid states physics up an error of order $ε$. If $ε$-dependent corrections to the flow are taken into account, the error is improved to order $ε^2$. We also discuss the propagation of the Wigner measure. The results are obtained as…
We investigate trace formulas for one-dimensional Schroedinger operators which are trace class perturbations of quasi-periodic finite-gap operators using Krein's spectral shift theory. In particular, we establish the conserved quantities for the solutions of the Korteweg-de Vries hierarchy in this class and relate them to the reflection coefficients via Abelian integrals on the underlying hyperelliptic Riemann surface.
We study the transition to the continuum of an initially bound quantum particle in $\RR^d$, $d=1,2,3$, subjected, for $t\ge 0$, to a time periodic forcing of arbitrary magnitude. The analysis is carried out for compactly supported potentials, satisfying certain auxiliary conditions. It provides complete analytic information on the time Laplace transform of the wave function. From this, comprehensive time asymptotic properties (Borel summable transseries) follow. We obtain in particular a criterion for whether the wave function gets fully delocalized (complete ionization). This criterion shows …
We consider 2- and 3-dimensional Schrödinger or generalized Schrödinger-Pauli operators with the non-degenerating magnetic field in the open domain under certain non-degeneracy assumptions we derive pointwise spectral asymptotics. We also consider asymptotics of some related expressions (see below). For all asymptotics loops rather than periodic trajectories play important role.
For certain one-dimensional Schroedinger-type difference operators with a complex potential, a "complete" set of exponentially decaying eigenvectors is shown to exist. "Completeness" entails that the parameters involved are obtained through evaluation of analytic functions on a Riemann surface.
We show coincidence of the two definitions of the integrated density of states (IDS) for a class of relativistic Schroedinger operators with magnetic fields and scalar potentials, the first one relying on the eigenvalue counting function of operators induced on open bounded sets with Dirichlet boundary conditions, the other one involving the spectral projections of the operator defined on the entire space. In this way one generalizes previous results for non-relativistic operators. The proofs needs the magnetic pseudodifferential calculus, as well as a Feynman-Kac-Ito formula for Levy processe…
Spectral and scattering theory at low energy for the relativistic Schroedinger operator are investigated. Some striking properties at thresholds of this operator are exhibited, as for example the absence of 0-energy resonance. Low energy behavior of the wave operators and of the scattering operator are studied, and stationary expressions in terms of generalized eigenfunctions are proved for the former operators. Under slightly stronger conditions on the perturbation the absolute continuity of the spectrum on the positive semi axis is demonstrated. Finally, an explicit formula for the action of…
We show that whole-line Schrödinger operators with finitely many bound states have no embedded singular spectrum. In contradistinction, we show that embedded singular spectrum is possible even when the bound states approach the essential spectrum exponentially fast. We also prove the following result for one- and two-dimensional Schrödinger operators, $H$, with bounded positive ground states: Given a potential $V$, if both $H\pm V$ are bounded from below by the ground-state energy of $H$, then $V\equiv 0$.
We study spectra of alloy-type random Schrödinger operators on metric graphs. For finite edge subsets of general graphs we prove a Wegner estimate which is linear in the volume (i.e. the number of edges) and the length of the considered energy interval. The single site potential of the alloy-type model needs to have fixed sign, but the considered metric graph does not need to have a periodic structure. The second result we obtain is an exhaustion construction of the integrated density of states for ergodic random Schrödinger operators on metric graphs with a $\ZZ^ν$-structure. For certain mode…
We classify the hulls of different limit-periodic potentials and show that the hull of a limit-periodic potential is a procyclic group. We describe how limit-periodic potentials can be generated from a procyclic group and answer arising questions. As an expository paper, we discuss the connection between limit-periodic potentials and profinite groups as completely as possible and review some recent results on Schroedinger operators obtained in this context.
