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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 83 (7 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
Spectral analysis on infinite Sierpinski gaskets, preprint
 198 JUN KIGAMI
"... We study the spectral properties of the Laplacian on infinite Sierpin ski gaskets. We prove that the Laplacian with the Neumann boundary condition has pure point spectrum. Moreover, the set of eigenfunctions with compact support is complete. The same is true if the infinite Sierpin ski gasket has ..."
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Cited by 61 (15 self)
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We study the spectral properties of the Laplacian on infinite Sierpin ski gaskets. We prove that the Laplacian with the Neumann boundary condition has pure point spectrum. Moreover, the set of eigenfunctions with compact support is complete. The same is true if the infinite Sierpin ski gasket has no boundary, but is false for the Laplacian with the Dirichlet boundary condition. In all these cases we describe the spectrum of the Laplacian and all the eigenfunctions with compact support. To obtain these results, first we prove certain new formulas for eigenprojectors of the Laplacian on finite Sierpin ski pregaskets. Then we prove that the spectrum of the discrete Laplacian on a Sierpin ski lattice is pure point, and the eigenfunctions are localized. 1998 Academic Press Key Words: fractals; fractal graphs; Laplacian; pure point spectrum; localization. 1.
Convergence of spectra of mesoscopic systems collapsing onto a graph
 J. Math. Anal. Appl
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Internal Lifshits Tails For Random Perturbations Of Periodic Schrödinger Operators
 Duke Mathematical J
, 1999
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The Weierstraß representation of closed surfaces in R³
 TR1] [TR2] A. TRAUTMAN. SPINORS AND THE DIRAC OPERATOR ON HYPERSURFACES I. GENERAL
, 1997
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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 44 (1 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 integrated density of states for random Schrödinger operators
"... We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is discussed i ..."
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Cited by 43 (3 self)
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We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is discussed in the second section. One central topic of this survey is the asymptotic behavior of the integrated density of states at the boundary of the spectrum. In particular, we are interested in Lifshitz tails and the occurrence of a classical and a quantum regime. In the last section we discuss regularity properties of the IDS. Our emphasis is on the discussion of fundamental problems and central ideas to handle them. Finally, we discuss further developments and problems
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 const ..."
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Cited by 39 (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 31 (18 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.