Invisibility and PT-symmetry

For a general complex scattering potential defined on a real line, we show that the equations governing invisibility of the potential are invariant under the combined action of parity and time-reversal (PT) transformation. We determine the PT-symmetric an well as non-PT-symmetric invisible configurations of an easily realizable exactly solvable model that consists of a two-layer planar slab consisting of optically active material. Our analysis shows that although PT-symmetry is neither necessary nor sufficient for the invisibility of a scattering potential, it plays an important role in the characterization of the invisible configurations. A byproduct of our investigation is the discovery of certain configurations of our model that are effectively reflectionless in a spectral range as wide as several hundred nanometers.Comment: 11 pages, 3 figures, revised version, accepted for publication in Phys.Rev.

Analyticity and uniform stability in the inverse spectral problem for Dirac operators

We prove that the inverse spectral mapping reconstructing the square integrable potentials on [0,1] of Dirac operators in the AKNS form from their spectral data (two spectra or one spectrum and the corresponding norming constants) is analytic and uniformly stable in a certain sense.Comment: 19 page

Spectral Singularities of Complex Scattering Potentials and Infinite Reflection and Transmission Coefficients at real Energies

Spectral singularities are spectral points that spoil the completeness of the eigenfunctions of certain non-Hermitian Hamiltonian operators. We identify spectral singularities of complex scattering potentials with the real energies at which the reflection and transmission coefficients tend to infinity, i.e., they correspond to resonances having a zero width. We show that a wave guide modeled using such a potential operates like a resonator at the frequencies of spectral singularities. As a concrete example, we explore the spectral singularities of an imaginary PT-symmetric barrier potential and demonstrate the above resonance phenomenon for a certain electromagnetic wave guide.Comment: Published versio

Inverse spectral problems for Sturm--Liouville operators with matrix-valued potentials

We give a complete description of the set of spectral data (eigenvalues and specially introduced norming constants) for Sturm--Liouville operators on the interval $[0,1]$ with matrix-valued potentials in the Sobolev space $W_2^{-1}$ and suggest an algorithm reconstructing the potential from the spectral data that is based on Krein's accelerant method.Comment: 39 pages, uses iopart.cls, iopams.sty and setstack.sty by IO

Multidimensional Borg-Levinson Theorem

We consider the inverse problem of the reconstruction of a Schr\"odinger operator on a unknown Riemannian manifold or a domain of Euclidean space. The data used is a part of the boundary $\Gamma$ and the eigenvalues corresponding to a set of impedances in the Robin boundary condition which vary on $\Gamma$. The proof is based on the analysis of the behaviour of the eigenfunctions on the boundary as well as in perturbation theory of eigenvalues. This reduces the problem to an inverse boundary spectral problem solved by the boundary control method

Inverse spectral problems for energy-dependent Sturm-Liouville equations

We study the inverse spectral problem of reconstructing energy-dependent Sturm-Liouville equations from their Dirichlet spectra and sequences of the norming constants. For the class of problems under consideration, we give a complete description of the corresponding spectral data, suggest a reconstruction algorithm, and establish uniqueness of reconstruction. The approach is based on connection between spectral problems for energy-dependent Sturm-Liouville equations and for Dirac operators of special form.Comment: AMS-LaTeX, 28 page

The Two-Spectra Inverse Problem for Semi-Infinite Jacobi Matrices in The Limit-Circle Case

We present a technique for reconstructing a semi-infinite Jacobi operator in the limit circle case from the spectra of two different self-adjoint extensions. Moreover, we give necessary and sufficient conditions for two real sequences to be the spectra of two different self-adjoint extensions of a Jacobi operator in the limit circle case.Comment: 26 pages. Changes in the presentation of some result

Exceptional points in quantum and classical dynamics

We notice that, when a quantum system involves exceptional points, i.e. the special values of parameters where the Hamiltonian loses its self-adjointness and acquires the Jordan block structure, the corresponding classical system also exhibits a singular behaviour associated with restructuring of classical trajectories. The system with the crypto-Hermitian Hamiltonian H = (p^2+z^2)/2 -igz^5 and hyper-ellictic classical dynamics is studied in details. Analogies with supersymmetric Yang-Mills dynamics are elucidated.Comment: References added. Final version to be published in J. Phys.

Incomplete inverse spectral and nodal problems for differential pencils

[[abstract]]We prove uniqueness theorems for so-called half inverse spectral problem (and also for some its modification) for second order differential pencils on a finite interval with Robin boundary conditions. Using the obtained result we show that for unique determination of the pencil it is sufficient to specify the nodal points only on a part of the interval slightly exceeding its half.[[notice]]補正完畢[[incitationindex]]SCI[[booktype]]紙本[[booktype]]電子

Self-assembly and anti-amyloid cytotoxicity activity of amyloid beta peptide derivatives

The self-assembly of two derivatives of KLVFF, a fragment Abeta(16-20) of the amyloid beta (Abeta) peptide, is investigated and recovery of viability of neuroblastoma cells exposed to Abeta is observed at sub-stoichiometric peptide concentrations. Fluorescence assays show that NH2-KLVFF-CONH2 undergoes hydrophobic collapse and amyloid formation at the same critical aggregation concentration (cac). In contrast, NH2-K(Boc)LVFF-CONH2 undergoes hydrophobic collapse at a low concentration, followed by amyloid formation at a higher cac. These findings are supported by the beta-sheet features observed by FTIR. Electrospray ionization mass spectrometry indicates that NH2-K(Boc)LVFF-CONH2 forms a significant population of oligomeric species above the cac. Cryo-TEM, used together with SAXS to determine fibril dimensions, shows that the length and degree of twisting of peptide fibrils seem to be influenced by the net peptide charge. Grazing incidence X-ray scattering from thin peptide films shows features of beta-sheet ordering for both peptides, along with evidence for lamellar ordering of NH2-KLVFF-CONH2. This work provides a comprehensive picture of the aggregation properties of these two KLVFF derivatives and show their utility, in unaggregated form, in restoring the viability of neuroblastoma cells against Abeta-induced toxicity