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Spectral Analysis and Applications to Soliton Equations

My current research area concerns mathematical physics and dynamical systems. I am interested in direct and inverse spectral theory for differential and difference operators in connection with completely integrable nonlinear dynamical systems (soliton equations).

Direct and inverse scattering theory (ABM, KG, IE, JM, GT)

Direct and inverse scattering theory for one-dimensional Schrödinger and Jacobi (discrete Schrödinger) operators is a classical topic in quantum mechanics. By comparing the perturbed motion of the Hamiltonian H = H₀ + V with the unperturbed motion generated by H₀ one tries to recover the potential V. The case where H₀ is the free motion is well understood (see e.g., [Te00], Chapter 10). So is the case where H₀ is a periodic operator (see e.g., [EMT06]). For the case where H asymptotically looks like different periodic operators in different directions, only partial results (see e.g., [EMT07]) were known. We gave a complete solution for periodic finite-gap operators in [BET07] and for the corresponding Jacobi operators in [EMT07a]. We are currently working on the general (infinite-gap) periodic case.

Inverse scattering transform for soliton equations (IE, JM, GT)

The results of the previous item can be applied to solve soliton equations via the inverse scattering transform (see e.g., [Te00], Chapter 13). We have done this for the Toda hierarchy in [EMT06a, EMT07a] and plan to extend these results to the Ablowitz-Ladik hierarchy.

Algebro-geometric solutions of soliton equations (FG, HH, JM, GT)

Soliton equations are an extremely important topic used to describe many physical phenomena. Algebro-geometric solutions of soliton equations are a class of solutions which can be constructed explicitly by means of tools from algebraic geometry (see e.g., [Te00], Chapter 13). We have determined all complex-valued quasi-periodic finite-gap solutions of Ablowitz-Ladik hierarchy in [GHMT07, GHMT07a]. Moreover, we have extended the results for the Toda hierarchy from [GHT06] to the Ablowitz-Ladik hierarchy [GHMT07b] and we investigated local conservation laws and the Hamiltonian formalism [GHMT07c]. We have summarized our finding in our recent research monograph [GHMT08].

Soliton asymptotics (IE, SK, AK, JM, IN, AS, GT)

The classical stability result for soliton equations states that from a small perturbation of the zero solution only a number of solitons persist for large times. We have shown how to include solitons in [KrT07c] (see also the recent reviews for the Toda [KrT08] and KdV [GrT08] equation). Moreover, it was believed that this classical result still holds for small perturbation of a periodic solution as well. In [KaT06], [KaT07] we show that the limit is much more complicated and can be described by tools from algebraic geometry. In addition, we have computed higher order asymptotics in [KaT08] and included solitons in [KrT08b]. The present results are only for the Toda equation, but we are working on extending it to the Korteweg-de Vires (KdV), Camassa-Holm (CH), Ablowitz-Ladik, and nonlinear Schrödinger (NLS) equations.

Oscillation theory (KA, DD, HK, GT)

Oscillation theory is an important tool for investigating the spectra of Sturm-Liouville (SL) operators (e.g., the one-dimensional Schrödinger equation from quantum mechanics). Classical oscillation theory relates the number of zeros of solutions to the number of eigenvalues of a Sturm-Liouville (SL) operator. In [GST96] we have shown how the number of zeros of the Wronski determinant of two solutions is related to the spectrum. In [KrT07] we generalized this result by relating the zeros of the Wronski determinant of two solutions of different SL operators to the spectral shift function of the SL operators. Extensions and applications are given in [KrT07a] and [KrT07b]. We are now working to extend these results to Jacobi and one-dimensional Dirac operators.

Eigenvalues of Sturm-Liouville operatos (GT, AZ)

Computation of the eigenvalues of a Sturm-Liouville below the essential spectrum can be effectively done via the SLEIGN package. Our aim is to extend SLEIGN to compute eigenvalues in essential spectral gaps based on [Te07].

Breath gas analysis (AA, HK, GT, ST, KU)

Breath analysis represents a new diagnostic technique that is without risk for the patient, even if repeated frequently, and can provide information beyond conventional analysis of blood and urine. Recent results suggest that detection of different kinds of cancer is possible by means of breath analysis in the very early stages of the disease. We want to implement a mathematical model describing the relationship between blood and breath gas concentrations of certain marker substances. Such models have been extensively studied in the past. However, previously the main focus has been on understanding the human respiratory system. In contradistinction, we want to focus on the reverse direction. More precisely, we want to use the well-established knowledge of the respiratory system to compute blood concentrations from gas concentrations for certain marker substances.

A list of publications including recent preprints is available.