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The Discrete Spectrum in the Gaps of the Continuous One for Non-Signdefinite Perturbations with a Large Coupling Constant
Abstract
Given two selfadjoint operators A and V=V
+ -V
-, we study the motion of the eigenvalues of the operator A(t)=A-tV as t increases. Let α>0 and let λ be a regular point for A. We consider the quantities N
+(λ,α), N
-(λ,α), N
0(λ,α) defined as the number of the eigenvalues of the operator A(t) that pass point λ from the right to the left, from the left to the right or change the direction of their motion exactly at point λ, respectively, as t increases from 0 to α>0. An abstract theorem on the asymptotics for these quantities is presented. Applications to Schrödinger operators and its generalizations are given.
... The study of the eigenvalues of Schrödinger operators below the essential spectrum goes back over fifty years to Bargmann [5], Birman [6], and Schwinger [43], and of power bounds on the eigenvalues to Lieb–Thirring [35, 36]. There has been considerably less work on eigenvalues in gaps—much of what has been studied followed up on seminal work by Deift and Hempel [23]; see [1, 2, 25, 26, 27, 28, 29, 32, 33, 40, 41, 42] and especially work by Birman and collaborators [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. Following Deift–Hempel, this work has mainly focused on the set of λ's so that some given fixed e in a gap of σ(A) is an eigenvalue of A + λB and the growth of the number of eigenvalues as λ → ∞ most often for closed intervals strictly inside the gap. ...
... Our goal here is to prove Theorem 1.1. We begin by recalling the version of the Birman–Schwinger principle for points in gaps, which is essentially the key to [1, 2, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 23, 25, 26, 27, 28, 29, 32, 33, 40, 41, 42]: Proposition 2.1. Let A be a bounded selfadjoint operator with (x, y) ∩ σ(A) = ∅. ...
... We abuse notation and write compact operators In the bounded case, we only considered intervals in the lower half of a gap since A → −A, B → −B flips half-intervals. But, as has been noted in the unbounded case (see, e.g., [11, 40]), there is now an asymmetry, so we will state separate results. We start with the bottom half case: Theorem 3.1. ...
We consider C=A+B where A is selfadjoint with a gap (a,b) in its
spectrum and B is (relatively) compact. We prove a general result allowing
B of indefinite sign and apply it to obtain a bound for
perturbations of suitable periodic Schrodinger operators and a (not
quite)Lieb-Thirring bound for perturbations of algebro-geometric almost
periodic Jacobi matrices.
... In this paper we supplement the results obtained in [15]. We remind the reader of the setting of the problem. ...
... In applications these relationships often allow us to apply already known results about asymptotics for N V λ α . Finally note that for asymptotics of order α p with p > 1, the relationships (1) and (2) were obtained in [15]. Now we extend the results obtained in [15] to the case 0 < p ≤ 1. ...
... Finally note that for asymptotics of order α p with p > 1, the relationships (1) and (2) were obtained in [15]. Now we extend the results obtained in [15] to the case 0 < p ≤ 1. In applications to Schrödinger operators this corresponds to the case of one and two dimensions. ...
. Given two selfadjoint operators A and V = V+ V , we study the motion of the eigenvalues of the operator A(t) = A tV as t increases. Let > 0 and let be a regular point for A. We consider the quantities N+ (V ; ; ); N (V ; ; ); N 0 (V ; ; ) dened as the number of the eigenvalues of the operator A(t) that pass point from the right to the left, from the left to the right or change the direction of their motion exactly at point , respectively, as t increases from 0 to > 0: We study asymptotic characteristics of these quantities as !1: In the present paper we extend the results obtained in [16] and give new applications to dierential operators. 0. Introduction 1. In this paper we supplement the results obtained in [16]. We remind the setting of the problem. Let A be a selfadjoint, semibounded from below operator in a Hilbert space H, let V be a selfadjoint nonsignde nite perturbation. Assume that the quadratic form of the operator V is compact with respect to the f...
... The asymptotics of sf(λ; M + tA, M) as t → ∞ and related issues have been extensively studied both for concrete differential operators M + tA and in an abstract setting; see e.g. the survey [16] for the history and a recent paper [17] for extensive bibliography. Most relevant to our approach are the operator theoretic constructions of M. Birman (see [7] and references therein) and O. Safronov [25,26,27]. We also note that there is a large family of index theorems (see e.g. ...
