ICMS

Inverse spectral problems in one dimension

Oct 17, 2011 - Oct 21, 2011

ICMS, 15 South College Street Edinburgh

This workshop is a satellite event of the semester on Inverse Problems at the Isaac Newton Institute.

It is planned that  one hour keynote talks will be given by:
J Behrndt (Technische Universitat Graz)
F Gesztesy (Missouri)
P Kurasov (Stockholm)
M Malamud (Donetsk)
S Naboko (St Petersburg)
B Rundell (Texas A&M University)
G Teschi (Vienna) 
R Weikard (Birmingham, Alabama)

There will be the opportunity for some contributed talks as well as time for smaller research group discussions.

Arrangements

Participation
This workshop is now closed.   If you have a enquiry please email Jane Walker.

Venue
The venue for this INI satellite workshop is ICMS, 15 South College Street, Edinburgh, EH8 9AA, (see this link and the 'Travel' section below for directions).  All lectures will be held in the Newhaven Lecture Theatre. To view this room and a list of the visual equipment available click here. In addition, two blackboards have recently been installed and a PC outputting to a high definition projector and audio system.

Wireless Access
The workshop venue, 15 South College Street, has wireless access throughout. On arrival at Registration you will be given instructions and a code for accessing the wireless network.  For those without laptops, there are computers available for you to check your emails.

UK Visas
If you are travelling from overseas you may require an entry visa. A European visa does not guarantee entry to the UK. Please use this link to the UK Visas site to find out if you need a visa and if so how to apply for one.

Travel
Information about international travel to the UK and Edinburgh is available here.   You may also find this map and  this link useful for the workshop.

A taxi directly from the airport to the city centre will cost approximately 20.00 - 25.00 GBPfor a one-way journey. There is also a bus service from the airport to the city centre.

Within the city, Lothian buses charge £1.30 for a single, £3.20 for a day ticket. Please note that the exact fare is required and no change is given.

If travelling by train, please note that Edinburgh has two railway stations - Waverley Railway Station being the main station and closest to the workshop venue at 15 South College Street. If you alight at Edinburgh Waverley, the workshop venue is an easy 10 minute walk over North and South Bridge. The second railway station is called Haymarket and is at the West End of the city centre.

If you are travelling directly to your accommodation you can take an AIRLINK bus from the airport to the city centre - the fare is £6 return and the journey takes about 30 minutes. From the city centre (Waverley Bridge where the bus terminates) walk to North Bridge (a two minute walk) and take bus number 14, 30, 33, or 48.  Ask the driver to let you off at Pollock Halls. The bus fare is £1.30 (you need the exact money). Or you can get a taxi for about £5.  It's a five-ten minute cab ride away.   This map showing the location of Pollock Halls may be helpful.  

Travel Insurance
Please note that it is your responsibility to have adequate travel insurance to cover medical and other emergencies that may occur on your trip.

Accommodation
Accommodation details have been sent out individually.   Please pick up your keys after 14.00 from the Reception Centre at Pollock Halls which is situated near the main entrance to the Pollock Halls complex on Holyrood Park Road.  The Reception Centre is staffed 24 hours a day.  If you experience delays during your travels, meaning you will not arrive on the intended date, please contact the accommodation staff on +44 131 651 2198 to let them know. It is approximately a 25-35 minute walk from Pollock Halls to the workshop venue at 15 South College Street. 

Catering Arrangements
All catering is provided as part of the workshop grant. Lunch will be provided in the Chapterhouse on Level 1 of the workshop venue on each day of the workshop.   Coffee and tea will be available during each of the breaks as listed in the Programme below.   The workshop dinner will take place on Thursday 20 October, 19.00, at the Magnum Restaurant, 1 Albany Street, Edinburgh, EH1 3PY.  