We investigate the operator $-Δ-αδ(x-Γ)$ in $L^2(\mathbb{R}^3)$, where $Γ$ is a smooth surface which is either compact or periodic and satisfies suitable regularity requirements. We find an asymptotic expansion for the lower part of the spectrum as $α\to\infty$ which involves a ``two-dimensional'' comparison operator determined by the geometry of the surface $Γ$. In the compact case the asymptotics concerns negative eigenvalues, in the periodic case Floquet eigenvalues. We also give a bandwidth estimate in the case when a periodic $Γ$ decomposes into compact connected components. Finally, we c…
We consider time-dependent nonlinear Schroedinger equations subject to smooth, lattice-periodic potentials plus additional confining potentials, slowly varying on the lattice scale. After an appropriate scaling we study the homogenization limit for vanishing lattice spacing. Assuming well prepared initial data, the resulting effective dynamics is governed by a homogenized nonlinear Schroedinger equation with an effective mass tensor depending on the initially chosen Bloch eigenvalue. The given results rigorously justify the use of the effective mass formalism for the description of Bose-Einste…
We show absence of positive eigenvalues for generalized 2-body hard- core Schroedinger operators under the condition of bounded strictly convex obstacles. A scheme for showing absence of positive eigenvalues for generalized N -body hard-core Schroedinger operators, N \geq 2, is presented. This scheme involves high energy resolvent estimates, and for N = 2 it is implemented by a Mourre commutator type method. A particular example is the Helium atom with the assumption of infinite mass and finite extent nucleus.
In this paper we consider the nonlinear one-dimensional time-dependent Schroedinger equation with a periodic potential and a local perturbation. In the limit of large periodic potential the time behavior of the wavefunction can be approximated, with a precise estimate of the remainder term, by means of the solution to the discrete nonlinear Schroedinger equation of the tight-binding model.
We consider a one-frequency, quasi-periodic, block Jacobi operator, whose blocks are generic matrix-valued analytic functions. We establish Anderson localization for this type of operator under the assumption that the coupling constant is large enough but independent of the frequency. This generalizes a result of J. Bourgain and S. Jitomirskaya on localization for band lattice, quasi-periodic Schroedinger operators.
We report localized nonlinear modes of the self-focusing and defocusing nonlocal nonlinear Schroedinger equation with the generalized PT-symmetric Scarf-II, Rosen-Morse, and periodic potentials. Parameter regions are presented for broken and unbroken PT-symmetric phases of linear bounded states and the linear stability of the obtained solitons. Moreover, we numerically explore the dynamical behaviors of solitons and find stable solitons for some given parameters.
We prove that the Sobolev norm for a 1-D periodic Schroedinger equation remain bounded under small resonant perturbations.
We construct quasi-periodic solutions to the lattice nonlinear random Schroedinger equation on a set of potentials of positive measure via using a Lyapunov-Schmidt decomposition and a multiscale Newton scheme.
Consider the H^{1/2}-critical Schroedinger equation with a cubic nonlinearity in R^3, i \partial_t ψ+ Δψ+ |ψ|^2 ψ= 0. It admits an eight-dimensional manifold of periodic solutions called solitons e^{i(Γ+ vx - t|v|^2 + α^2 t)} ϕ(x-2tv-D, α), where ϕ(x, α) is a positive ground state solution of the semilinear elliptic equation -Δϕ+ α^2ϕ= ϕ^3. We prove that in the neighborhood of the soliton manifold there exists a H^{1/2} real analytic manifold N of asymptotically stable solutions of the Schroedinger equation, meaning they are the sum of a moving soliton and a dispersive term. Furthermore, a sol…
We consider one-dimensional difference Schroedinger equations on the discrete line with a potential generated by evaluating a real-analytic potential function V(x) on the one-dimensional torus along an orbit of the shift x-->x+nw. If the Lyapunov exponent is positive for all energies and w, then the integrated density of states is absolutely continuous for almost every w. In this work we establish the formation of a dense set of gaps in the spectrum. Our approach is based on an induction on scales argument, and is therefore both constructive as well as quantitative. Resonances between eigenfun…
Consider the Schroedinger operator with semiclassical parameter h, in the limit where h goes to zero. When the involved long-range potential is smooth, it is well known that the boundary values of the operator's resolvent at a positive energy E are bounded by O(1/h) if and only if the associated Hamilton flow is non-trapping at energy E. In the present paper, we extend this result to the case where the potential may possess Coulomb singularities. Since the Hamilton flow then is not complete in general, our analysis requires the use of an appropriate regularization.
