## ON THE SPECTRUM OF THE DIRICHLET LAPLACIAN IN A NARROW STRIP, (705)

### BibTeX

@MISC{Friedl705onthe,

author = {Leonid Friedl and Michael Solomyak},

title = {ON THE SPECTRUM OF THE DIRICHLET LAPLACIAN IN A NARROW STRIP,},

year = {705}

}

### OpenURL

### Abstract

There are several reasons why the study of the spectrum of the Laplacian in a narrow neighborhood of an embedded graph is interesting. The graph can be embedded into a Euclidean space or it can be embedded into a manifold. In his pioneering work [3], Colin de Verdière

### Citations

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Perturbation theory for linear operators. Second edition. Grundlehren der Mathematischen Wissenschaften, Band 132
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Citation Context ...tials V such that V (x) ≥ V0(x) on R. Proof. 1◦ Under the assumptions of proposition the strong convergence → Z−1 is well known. For instance, it follows from theorem Z −1 V,Iǫ V VIII.1.5 in the book =-=[5]-=-. Its assumptions are evidently satisfied if we take into account that C∞ 0 (R) is a core for the operator ZV . For each ǫ we have dV,Iǫ ⊂ dV and zV,Iǫ[u] = zV [u] for every u ∈ dV,ǫ. By the definitio... |

98 |
Spectral theory of self-adjoint operators in Hilbert space
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(Show Context)
Citation Context ...account that C∞ 0 (R) is a core for the operator ZV . For each ǫ we have dV,Iǫ ⊂ dV and zV,Iǫ[u] = zV [u] for every u ∈ dV,ǫ. By the definition of inequalities between self-adjoit operators (see e.g. =-=[1]-=-, section 10.2.3), this means that ZV,Iǫ ≥ ZV and, by theorem 10.2.6 from [1], Z −1 ≤ Z−1 V,Iǫ V . Since V (x) → ∞ as |x| → ∞, the operator Z −1 V is compact. Now, we get the statement 1 ◦ by applying... |

46 | Variational problems on multiply connected thin strips I: Basic estimates and convergence of the Laplacian spectrum, Archive for Rational Mechanics and Analysis
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Citation Context ...ositive eigenvalue of the Laplacian on (M, g) equals N. Recent interest to the problem is, in particular, motivated by possible applications to mesoscopic systems. Rubinstein and Schatzman studied in =-=[10]-=- eigenvalues of the Neumann Laplacian in a narrow strip surrounding an embedded planar graph. The strip has constant width ǫ everywhere except neighborhoods of vertices. Under some assumptions on the ... |

46 |
Trace Ideals and Their
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(Show Context)
Citation Context ..., this means that ZV,Iǫ ≥ ZV and, by theorem 10.2.6 from [1], Z −1 ≤ Z−1 V,Iǫ V . Since V (x) → ∞ as |x| → ∞, the operator Z −1 V is compact. Now, we get the statement 1 ◦ by applying theorem 2.16 in =-=[11]-=- (which is an analogue of the classical Lebesgue theorem on dominated convergence). In particular, the theorem says that if Tǫ, 0 < ǫ < ǫ0 is a family of compact, self-adjoint operators such that Tǫ →... |

27 |
The essential spectrum of Schrödinger
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Citation Context ...hes an isospectral family in the norm topology. A similar effect, in a more complicated problem of the behavior of the essential spectra of certain operator families, was studied by Last and Simon in =-=[7]-=-. We will show now that theorems 1.2 and 1.3 imply theorem 1.1. Indeed, the non-zero eigenvalues of the operator Q−1 ǫ ⊕0 are the same as those of Q−1 ǫ . By theorem 1.2 we have for all j ∈ N and ǫ < ... |

27 | Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case
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(Show Context)
Citation Context ... is not constant. The Dirichlet boundary condition turns out to be more complicated than the Neumann condition. Eigenvalues of a domain of width ǫ are bounded from below by π 2 /ǫ 2 . Post studied in =-=[9]-=- eigenvalues λj(ǫ) of the Dirichlet Laplacian in a neighborhood of a planar graph that has constant width ǫ near the edges and that narrows down toward the vertices. He proved that λj(ǫ) − π 2 /ǫ 2 co... |

3 |
Double operator integrals in Hilbert space, Int. Equat. Oper. Theory 47 (2003), 131–168. 49 Yu.B. Farforovskaya, The connection of the Kantorovich-Rubinshtein metric for spectral resolutions of selfadjoint operators with functions of operators, Vestnik
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- 1968
(Show Context)
Citation Context ...al Theorem that φ(S) = (·, e)e; φ(T) = (·, f)f, and therefore the operator K can be represented as K = φ(S) − φ(T). This representation allows us to apply the theory of double operator integrals (see =-=[2]-=-, and especially section 8 therein.) In particular, we conclude from theorems 8.1 and 8.3 that ‖K‖ ≤ C‖S − T‖, C = C(φ). Since rankK ≤ 2, we conclude that also (5.4) ‖K‖HS ≤ √ 2 ‖K‖ ≤ C √ 2 ‖S − T‖.1... |

3 |
The size of the first eigenfunction of a convex planar domain
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Citation Context ... formula for Pǫ. Similarly, theorem 1.3 yields ∫ ∣ (1.11) ∣ ∣ I ˜ Ψj,ǫ(x) − ǫ −α/2 Xj(xǫ −α ) ∣ 2 dx → 0. where Xj is the j-s normalized eigenfunction of the operator H. Grieser and Jerison proved in =-=[4]-=- much stronger an estimate for the first eigenfunction in a convex, narrow domain (in our setting, the function h(x) is concave.) Similar problems are discussed in a survey paper [8] by Nazarov. In th... |

1 |
de Verdière, Sur la mutiplicité de la première valeare propre non nulle du laplacien
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(Show Context)
Citation Context ... spectrum of the Laplacian in a narrow neighborhood of an embedded graph is interesting. The graph can be embedded into a Euclidean space or it can be embedded into a manifold. In his pioneering work =-=[3]-=-, Colin de Verdière used Riemannian metrics concentrated in a small neighborhood of a graph to prove that for every manifold M of dimension greater than two and for every positive number N there exist... |

1 |
Localization effects for eigenfunctions near to the edge of a thin domain, Mathematica Bohemica 127
- Nazarov
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(Show Context)
Citation Context ...Jerison proved in [4] much stronger an estimate for the first eigenfunction in a convex, narrow domain (in our setting, the function h(x) is concave.) Similar problems are discussed in a survey paper =-=[8]-=- by Nazarov. In the next three sections we prove theorems 1.2 and 1.3. In section 5 we explain the derivation of the inequalities (1.10) and (1.11), and in the last section 6 we describe possible exte... |