Programme

Monday 17 October

09.15 – 10.00	Registration in the Chapterhouse
10.00 – 11.00	Fritz Gesztesy (University of Missouri)  On boundary data maps and some of their applications
11.00 – 12.00	Discussion focussed on the talk  (in the Chapterhouse with tea and coffee)
12.00 – 12.30	Rami Band (University of Bristol) Quantum graphs which sound the same
12.30 – 14.30	Lunch served in the Chapterhouse
14.30 – 15.30	Heinz Langer (Technische Universitatet Wien) An inverse spectral problem and extrapolation of a continuous second order stationary process from a finite time interval
15.30 – 16.30	Discussion focussed on the talk  (in the Chapterhouse with tea and coffee)
16.30 – 18.00	Time for small group discussions
18.30	Building closes

 

Tuesday 18 October

09.30 – 10.30	Jussi Behrndt  (Technische Universitat Graz) An inverse problem of Calderon type
10.30 – 11.30	Discussion focussed on the talk  (in the Chapterhouse with tea and coffee)
11.30 – 12.30	Time for small group discussions
12.00 – 12.30	Rostylav Hryniv  (University of Rzeszów) Inverse scattering for non-classical impedance Schroedinger operators
12.30 – 14.00	Lunch served in the Chapterhouse
14.00 – 14.30	Harald Woracek  (Technische Universitatet Wien) A local inverse spectral theorem for Hamiltonian systems
14.30 – 15.30	Pavel Kurasov (Stockholm University) Inverse problems for quantum graphs and periodic Schroedinger operators
15.30 – 16.30	Discussion focussed on the talk  (in the Chapterhouse with tea and coffee)
16.30 – 18.00	Time for small group discussions
18.30	Building closes

 

Wednesday 19 October

09.30 – 10.30	Christer Bennewitz  (Lund University) Inverse scattering for left definite Sturm-Liouville equations
10.30 – 11.30	Discussion focussed on the talk  (in the Chapterhouse with tea and coffee)
11.30 – 12.30	Time for small group discussions
12.00 – 12.30	Jonathan  Eckhardt (Universität Wien) Singular left-definite Sturm-Liouville problems
12.30 – 14.00	Lunch served in the Chapterhouse
14.00 – 15.00	Gerald Teschl  (Universität Wien) Inverse problems for Jacobi operators
15.00 – 16.00	Rudi Weikard  (University of Alabama Birmingham) On the stability of inverse resonance problems
16.00 – 16.30	Discussion focussed on the talk  (in the Chapterhouse with tea and coffee)
16.30 – 18.00	Time for small group discussions
18.30	Building closes

 

Thursday 20 October

09.30 – 10.30	M.  Malamud (National Academy of Sciences of Ukraine) On unique recovery of the first order system by a part of its monodromy matrix
10.30 – 11.30	Discussion focussed on the talk  (in the Chapterhouse with tea and coffee)
11.30 – 12.30	Time for small group discussions
12.00 – 12.30	Alexei Rybkin  (University of Alaska Fairbanks) On the inverse scattering problem with a partially known potential
12.30 – 14.00	Lunch served in the Chapterhouse
14.00 – 14.30	Vyacheslav Pyvovarchyk  (South Ukraniaian National Pedagogical University) On Hochstadt-Lieberman theorems
14.30 – 15.30	Sergey Naboko   (St Petersburg State University) Constructive examples of unbounded Hermition Jacobi matrices and the Last-Simon approach for the essential spectra calculation
15.30 – 16.30	Discussion focussed on the talk  (in the Chapterhouse with tea and coffee)
16.30 – 17.30	Bill Rundell (Texas A&M University) The two spectrum problem with fixed boundary conditions
17.30 – 18.00	Time for small group discussions
18.30	Building closes
19.00 – 22.00	Workshop dinner at the Magnum Restaurant, 1 Albany Street, Edinburgh

 

Presentations:

Presentation Details
Band, Rami
Quantum graphs which sound the same.
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In 1966 Marc Kac asked 'Can one hear the shape of a drum?'. The answer was found only in 1992, when Gordon et al. found a pair of drums with the same spectrum. The study of isospectrality and inverse problems is obviously not limited to drums and treats various objects such as molecules, quantum dots and graphs. We present a new method for the construction of isospectral objects and demonstrate its strength on quantum graphs. We show that this method enables one to produce new examples of isospectrality. This is joint work with Ori Parzanchevski and Gilad Ben-Shach.
Behrndt, Jussi
An inverse problem of Calderon type
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A generalized variant of the Calderon problem from electrical impedance tomography with partial data for anisotropic Lipschitz conductivities is considered. The following two results are discussed: (i) The self-adjoint Dirichlet operator associated with an elliptic differential expression on a bounded Lipschitz domain is determined uniquely up to unitary equivalence by the knowledge of the Dirichlet-to-Neumann map on an open subset of the boundary, and (ii) the Dirichlet operator can be reconstructed from the residuals of the Dirichlet-to-Neumann map on this subset. The lecture is based on joint work with Jonathan Rohleder.
Bennewitz, Christer
Inverse scattering for left definite Sturm-Liouville equations
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This talk will discuss inverse scattering for left definite Sturm-Liouville equations. An important reason for considering this is the fact that the spectral problem associated with the Camassa-Holm equation is of this kind. The Camassa-Holm equation is an integrable system in a similar sense as the KdV equation is. Among several other things it may be used to model shallow water waves, and in contrast to the KdV equation it can also model breaking waves. We will define the scattering matrix and discuss some of its properties, and then prove a uniqueness theorem for inverse scattering of left definite equations. There are several difficulties to be overcome in comparison to the case of the Schrödinger equation on the line. Our proof is based on the so called Jost transform associated with our equation. This is a generalization of the usual Fourier transform, and we will prove a Paley-Wiener type theorem for the Jost transform which is crucial for our approach. This reports on work joint with Malcolm Brown in Cardiff and Rudi Weikard in Birmingham, AL.
Eckhardt, Jonathan
Singular left-definite Sturm-Liouville problems
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Inverse spectral theory for left-definite Sturm-Liouville problems recently gained attention due to its applicabilty to the Camassa-Holm equation. We develop Weyl-Titchmarsh theory for left-definite Sturm-Liouville problems with two singular endpoints and give an inverse uniqueness theorem for the associated spectral measure. The results are applied to solve the inverse problem for the isospectral problem of the Camassa-Holm equation for the class of finite signed measure weights.
Gesztesy, Fritz
On boundary data maps and some of their applications
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We present a discussion of boundary data maps (i.e., energy-dependent Robin-to-Robin maps, a natural extension of the Dirichlet-to-Neumann map) and some of its applications. The latter involve, symmetrized perturbation determinants, Krein-type resolvent formulas and associated trace formulas, and aspects of inverse spectral theory. This is based on various joint work with S. Clark, M. Mitrea, R. Nichols, and M. Zinchenko
Hryniv, Rostyslav
Inverse scattering for non-classical impedance Schrodinger operators
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We aim to review recent progress in the direct and inverse scattering theory for one-dimensional Schrodinger operators on the line in the impedance form, [H = -frac1{p^2(x)}frac{d}{dx}p^2(x) frac{d}{dx}.] If p is smooth enough, the operator H is unitarily equivalent to a Schrodinger operator in potential form with potential~q(x) = p''(x)/p(x); however, such a reduction is impossible when e.g. p is discontinuous. Two classes of non-smooth impedance functions~p will be considered. Absolutely continuous impedances correspond to singular Miura potentials that are distributions from W^{-1}_{2,mathrm{loc}}(mathbb{R}); nevertheless, most of the classic scattering theory for Schrodinger operators with Faddeev--Marchenko potentials is carried over to this singular setting. The second class consists of discontinuous impedances arising e.g. in the study of electromagnetic wave propagation in stratified media. Some partial results for a related problem on scattering in media with discontinuous wave speed propagation have been obtained by Aktosun, Klaus, van der Mee, Sabatier a.o.; however, no complete solution of the inverse scattering problem is available yet. We shall demonstrate that the corresponding Schr"odinger operators possess rather unusual scattering properties. For instance, in the model case of piece-wise constant impedance functions with discontinuities on a periodic lattice the corresponding reflection coefficients are periodic; the inverse scattering problem has then a complete solution. The talk is based on the joint work with S. Albeverio (Bonn, Germany), Ya. Mykytyuk (Lviv, Ukraine), and P. Perry (Kentucky, USA).
Kurasov, Pavel
Inverse problems for quantum graphs and periodic Schr"odinger operators
