David W. Pravica

Spectral Analysis for Optical Fibres and Stratified Fluids I: The Limiting Absorption Principle

The wave equation (∂t2+H)Ψ=0 withH=−c2(x,y)ρ(x,y)▽z·ρ(x,y)−1▽z, z=(x,y),xϵRk,yϵRm, describes the propagation of acoustical waves in stratified fluids (m = 1, k = 2), as well as electromagnetic waves in layered dielectric media (m = 1, k = 2), and in optical fibres (m = 2, k = 1,and p(x, y) = 1). The density p(x, y) and signal speed c(x, y) are (possibly long range) perturbations of a density p0(y) and speed c0(y), i.e., (p(x, y) − p0(y)), (c(x,y) − c0(y)) → 0 as ¦z¦→ ∞. Positive commutator methods are used to show that the spectrum of H is absolutely continuous, except possibly for a sequence of isolated eigenvalues of finite multiplicity that can accumulate only at zero and ∞. Away from those eigenvalues a limiting absorption principle for H is established: it is proven that the resolvent (H − k)−1 of H has a norm limit as an operator between L2(Rk+m,(1+z2)α2 (c2ρ)−1 dnz) and L2(Rk+m, (1+z2)−α2 (c2ρ)−1 dnz) for α >12 as kϵC\R+ approaches the real axis either from above or from below.

Top resonances of a black hole

The wave equation ofofφ = 0 is studied in the exterior of a Schwarzschild black hole, r > 2M. By assuming stationary-spherically symmetric solutions ψ= e iω t r -1 φ(r) Y l (Ω), the wave equation reduces to the Schrodinger equation (- ∂ r* 2 +V(r,l))φ= ω 2 φ, where r* = r + 2M ln(r/2M - 1). The potential V has a single maximum near r = 3M for each l]N, so a family of top resonances is expected to exist. It is demonstrated that there are spectral resonances z l k ∼ V max l - il(2k+1) 27 M 2 , where kεN0 is a parameter of the harmonic oscillator, and where V max l ≡V( r max l ,l)∼ l(l+1) 27 M 2 + 2 81 M 2 . The resonant states orbit near the radius rlmax ∼ 3M(1-(3l)-2) for large l > 0. A modification of the standard complex scaling technique is required for the analysis.

Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function

The family of nth order q-Legendre polynomials are introduced. They are shown to be obtainable from the Jacobi theta function and to satisfy recursion relations and multiplicatively advanced differential equations (MADEs) that are analogues of the recursion relations and ODEs satisfied by the nth degree Legendre polynomials. The nth order q-Legendre polynomials are shown to have vanishing kth moments for , as does the nth degree truncated Legendre polynomial. Convergence results are obtained, approximations are given, a reciprocal symmetry is shown, and nearly orthonormal frames are constructed. Conditions are given under which a MADE remains a MADE under inverse Fourier transform. This is used to construct new wavelets as solutions of MADEs.

Validity of a closed-form diffusion solution in P1 approximation for reflectance imaging with an oblique beam of arbitrary profile.

Determination of optical parameters of turbid media from reflectance image data is an important class of inverse problems due to its potential for noninvasive characterization of materials and biological tissues, which demands rapid modeling tools to generate calculated images. We treat the problem of reflectance imaging with homogeneous semi-infinite turbid media as a boundary-value problem of diffusion type in the P1 approximation to the radiative transfer equation. A closed-form solution has been obtained for an oblique incident beam of arbitrary profile and its accuracy has been examined against a Monte Carlo method and measured data. We find that the diffusion solution provides a sufficiently accurate tool to rapidly calculate reflectance images for samples of large or moderate scattering albedo illuminated by a beam of arbitrary profile as long as the anisotropy factor remains less than 0.7 and single scattering albedo larger than 0.8. The closed-form solution can thus be used as a part of a forward modeling toolbox to determine optical parameters from reflectance image data in combination with other method such as the Monte Carlo simulation.

Advanced Models for Tsunami and Rogue Waves

A wavelet , that satisfies the q-advanced differential equation for , is used to model N-wave oscillations observed in tsunamis. Although q-advanced ODEs may seem nonphysical, we present an application that model tsunamis, in particular the Japanese tsunami of March 11, 2011, by utilizing a one-dimensional wave equation that is forced by . The profile is similar to tsunami models in present use. The function is a wavelet that satisfies a q-advanced harmonic oscillator equation. It is also shown that another wavelet, , matches a rogue-wave profile. This is explained in terms of a resonance wherein two small amplitude forcing waves eventually lead to a large amplitude rogue. Since wavelets are used in the detection of tsunamis and rogues, the signal-analysis performance of and is examined on actual data.

