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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] (see also [EGLT14]) and for the corresponding Jacobi operators in [EMT07a, EMT12]. Extensions to the general (infinite-gap) periodic case were given in [Gr11, Gr13].

Inverse scattering transform for soliton equations (IE, KG, 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 for the Korteweg-de Vries (KdV) equation in [EGrT09, ET11].

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, AM, 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 included solitons in [KrT08b]. These results were for the Toda equation, the case of the KdV equation is given in [MLT12]. Asymptotics for the KdV equation with a steplike initial condition were given in [EGKT13] and for the Toda lattice in [EMT13].

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]. Extensions to Jacobi operators were given in [AT09, A14].

Eigenvalues of Sturm-Liouville operatos (GT)

Computation of the eigenvalues of a Sturm-Liouville below the essential spectrum can be effectively done via the SLEIGN package. In [Te07] we presented an algorithm to compute eigenvalues in essential spectral gaps.

Singular Weyl-Titchmarsh-Kodaira theory (AB, RB, JE, AK, AS, GT)

In [KST10, KST12, KT11, KST12b] we developed an analog of Weyl-Titchmarsh-Kodaira theory for Schrödinger operators with strongly singular potentials. In addition to the general theory, our main focus in these papers has been the application to spherically symmetric Schrödinger operators. However, our theory is also applicable to perturbations of the harmonic oscillator or to the type of Sturm-Liouville operators which arise in the Lax pair of the dispersionless Camassa-Holm equation. In particular, it was conjectured by McKean that decaying initial conditions of this equation asymptotically split into an infinite number of solitons. we have proven this conjecture in [ET13, ET14]. Extensions to Jacobi operators are given in [ET13] and to Dirac operators in [BEKT14, EKT14, BEKT14]. Finally, we investigated spectral asymptotics for canonical systems [EKT15] (see also [LTW15]).

Sturm-Liouville operators with measure-valued coefficients (JE, AK, FG, MM, RN, GT)

Sturm-Liouville operators with measure-valued coefficients arise in many applications. The one-dimensional Schrödinger equation with a Dirac delta potential being the most prominent example which can be found in any textbook on quantum mechanics. In [ET13] we have developed a general theory which covers many important examples as a special case: Jacobi operators, SL operators on time scales [ET13, ET13b], delta interactions, delta prime interactions [EKMT14], Krein strings, etc. An alternate approach based on the connections with supersymmetric Dirac operators was developed in [EGNT13, EGNT13, EGNT14, EGNST14b].

Dispersion estimates (IE, EK, VM)

Dispersion estimates for one-dimensional Schrödinger and Klein-Gordon equations play an important role not only in understanding the dynamics of these equations, but also in the investigation of associated nonlinear evolution equations. So far we have improved the known estimates for the continuous and discrete case [EKT14, EKMT14].

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. [KKUKTTMSHA09, KUTTKHA11, KKUMKTTHA10, KKTUTMHA11, KUTAT11, KUTTMKHA12, KKFHUTHAH12, KMUTKSAB12, UKMJKTAT14]. The potential use for urban search and recue operations was investigated in [UMUTA15].

A list of publications including recent preprints is available.