Mathematics

We establish Anderson localization for general analytic k -frequency quasi-periodic operators on Z d for \textit{arbitrary} k,d .

We consider non-self-adjoint Schrödinger operators H c =−Δ+ V c (resp. H r =−Δ+ V r ) acting in L 2 ( R d ) , d≥1 , with dilation analytic complex (resp. real) potentials. We were able to find out perhaps a new application of dilation analytic method in \cite{So1} (N. Someyama, "Number of Eigenvalues of Non-self-adjoint Schrödinger Operators with Dilation Analytic Complex Potentials," Reports on Mathematical Physics, Volume 83, Issue 2, pp.163-174 (2019).). We give a Lieb--Thirring type estimate on resonance eigenvalues of H c in the open complex sector and that on embedded eigenvalues of H r in the same way as \cite{So1}. To achieve that, we derive Lieb--Thirring type inequalities for isolated eigenvalues of H on several complex subplanes.

A new idea to approximate the second eigenfunction and the second eigenvalue of p -Laplace operator is given. In the case of the Dirichlet boundary condition, the scheme has the restriction that the positive and the negative part of the second eigenfunction have equal L p -norm, however, in the case of Neumann boundary condition, our algorithm has not such restriction. Our algorithm generates a descending sequence of positive numbers that converges to the second eigenvalue. We give various examples and computational tests.

It is well known that the matrix exponential of a non-normal matrix can exhibit transient growth even when all eigenvalues of the matrix have negative real part, and similarly for the powers of the matrix when all eigenvalues have magnitude less than 1. Established conditions for the existence of these transient effects depend on properties of the entire matrix, such as the Kreiss constant, and can be laborious to use in practice. In this work we develop a relationship between the invariant subspaces of the matrix and the existence of transient effects in the matrix exponential or matrix powers. Analytical results are obtained for two-dimensional invariant subspaces and Jordan subspaces, with the former causing transient effects when the angle between the subspace's constituent eigenvectors is sufficiently small. In addition to providing a finer-grained understanding of transient effects in terms of specific invariant subspaces, this analysis also enables geometric interpretations for the transient effects.

We consider the Dirac system on the interval [0,1] with a spectral parameter μ∈C and a complex-valued potential with entries from L p [0,1] , where 1≤p<2 . We study the asymptotic behavior of its solutions in a stripe |Imμ|≤d for μ→∞ . These results allows us to obtain sharp asymptotic formulas for eigenvalues and eigenfunctions of Sturm--Liouville operators associated with the aforementioned Dirac system.

For Jacobi parameters belonging to one of the three classes: asymptotically periodic, periodically modulated and the blend of these two, we study the asymptotic behavior of the Christoffel functions and the scaling limits of the Christoffel-Darboux kernel. We assume regularity of Jacobi parameters in terms of the Stolz class. We emphasize that the first class only gives rise to measures with compact supports.

We study orthogonal polynomials with periodically modulated Jacobi parameters in the case when 0 lies on the soft edge of the spectrum of the corresponding periodic Jacobi matrix. We determine when the orthogonality measure is absolutely continuous and we provide a constructive formula for it in terms of the limit of Turán determinants. We next consider asymptotics of the solutions of associated second order difference equation. Finally, we study scaling limits of the Christoffel--Darboux kernel.

We study the zero set of "generic" Laplace eigenfunctions on the standard two dimensional flat torus T 2 = R 2 / Z 2 . We find the asymptotic nodal length in any ball of radius larger than the Planck-scale. In particular, we prove that the nodal set equidistributes on T 2 . Moreover, we show that the nodal set equidistributes at Planck-scale around almost every point for some eigenfunctions and give examples of eigenfunctions whose nodal set fails to equidistribute at Planck-scale. The proof is based on Bourgain's de-randomisation and the main new ingredient is the integrability of arbitrarily large powers of the logarithm of "generic" eigenfunctions, based on the work of Nazarov, and some arithmetic ingredients called semi-correlations.

The periodic Schrodinger operator H on a discrete periodic graph is considered. We estimate the discrete spectrum of the perturbed operator H − (t)=H−tV , t>0 , where the potential V ge0 is decreasing and t>0 is the coupling constant. If the potential has a power asymptotics at infinity, then we obtain asymptotics of the discrete spectrum of operator H − (t) with a large coupling constant.

Whispering gallery modes [WGM] are resonant modes displaying special features: They concentrate along the boundary of the optical cavity at high polar frequencies and they are associated with complex scattering resonances very close to the real axis. As a classical simplification of the full Maxwell system, we consider two-dimensional Helmholtz equations governing transverse electric [TE] or magnetic [TM] modes. Even in this 2D framework, very few results provide asymptotic expansion of WGM resonances at high polar frequency m→∞ for cavities with radially varying optical index. In this work, using a direct Schrödinger analogy we highlight three typical behaviors in such optical micro-disks, depending on the sign of an effective curvature that takes into account the radius of the disk and the values of the optical index and its derivative. Accordingly, this corresponds to abruptly varying effective potentials (step linear or step harmonic) or more classical harmonic potentials, leading to three distinct asymptotic expansions for ground state energies. Using multiscale expansions, we design a unified procedure to construct families of quasi-resonances and associate quasi-modes that have the WGM structure and satisfy eigenequations modulo a super-algebraically small residual O( m −∞ ) . We show using the black box scattering approach that quasi-resonances are O( m −∞ ) close to true resonances.