... The key estimates. Here we state and prove a result (see (4.10), (4.11) below) which is a slight generalisation of [26] (see also [25]). Let H 0 be a lower semi-bounded self-adjoint operator in H; choose γ ∈ R such that H 0 + γI I. Let V + 0 and V − 0 be selfadjoint operators in H which are form-compact with respect to H 0 . ...
... Letting a → 0, we obtain The estimates (4.10), (4.11) were obtained in [26] (see also [25]) by a different method. These estimates were then applied in [25,26] to various cases when H 0 is a differential operator and V is the operator of multiplication by a function from an appropriate L q class. ...
Using the notion of spectral flow, we suggest a simple approach to various
asymptotic problems involving eigenvalues in the gaps of the essential spectrum
of self-adjoint operators. Our approach uses some elements of the spectral
shift function theory. Using this approach, we provide generalisations and
streamlined proofs of two results in this area already existing in the
literature. We also give a new proof of the generalised Birman-Schwinger
principle.
... 1) There are a lot of papers about the discrete spectrum in a gap of the Schrödinger operator with a periodic potential perturbed by a decaying potential, see [10,11,43] and the references therein. 2) Homogenization theory has been used to study periodic elliptic operators near spectral band edges, see [3,4,6,9,12,24] and the references therein. ...
We consider Laplacians on periodic both discrete and metric equilateral
graphs. Their spectrum consists of an absolutely continuous part (which is a
union of non-degenerate spectral bands) and flat bands, i.e., eigenvalues of
infinite multiplicity. We estimate effective masses associated with the ends of
each spectral band in terms of geometric parameters of graphs. Moreover, in the
case of the beginning of the spectrum we determine two sided estimates of the
effective mass in terms of geometric parameters of graphs. The proof is based
on the Floquet theory, the factorization of fiber operators, the perturbation
theory and the relation between effective masses for Laplacians on discrete and
metric graphs, obtained in our paper.
... Remark 2. The signs ascribed to ITEs in Theorem 1.1 resemble the standard procedure used in the definition of spectral flows (see, for example, [1], [26], [32]). The principal difference is that this sign depends not on the direction of motion of ITEs, but the direction of motion of another object: the eigenvalues of the scattering matrix related to the ITEs. ...
We consider the interior transmission eigenvalue (ITE) problem, which arises
when scattering by inhomogeneous media is studied. The ITE problem is not
self-adjoint. We show that positive ITEs are observable together with plus or
minus signs that are defined by the direction of motion of the corresponding
eigenvalues of the scattering matrix (when the latter approach {\bfz=1)}. We
obtain a Weyl type formula for the counting function of positive ITEs, which
are taken together with ascribed signs.
... It is described, for example, in [17]. Here is another main result of this paper: One part of our proof involves the general theory of eigenvalues in gaps, a subject with considerable literature (see [1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 23, 25, 26, 27, 28, 33, 35, 38, 45, 46, 47, 50, 51]). We will find a general Birman–Schwinger-type bound that could also be used to simplify many of these earlier works. ...
We prove bounds of the form \sum_{e\in I\cap\sigma_\di (H)} \dist
(e,\sigma_\e (H))^{1/2} \leq L^1-norm of a perturbation, where I is a gap.
Included are gaps in continuum one-dimensional periodic Schr\"odinger operators
and finite gap Jacobi matrices where we get a generalized Nevai conjecture
about an condition implying a Szeg\H{o} condition. One key is a general
new form of the Birman--Schwinger bound in gaps.
... (0.9) 1 We write p V in (0.4) in order to make our notation coherent with that of 3, 27]. 2 De nition (0.7) is correct due to the analyticity of n (H 0 ? V ) in | see 27] for the details. 3 The eigenvalues which \turn" at the point , do not enter the expression (0.7) | see 27,28]. ...
We consider the spectral shift function (; H 0 Gamma ffV; H 0 ), where H 0 is a Schrodinger operator with a variable Riemannian metric and an electro-magnetic field and V is a perturbation by a multiplication operator. We prove the Weyl type asymptotic formula for (; H 0 Gamma ffV ; H 0 ) in the large coupling constant limit ff !1. 0 Introduction 0.1 Let H 0 = Gamma4 in L 2 (R d ), d 1 and let V = V (x), x 2 R d , be a perturbation potential which decays rapidly enough as jxj !1. For a coupling constant ff ? 0 and a spectral parameter ! 0 denote N(; ff) = #fn 2 N j n (H 0 Gamma ffV ) ! g; (0.1) where n (H 0 Gamma ffV ) are the eigenvalues of H 0 Gamma ffV , numbered with the multiplicities taken into account. Under certain restrictions on V (which are different for d = 1, d = 2 and d 3), the following Weyl type asymptotic formula is valid (see, e.g., [24, 4] and references therein): lim ff!1 ff Gammad=2 N(; ff) = (2) Gammad ! d Z V d=2 + (x)dx; ! 0; (0.2) w...
