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82
Convergence of Spectra of Mesoscopic Systems Collapsing Onto a Graph.
 J. Math. Anal. Appl
, 1999
"... Let M be a finite graph in the plane and M " be a domain that looks like the "fattened graph M (exact conditions on the domain are given). It is shown that the spectrum of the Neumann Laplacian on M " converges when " ! 0 to the spectrum of an ODE problem on M . Presence of an electromagnetic f ..."
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Cited by 55 (2 self)
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Let M be a finite graph in the plane and M " be a domain that looks like the "fattened graph M (exact conditions on the domain are given). It is shown that the spectrum of the Neumann Laplacian on M " converges when " ! 0 to the spectrum of an ODE problem on M . Presence of an electromagnetic field is also allowed. Considerations of this kind arise naturally in mesoscopic physics and other areas of physics and chemistry. The results of the paper extend the ones previously obtained by J. Rubinstein and M. Schatzman. 2000 MSC: 35Q40, 35P15, 35J10, 81V99 Key words and phrases: mesoscopic system, Schrodinger operator, spectrum 1 Introduction In recent years one has witnessed growing interest in spectral theory of differential (versus difference) operators on graphs. Although probably one of the first such studies was done in physical chemistry [47], the main thrust 1 in this direction came from the mesoscopic physics [29]. Recent progress in nanotechnology and microelectronics en...
Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs
 J. Phys. A: Math. Gen
"... The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics, and other areas. A Schnol type theorem is proven that allows one ..."
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Cited by 45 (5 self)
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The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics, and other areas. A Schnol type theorem is proven that allows one to detect that a point λ belongs to the spectrum when a generalized eigenfunction with an subexponential growth integral estimate is available. A theorem on spectral gap opening for “decorated ” quantum graphs is established (its analog is known for the combinatorial case). It is also shown that if a periodic combinatorial or quantum graph has a point spectrum, it is generated by compactly supported eigenfunctions (“scars”). 1
Internal Lifshits Tails For Random Perturbations Of Periodic Schrödinger Operators
"... . Let H be a \Gammaperiodic Schrodinger operator acting on L 2 (R d ) and consider the random Schrodinger operator H! = H + V! where V! (x) = X fl2\Gamma ! fl V (x \Gamma fl) (here V is a positive potential and (! fl ) fl2\Gamma a collection of positive i.i.d random variables). We prove tha ..."
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Cited by 42 (7 self)
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. Let H be a \Gammaperiodic Schrodinger operator acting on L 2 (R d ) and consider the random Schrodinger operator H! = H + V! where V! (x) = X fl2\Gamma ! fl V (x \Gamma fl) (here V is a positive potential and (! fl ) fl2\Gamma a collection of positive i.i.d random variables). We prove that, at the edge of a gap of H that is not filled in for H! , the integrated density of states of H! has a Lifshits tail behaviour if and only if the integrated density of states of H is nondegenerate. R' esum' e. Soient H un op'erateur de Schrodinger \Gammap'eriodique agissant sur L 2 (R d ), V un potentiel positif et (! fl ) fl2Z d une famille de variables al'eatoires i.i.d positives. Consid'erons l'op'erateur de Schrodinger al'eatoire H! = H + V! o`u V! (x) = X fl2Z d ! fl V (x \Gamma fl). On montre que, au bord d'une lacune spectrale de H qui n'est pas combl'e pour H! la densit'e d"etats int'egr'ee de H! a un comportement asymptotique de Lifshits si et seulement si la densit'e d"...
Localization of Classical Waves I: Acoustic Waves.
 Commun. Math. Phys
, 1996
"... We consider classical acoustic waves in a medium described by a position dependent mass density %(x). We assume that %(x) is a random perturbation of a periodic function % 0 (x) and that the periodic acoustic operator A 0 = \Gammar \Delta 1 %0 (x) r has a gap in the spectrum. We prove the existe ..."
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Cited by 37 (0 self)
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We consider classical acoustic waves in a medium described by a position dependent mass density %(x). We assume that %(x) is a random perturbation of a periodic function % 0 (x) and that the periodic acoustic operator A 0 = \Gammar \Delta 1 %0 (x) r has a gap in the spectrum. We prove the existence of localized waves, i.e., finite energy solutions of the acoustic equations with the property that almost all of the wave's energy remains in a fixed bounded region of space at all times, with probability one. Localization of acoustic waves is a consequence of Anderson localization for the selfadjoint operators A = \Gammar \Delta 1 %(x) r on L 2 (R d ). We prove that, in the random medium described by %(x), the random operator A exhibits Anderson localization inside the gap in the spectrum of A 0 . This is shown even in situations when the gap is totally filled by the spectrum of the random operator; we can prescribe random environments that ensure localization in almost the wh...
The Weierstraß representation of closed surfaces in R 3
 Tr1] [Tr2] A. Trautman. Spinors and the Dirac operator on hypersurfaces I. General
, 1997
"... The present article is a sequel to [19, 20]. The results presented here extend onto general surfaces the results obtained in [20] for surfaces of revolution and were exposed in a lot of talks of the author during the last year being at the end were ..."