MURMURS AND NOISE CAUSED BY ARTERIAL NARROWING – THEORY AND CLINICAL PRACTICE

Arterial narrowing can cause an audible whirling in the blood flow. We propose diagnosing such narrowing by simply recording that sound and analyzing its spectrum. We show how the Navier-Stokes equation for flow through a narrowing can be turned into a Schrodinger type equation. The complex eigenvalues of the latter equation give the frequencies and decay rates of the vortices present in the whirling pattern. Our diagnosis is based on understanding the relation between features in the sound spectrum and the severity of the narrowing. Today the most commonly used method of diagnosis is duplex ultrasound. In a small clinical trial our method appears to be as good as duplex ultrasound.

Bounds on resonances for the Laplacian on perturbations of half-space

The resonances of the Laplacian on perturbations of half-spaces of dimensions greater than or equal to two, with either Dirichlet or Neumann boundary conditions, are studied. An upper bound for the resonance counting function is proven. If the domain has an elliptic, nondegenerate, nonglancing periodic billiard trajectory, it is shown that there exists a sequence of resonances that converge to the real axis.

Spectral Resonances which Become Eigenvalues

The stationary Schrödinger equation is ∂x2φ + λV(x)φ=zφ for φ∈ℒ2(R+,dx). If the potential is bounded below, singular only atx=0, negative on some compact interval and behaves likeV(x)∼1/xμ asx→∞ with 2≧μ>0, then the system admits shape resonances which continuously become eigenvalues as λ increases. Here λ>0 and for μ=2 a sufficiently large λ is required. Exponential bounds are obtained on Im(z) as λ approaches a threshold. The group velocity near threshold is also estimated.

Low energy resolvent bounds for elliptic operators: an application to the study of waves in stratified media and fiber optics

For the positive self-adjoint operators H = -{partial_derivative}{sub igij}(x){partial_derivative} + q(x) on H {triple_bond} L{sup 2}(R{sup n}) with n {ge} 3, it is shown that {parallel}({vert_bar}x{vert_bar}){sup {minus}1}({radical}H - z){sup {minus}1}({vert_bar}x{vert_bar} + 1){sup {minus}1}{parallel} {le} C/Re(z){sup 2} as Re(z){yields}0{sup +}, by imposing several conditions on g{sub ij} and q. In the special case g{sub ij} = c{sup 2}{delta}{sub ij} these conditions reduce to {vert_bar}x{vert_bar}{center_dot}{del}c, (x{sup 2} + 1){sup 1+{var_epsilon}}({vert_bar}q(x){vert_bar} + {vert_bar}x{center_dot}{del}q{vert_bar}) {epsilon} L{sup {infinity}} with the nontrapping condition (c - x{center_dot} {Delta}c) {ge} kc, and a positivity condition C(x{sup 2} + 1){sup {minus}1} {le} 4p{vert_bar}x{vert_bar}{sup {minus}2} -(N - 2){sup 2}(-q){sub +}, for some k, C, p > 0. Results are applied to the stratified wave equation ({partial_derivative}{sub t}{sup 2} - c{sup 2}(y){Delta}{sub z}){psi} = 0, where z = x{circle_plus} y {epsilon} R{sup k} {circle_plus} R{sup m} with n = k + m, and {vert_bar}y{vert_bar}(y{center_dot}{del}{sub y}c){epsilon} L{sup {infinity}}(R{sup m}). In all cases the condition (c-y{center_dot}{del}{sub y}c) {le} kc leads to a local-energy decay estimate for {psi}(z,1). 11 refs.

Propagation estimates for dispersive wave equations: Application to the stratified wave equation

The plane-stratified wave equation (∂t2+H)ψ=0 with H=−c(y)2∇z2 is studied, where z=x⊕y, x∈Rk, y∈R1 and |c(y)−c∞|→0 as |y|→∞. Solutions to such an equation are solved for the propagation of waves through a layered medium and can include waves which propagate in the x-directions only (i.e., trapped modes). This leads to a consideration of the pseudo-differential wave equation (∂t2+ω(−Δx))ψ=0 such that the dispersion relation ω(ξ2) is analytic and satisfies c1⩽ω′(ξ2)⩽c2 for c*>0. Uniform propagation estimates like ∫|x|⩽|t|αE(UtP±φ0)dkx⩽Cα,β(1+|t|)−β∫E(φ0)dkx are obtained where Ut is the evolution group, P± are projection operators onto the Hilbert space of initial conditions φ∈H and E(⋅) is the local energy density. In special cases scattering of trapped modes off a local perturbation satisfies the causality estimate ‖P+ρΛjSP−ρΛk‖⩽Cνρ−ν for each ν<1/2. Here P+ρΛj (P−ρΛk) are remote outgoing/detector (incoming/transmitter) projections for the jth (kth) trapped mode. Also Λ⋐R+ is compact, so the projections lo...