The analysis of eigenvalues of Laplace and Schrödinger operators is an important and classical topic in mathematical physics with many applications. This book presents a thorough introduction to the area, suitable for masters and graduate students, and includes an ample amount of background material on the spectral theory of linear operators in Hilbert spaces and on Sobolev space theory. Of particular interest is a family of inequalities by Lieb and Thirring on eigenvalues of Schrödinger operators, which they used in their proof of stability of matter. The final part of this book is devoted to the active research on sharp constants in these inequalities and contains state-of-the-art results, serving as a reference for experts and as a starting point for further research.
Given two selfadjoint operators A and V = V+ − V−, we study the motion of the eigenvalues of the operator A(t) = A − tV as t increases. Let α > 0 and let λ be a regular point for A. We consider the quantity N(λ, A, W+, W−, α) defined as the difference between the number of the eigenvalues of A(t) that pass the point λ from right to left and the number of the eigenvalues passing λ from left to right as t increases from 0 to α. We study the asymptotic behavior of N(λ, A, W+, W−, α) as α → ∞. Applications to Schrödinger and Dirac operators are given.
In the large-coupling constant limit we obtain an asymptotic expansion in powers of
m - \frac1 d \mu ^{{ - \frac{1} {\delta }}}
of the derivative of the spectral shift function corresponding to
( - D+ mW(x), - D ) {\left( { - \Delta + \mu W(x), - \Delta } \right)}
. Here the potential W(x) is positive and
W(x) ~ w0 (\fracx |x|)|x| - d W(x) \sim \omega _{0} (\frac{x} {{|x|}})|x|^{{ - \delta }}
near infinity for some δ> n and ω0 ∈C∞ (Sn-1).
In this paper we consider a one-dimensional Schrödinger eigenvalue problem with a potential consisting of a periodic part together with a compactly supported defect potential. Such problems arise as models in condensed matter to describe color in crystals as well as in engineering to describe optical photonic structures. We are interested in studying the existence of point eigenvalues in gaps in the essential spectrum, and in particular in counting the number of such eigenvalues. We use a homotopy argument in the width of the potential to count the eigenvalues as they arecreated, essentially by computing a Maslov index. As a consequence of this we prove the following significant generalization of Zheludev's theorem: the number of point eigenvalues in a gap in theessential spectrum is exactly 1 for a sufficiently large gap number unless a certain Diophantineapproximation problem has solutions, in which case there exists a subsequence of gaps containing0, 1, or 2 eigenvalues. We state some conditions under which the solvability of the Diophantineapproximation problem can be established.
In this paper we consider a Schrodinger eigenvalue problem with a potential consisting of a periodic part together with a compactly supported defect potential. Such problems arise as models in condensed matter to describe color in crystals as well as in engineering to describe optical photonic structures. We are interested in studying the existence of point eigenvalues in gaps in the essential spectrum, and in particular in counting the number of such eigenvalues. We use a homotopy argument in the width of the potential to count the eigenvalues as they are created. As a consequence of this we prove the following significant generalization of Zheludev's theorem: the number of point eigenvalues in a gap in the essential spectrum is exactly one for sufficiently large gap number unless a certain Diophantine approximation problem has solutions, in which case there exists a subsequence of gaps containing 0,1 or 2 eigenvalues. We state some conditions under which the solvability of the Diophantine approximation problem can be established.