In this article we study limits of Wigner distributions (the so-called semiclassical measures) corresponding to sequences of solutions to the semiclassical Schroedinger equation at times scales $α_{h}$ tending to infinity as the semiclassical parameter $h$ tends to zero (when $α_{h}=1/h$ this is equivalent to consider solutions to the non-semiclassical Schreodinger equation). Some general results are presented, among which a weak version of Egorov's theorem that holds in this setting. A complete characterization is given for the Euclidean space and Zoll manifolds (that is, manifolds with perio…
In this paper we prove that the initial-boundary value problem for the forced non-linear Schroedinger equation with a potential on the half-line is locally and (under stronger conditions) globally well posed, i.e. that there is a unique solution that depends continuously on the force at the boundary and on the initial data. We allow for a large class of unbounded potentials. Actually, for local solutions we have no restriction on the grow at infinity of the positive part of the potential, and for global solutions very mild assumptions that allow, for example, for exponential grow.
We analyse a nonlinear Schrödinger equation for the time-evolution of the wave function of an electron beam, interacting selfconsistently through a Hartree-Fock nonlinearity and through the repulsive Coulomb interaction of an atomic nucleus. The electrons are supposed to move under the action of a time dependent, rapidly periodically oscillating electromagnetic potential. This can be considered a simplified effective single particle model for an X-ray Free Electron Laser (XFEL). We prove the existence and uniqueness for the Cauchy problem and the convergence of wave-functions to corresponding …
A relation between the Schroedinger wave functional and the Clifford-valued wave function which appears in what we call precanonical quantization of fields and fulfills a Dirac-like generalized covariant Schroedinger equation on the space of field and space-time variables is discussed. The Schroedinger wave functional is argued to be the trace of the positive frequency part of the continual product over all spatial points of the values of the aforementioned wave function restricted to a Cauchy surface. The standard functional differential Schroedinger equation is derived as a consequence of th…
The quantum-mechanical scattering on a compact Riemannian manifold with semi-axes attached to it (hedgehog-shaped manifold) is considered. The complete description of the spectral structure of Schroedinger operators on such a manifold is done, the proof of existence and uniqueness of scattering states is presented, an explicit form for the scattering matrix is obtained and unitary nature of this matrix is proven. It is shown that the positive part of the spectrum of the Schroedinger operator on the initial compact manifold as well as the spectrum of a point perturbation of such an operator may…
We analyze two-dimensional Schrödinger operators with the potential $|xy|^p - λ(x^2+y^2)^{p/(p+2)}$ where $p\ge 1$ and $λ\ge 0$, which exhibit an abrupt change of its spectral properties at a critical value of the coupling constant $λ$. We show that in the supercritical case the spectrum covers the whole real axis. In contrast, for $λ$ below the critical value the spectrum is purely discrete and we establish a Lieb-Thirring-type bound on its moments. In the critical case the essential spectrum covers the positive halfline while the negative spectrum can be only discrete, we demonstrate numeric…
The two main results of the article are concerned with Anderson Localization for one-dimensional lattice Schroedinger operators with quasi-periodic potentials with d frequencies. First, in the case d = 1 or 2, it is proved that the spectrum is pure-point with exponentially decaying eigenfunctions for all potentials (defined in terms of a trigonometric polynomial on the d-dimensional torus) for which the Lyapounov exponents are strictly positive for all frequencies and all energies. Second, for every non-constant real-analytic potential and with a Diophantine set of d frequencies, a lower bound…
The subject of this work are random Schroedinger operators on regular rooted tree graphs $\T$ with stochastically homogeneous disorder. The operators are of the form $H_λ(ω) = T + U + λV(ω)$ acting in $\ell^2(\T)$, with $T $ the adjacency matrix, $U$ a radially periodic potential, and $V(ω)$ a random potential. This includes the only class of homogeneously random operators for which it was proven that the spectrum of $H_λ(ω)$ exhibits an absolutely continuous (ac) component; a results established by A. Klein for weak disorder, in case U=0 and $V(ω)$ given by iid random variables on $\T$. Our m…
We consider two-dimensional Schroedinger equations with honeycomb potentials and slow time-periodic forcing of the form: $$iψ_t (t,x) = H^\varepsilon(t)ψ=\left(H^0+2i\varepsilon A (\varepsilon t) \cdot \nabla \right)ψ,\quad H^0=-Δ+V (x) .$$ The unforced Hamiltonian, $H^0$, is known to generically have Dirac (conical) points in its band spectrum. The evolution under $H^\varepsilon(t)$ of {\it band limited Dirac wave-packets} (spectrally localized near the Dirac point) is well-approximated on large time scales ($t\lesssim \varepsilon^{-2+}$) by an effective time-periodic Dirac equation with a ga…
We prove that the Floquet spectrum of a class of time-periodic Schroedinger equations under a a mildly nonlinear resonant forcing is purely absolutely continuous.
