On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators

We provide a comprehensive analysis of matrix–valued Herglotz functions and illustrate their applications in the spectral theory of self–adjoint Hamiltonian systems including matrix–valued Schrodinger and Dirac–type operators. Special emphasis is devoted to appropriate matrix–valued extensions of the well–known Aronszajn–Donoghue theory concerning support properties of measures in their Nevanlinna–Riesz–Herglotz representation. In particular, we study a class of linear fractional transformations MA(z) of a given n × n Herglotz matrix M(z) and prove that the minimal support of the absolutely continuous part of the measure associated to MA(z) is invariant under these linear fractional transformations.



Additional applications discussed in detail include self–adjoint finite–rank perturbations of self–adjoint operators, self–adjoint extensions of densely defined symmetric linear operators (especially, Friedrichs and Krein extensions), model operators for these two cases, and associated realization theorems for certain classes of Herglotz matrices.

On Matrix–Valued Herglotz Functions

For a large class ofN-body potentialsV we prove that if ϕ is an eigenfunction of −δ+V with eigenvalueE then sup{α2+E:α≧0, exp(α|x|)ϕ∈L2} is either a threshold or +∞. Consequences of this result are the absence of positive eigenvalues and “optimal”L2-exponential lower bounds.

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Exponential bounds and absence of positive eigenvalues for $N$-body Schrödinger operators

Using control of the growth of the transfer matrices, wediscuss the spectral analysis of continuum and discrete half-line Schrodinger operators with slowly decaying potentials. Among our results we show if , where W has compact support and , then H has purely a.c. (resp. purely s.c.) spectrum on (0,∞) if ). For λn

 {-1/2}
 a

 n
 potentials, where a

 n
 are independent, identically distributed random variables with E(a

 n
 ) = 0, E(a

 2

 n)=1, and λ < 2, we find singular continuous spectrum with explicitly computable fractional Hausdorff dimension.

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Modified Prüfer and EFGP Transforms and the Spectral Analysis of One-Dimensional Schrödinger Operators

A photonic crystal, or photonic band gap (PBG) optical material, is an artificially created periodic low-loss dielectric medium in which electromagnetic waves of certain frequencies cannot propagate The range of the prohibited frequencies is called the complete band gap A simple example of such a medium is a dielectric background material with a periodic array of air bubbles The reason why the band gap arises (if it does) is the coherent multiple scattering of waves and destructive interference To put it simply, if a wave of a prohibited frequency somehow managed to propagate in the medium, it would reflect and self-interfere in such a way that it would cancel itself completely It is expected that industrial manufacturing of photonic crystals will bring about a new technological revolution in optics, computing, information transmission, and other areas The idea of photonic crystals was coined in [107, 192], though simpler versions of such materials like layered media and optical gratings have been known for a long time We will not dwell much on the physics aspects of this field of research, since the reader can refer to the surveys and proceedings [33, 101, 108, 134, 154, 156, 167, 168, 182, 189] devoted to this topic, and especially to the lovely book [106] The bibliography [153] and the collection of photonic crystal links [190] are also very useful The area of photonic crystal research presents a bonanza of beautiful, important, and hard problems for a mathematician, most of which are still unexplored or explored only tangentially The number of mathematics publications dealing with PBG materials is growing (see [3, 7, 8, 21, 41, 42, 44, 45, 50, 51], [61]–[81], [86, 87, 100, 126, 127, 155, 188]) but is probably still not sufficient We hope that

M. S. P. Eastham

Papers

Schrödinger-type operators with continuous spectra