Introduction. Eigenvalue estimates for operators of mathematical physics play an important part in the study of asymptotic properties of the discrete spectrum, scattering theory, stability of matter etc. Among such results, estimates having the semiclassical order in the coupling constant are of special interest. The best known example here is the CLR-estimate for the Schrodinger operator H(V ) = GammaDelta Gamma V with the real potential V 2 L 1;loc : N 0 (H(V )) C(d) Z R d V+ (x) d 2 dx; d 3: (1) Here and below we denote by N fi (H) the number of eigenvalues of a selfadjoint operator H<F14
. Given two selfadjoint operators H 0 and V = V+ Gamma V Gamma , we study the motion of the spectrum of the operator H(ff) = H 0 + ffV as ff increases. Let be a real number. We consider the quantity (; H(ff); H 0 ) defined as a generalization of Krein's spectral shift function of the pair H(ff); H 0 . We study the asymptotic behavior of (; H(ff); H 0 ) as ff ! 1: Applications to differential operators are given. 0. Introduction Let H 0 = H 0 be a selfadjoint operator and let V = V be of the form V = V+ Gamma V Gamma : (1) We put H(ff) = H 0 + ffV; ff ? 0 and denote by (; ff) = (; H(ff); H 0 ) the "generalized spectral shift function" of the pair H(ff); H 0 which coincides with the Kreins spectral shift function if V is of trace class. For trace class operators V the function (Delta; ff) 2 L 1 can be defined by the relation Tr[OE(H(ff)) Gamma OE(H 0 )] = Z +1 Gamma1 (; ff)OE 0 ()d; called the trace formula. ( Here OE is an arbitrary function on R of a suitable clas...
We consider eigenvaluesEλ of the HamiltonianHλ=−Δ+V+λW,W compactly supported, in the λ→∞ limit. ForW≧0 we find monotonic convergence ofEλ to the eigenvalues of a limiting operatorH∞ (associated with an exterior Dirichlet problem), and we estimate the rate of convergence for 1-dimensional systems. In 1-dimensional systems withW≦0, or withW changing sign, we do not find convergence. Instead, we find a cascade phenomenon, in which, as λ→∞, each eigenvalueEλ stays near a Dirichlet eigenvalue for a long interval (of lengthO(
)) of the scaling range, quickly drops to the next lower Dirichlet eigenvalue, stays there for a long interval, drops again, and so on. As a result, for most large values of λ the discrete spectrum ofHλ is close to that ofE∞, but when λ reaches a transition region, the entire spectrum quickly shifts down by one. We also explore the behavior of several explicit models, as λ→∞.
We provide an alternative proof of the main result of Deift and Hempel [1] on the existence of eigenvalues ofv-dimensional Schrödinger operatorsH
λ=H
0+λW in spectral gaps ofH
0.
The authors prove a number of results asserting the existence of eigenvalues of the Schrödinger operatorH−λW in a gap of σ(H), whereH=−Δ+V andV andW are bounded. The existence of these eigenvalues is an important element in the theory of the colour of crystals. The basic theorems are proved in ℝv
; stronger results forv=1 are also presented.
The authors study the eigenvalue branches of the Schrödinger operatorH — λW in a gap of σ(H). In particular, they consider questions of asymptotic distribution of eigenvalues and bounds on the number of branches. They also address the completeness problem.
We use Brownian motion ideas to study Schrödinger operators H = built-1 2Δ + V on Lp(Rv). In particular: (a) We prove that limt→∞ t-1In ∥ e-tH ∥p,p is p-independent for a very large class of V's where ∥ A ∥p,p = norm of A as an operator from Lpto Lp. (b) For v ≥ 3 and V ε{lunate} Lv 2 - ε{lunate} ∩ Lv 2 + ε{lunate}, we show that sup ∥ e-tH ∥∞,∞ < ∞ if and only if H has no negative eigenvalues or zero energy resonances. (c) We relate the "localization of binding" recently noted by Sigal to Brownian hitting probabilities.
Continuing earlier investigations we show the existence of eigenvalues of Schrödinger operators H − λW, λ ϵ , inside a spectral gap of H = −Δ + V, under suitable conditions on V and W. Our method of proof is based on a decoupling of regions in v by means of Dirichlet or Neumann boundary conditions, where, in particular, the use of Neumann boundary conditions in the decoupling process leads to a new asymptotic results for eigenvalue counting functions.
Traducción del ruso Incluye bibliografía e índice
On the asymptotic distribution of eigenvalues in gaps In: Proceedings of the IMA-workshop on quasiclassical methods
- R Hempel
Hempel, R.: On the asymptotic distribution of eigenvalues in gaps. In: Proceedings of the IMA-workshop on quasiclassical methods, May 1995-IMA Vl in Math. and its Appl. Berlin–Heidelberg–New York: Springer-Verlag, to appear