In this paper we consider stationary solutions to the nonlinear one-dimensional Schroedinger equation with a periodic potential and a Stark-type perturbation. In the limit of large periodic potential the Stark-Wannier ladders of the linear equation become a dense energy spectrum because a cascade of bifurcations of stationary solutions occurs when the ratio between the effective nonlinearity strength and the tilt of the external field increases.
Let $λ_i(Ω,V)$ be the $i$th eigenvalue of the Schrödinger operator with Dirichlet boundary conditions on a bounded domain $Ω\subset \R^n$ and with the positive potential $V$. Following the spirit of the Payne-Pólya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential $V_\star$, we prove that $λ_2(Ω,V) \le λ_2(S_1,V_\star)$. Here $S_1$ denotes the ball, centered at the origin, that satisfies the condition $λ_1(Ω,V) = λ_1(S_1,V_\star)$. Further we prove under the same convexity assumptions on a spherically symmetric potential $V$, that $λ_2(B_R, V) /…
For a class of discrete quasi-periodic Schroedinger operators defined by covariant re- presentations of the rotation algebra, a lower bound on phase-averaged transport in terms of the multifractal dimensions of the density of states is proven. This result is established under a Diophantine condition on the incommensuration parameter. The relevant class of operators is distinguished by invariance with respect to symmetry automorphisms of the rotation algebra. It includes the critical Harper (almost-Mathieu) operator. As a by-product, a new solution of the frame problem associated with Weyl-Heis…
This article is a review of results on the nonlinear Schroedinger / Gross-Pitaevskii equation (NLS / GP). Nonlinear bound states and aspects of their stability theory are discussed from variational and bifurcation perspectives. Nonlinear bound states, in the particular cases where the potential is a single-well, double-well potential and periodic potential, are discussed in detail. We then discuss particle-like dynamics of solitary wave solutions interacting with a potential over a large, but finite, time interval. Finally we turn to the very long time behavior of solutions. We focus on the im…
Bound states in soft quantum layers
本文提出基于曲面平行坐标的方法研究依赖到曲面距离的约束势下三维薛定谔算子，在曲面渐近平坦时给出本质谱位置估计，并证明当曲面在紧集外为严格正总高斯曲率的旋转曲面时算子具有无穷多个离散特征值。
We describe a broad class of bounded non-periodic potentials in one-dimensional stationary quantum mechanics having the same spectral properties as periodic potentials. The spectrum of the corresponding Schroedinger operator consists of a finite or infinite number of allowed bands separated by gaps. In this letter we consider the simplest class of potentials, whose spectra consist of an interval on the negative semiaxis and the entire positive axis. The potentials are reflectionless, and a particle with positive energy moves freely in both directions. The potential is constructed as a limit of…
Numerical Solutions of the spectral problem for arbitrary self-adjoint extensions of the 1D Schroedinger equation
提出一种数值算法，用于求解一维正则薛定谔算子任意自伴延拓的谱问题，该算法基于非局部边界有限元方法，并证明了收敛性，通过数值实验验证了边界条件扰动导致负基态绝对值增大和边界态局域化。
The article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co-compact free action of the integer lattice Z^n. It is known that a local perturbation of the operator might embed an eigenvalue into the continuous spectrum (a feature uncommon for periodic elliptic operators of second order). In all known constructions of such examples, the corresponding eigenfunction is compactly supported. One wonders whether this must always be the case. The paper answers this question affirmatively. What is more surpr…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。