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Cited by 35 (2 self)
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The present article is a sequel to [19, 20]. The results presented here extend onto general surfaces the results obtained in [20] for surfaces of revolution and were exposed in a lot of talks of the author during the last year being at the end were
BandGap Structure Of Spectra Of Periodic Dielectric And Acoustic Media. I. Scalar Model
 I. Scalar model, SIAM J. Appl. Math
, 1996
"... . We investigate the bandgap structure of the spectrum of secondorder partial differential operators associated with the propagation of waves in a periodic twocomponent medium. The medium is characterized by a realvalued positiondependent periodic function "(x) that is the dielectric constant f ..."
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Cited by 25 (5 self)
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. We investigate the bandgap structure of the spectrum of secondorder partial differential operators associated with the propagation of waves in a periodic twocomponent medium. The medium is characterized by a realvalued positiondependent periodic function "(x) that is the dielectric constant for electromagnetic waves and mass density for acoustic waves. The imbedded component consists of a periodic lattice of cubes where "(x) = 1. The value of "(x) on the background is assumed to be greater than 1. We give the complete proof of existence of gaps in the spectra of the corresponding operators provided some simple conditions imposed on the parameters of the medium. Key words: propagation of electromagnetic and acoustic waves, bandgap structure of the spectrum, periodic dielectrics, periodic acoustic media. AMS subject classification. 35B27, 73D25, 78A45. 1. INTRODUCTION. One of the main observations in the quantum theory of solids is that the energy spectrum of an electron in a ...
On the Stability of Periodic Travelling Waves With Large Spatial Period
, 1999
"... In many circumstances, a pulse to a partial differential equation (PDE) on the real line is accompanied by periodic wave trains that have arbitrarily large period. It is then interesting to investigate the PDE stability of the periodic wave trains given that the pulse is stable. Using the Evans func ..."
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Cited by 23 (15 self)
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In many circumstances, a pulse to a partial differential equation (PDE) on the real line is accompanied by periodic wave trains that have arbitrarily large period. It is then interesting to investigate the PDE stability of the periodic wave trains given that the pulse is stable. Using the Evans function, Gardner has demonstrated that every isolated eigenvalue of the linearization about the pulse generates a small circle of eigenvalues for the linearization about the periodic waves. In this article, the precise location of these circles is determined. It is demonstrated that the stability properties of the periodic waves depend on certain decay and oscillation properties of the tails of the pulse. As a consequence, periodic waves with long wavelength typically destabilize at homoclinic bifurcation points at which multihump pulses are created. That is in contrast to the situation for the underlying pulses whose stability properties are not affected by these bifurcations. The proof uses LyapunovSchmidt reduction and relies on the existence of exponential dichotomies. The approach is also applicable to periodic waves with large spatial period of elliptic problems on R or on unbounded cylinders R with bounded.
Absolute Continuity Of The Periodic Magnetic Schrödinger Operator
 Invent. Math
, 1997
"... . We prove that the spectrum of the Schrodinger operator with periodic electric and magnetic potentials is absolutely continuous 1. Introduction Consider on L 2 (R d ); d 1; a selfadjoint differential operator A(x; D), D = \Gammair with coefficients periodic with respect to a lattice \Gamma. ..."
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Cited by 22 (0 self)
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. We prove that the spectrum of the Schrodinger operator with periodic electric and magnetic potentials is absolutely continuous 1. Introduction Consider on L 2 (R d ); d 1; a selfadjoint differential operator A(x; D), D = \Gammair with coefficients periodic with respect to a lattice \Gamma. The spectral analysis of this operator is based on the socalled Floquet decomposition (see for example [7] and [9]). This means that A(x; D) can be represented as a direct integral of the operator family A k = A(x; D+ k) operating on functions defined on the torus T = R d =\Gamma, in the quasimomentum k over the unit cell ~\Omega of the dual lattice. Under the assumption that each A k has a compact resolvent, the spectrum of the initial operator A(x; D) is given by the union of spectral bands I l = [ k2 ~\Omega E l (k); the functions E l (k) being the discrete eigenvalues of A k . If one of the functions E l is constant in k, then the associated band I l degenerates into a point, wh...
Localization of Classical Waves II: Electromagnetic Waves.
 Commun. Math. Phys
, 1997
"... We consider electromagnetic waves in a medium described by a position dependent dielectric constant "(x). We assume that "(x) is a random perturbation of a periodic function " 0 (x) and that the periodic Maxwell operator M 0 = r \Theta 1 " 0 (x) r \Theta has a gap in the spectrum, were r \Thet ..."
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Cited by 20 (0 self)
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We consider electromagnetic waves in a medium described by a position dependent dielectric constant "(x). We assume that "(x) is a random perturbation of a periodic function " 0 (x) and that the periodic Maxwell operator M 0 = r \Theta 1 " 0 (x) r \Theta has a gap in the spectrum, were r \Theta \Psi = r\Theta\Psi. We prove the existence of localized waves, i.e., finite energy solutions of Maxwell's equations with the property that almost all of the wave's energy remains in a fixed bounded region of space at all times. Localization of electromagnetic waves is a consequence of Anderson localization for the selfadjoint operators M = r \Theta 1 "(x) r \Theta . We prove that, in the random medium described by "(x), the random operator M exhibits Anderson localization inside the gap in the spectrum of M 0 . This is shown even in situations when the gap is totally filled by the spectrum of the random operator; we can prescribe random environments that ensure localization in almo...