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The inverse problem to reconstruct potential on a metric graph (with cycles) from the Titchmarsh-Weyl matrix function is discussed. Our attention will be focused on the role of vertex matching conditions.
Langer, Heinz
An inverse spectral problem and extrapolation of a continuous second order stationary process from a finite time interval
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The problem of extrapolation of a second order stationary process from a finite time interval can be solved by means of results on an inverse spectral problem for canonical differential systems. The method is based on M.G.Krein's results about the extension of positive definite functions.
Malamud, Mфкл
On unique recovery of the first order system by a part of its monodromy matrix
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We present different results and discuss different approaches to the problem of unique recovery of the first order system by a part of its monodromy matrix. We present necessary and sufficient conditions of such recovery.
Naboko, Sergey
Constructive examples of unbounded Hermitian Jacobi Matrices and the Last-Simon approach for the essential spectra calculation
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Constructive examples of unbounded Hermitian Jacobi Matrices with a few gaps in the essential spectrum to be considered.The main attention will be paid to the analysis of a "new" essential spectrum appearing due to the interlacing of a few regimes procedure of the construction.A sort of generalization of the Last-Simon approach for the essential spectra determination for unbounded Jacobi Matrices is one of the main goals of the presentation.
Pyvovarchyk, Vyacheslav
On Hochstadt-Lieberman theorem
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A method of recovering the potential of the Sturm-Liouville equation on a half-interval using known potential on another half-interval and the spectrum of the boundary value problem on the whole interval is proposed.
Rundell, Bill
The two spectrum problem with fixed boundary conditions
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To be announced
Rybkin, Alexei
On the inverse scattering problem with a partially known potential
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We are concerned with the inverse scattering problem for the full line Schrödinger operator -∂_{x}²+q(x) with a step-like potential q a priori known on R₊=(0,∞). Assuming q|_{R₊} is known and short range, we show that the unknown part q|_{R₋} of q can be recovered by q|_{R₋}(x)=-2∂_{x}²logdet(1+(1+M_{x}⁺)⁻¹G_{x}), where M_{x}⁺ is the classical Marchenko operator associated to q|_{R₊} and G_{x} is a trace class integral Hankel operator. The kernel of G_{x} is explicitly constructed in term of the difference of two suitably defined reflection coefficients. Since q|_{R₋} is not assumed to have any pattern of behaviour at -∞, defining and analysing scattering quantities becomes a serious issue. Our analysis is based upon some subtle properties of the Titchmarsh-Weyl m-function associated with R₋. This is joint work with Odile Bastille.
Teschl, Gerald
Inverse problems for Jacobi operators
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I will review some well-known inverse spectral results for Jacobi operators. I will also talk about applications to integrable systems (the Toda lattice) and some open problems.
Weikard, Rudi
On the stability of inverse resonance problems
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Inverse spectral and scattering problems are a classical subject in mathematical physics. In this talk a particular variant, the inverse resonance problem, is presented. Since, in practical settings, one can typically not expect to obtain all the necessary data and since, in any case, recovery algorithms cannot make use of all data even if they were available, we investigate which information may be contained from finite noisy data in a variety of settings. A number of open problems is presented.
Woracek, Harald
A local inverse spectral theorem for Hamiltonian systems
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We consider $2!times!2$--Hamiltonian systems of the form $y'(t)=zJH(t)y(t)$, $tin[s_-,s_+)$. If a system of this form is in the limit point case, an analytic function is associated with it, namely its Titchmarsh--Weyl coefficient $q_H$. The (global) uniqueness theorem due to L.~de~Branges says that the Hamiltonian $H$ is (up to reparameterization) uniquely determined by the function $q_H$. In the present paper we give a local uniqueness theorem: if the Titchmarsh--Weyl coefficients $q_{H_1}$ and $q_{H_2}$ corresponding to two Hamiltonian systems are exponentially close, then the Hamiltonians $H_1$ and $H_2$ coincide (up to reparameterization) up to a certain point of their domain, which depends on the quantitative degree of exponential closeness of the Titchmarsh--Weyl coefficients. [joint work with Matthias Langer]

Participants

Name	Institution
Band, Rami	University of Bristol
Behrndt, Jussi	Technische Universität Graz
Bennewitz, Christer	Lund University
Bögli, Sabine	University of Bern
Boman, Jan	Stockholm University
Brown, Malcolm	Cardiff University
Clark, Stephen	Missouri University of Science and Technology
Cunningham, Laura	University of Strathclyde
Eckhardt, Jonathan	Universität Wien
Gesztesy, Fritz	University of Missouri
Harmer, Mark	Bahrain Polytechnic
Hryniv, Rostyslav	University of Rzeszów
Klaus, Martin	Virginia Tech
Kurasov, Pavel	Stockholm University
Langer, Heinz	Vienna University of Technology
Langer, Matthias	University of Strathclyde
Luger, Annemarie	Stockholm University
Malamud, Mфкл	Institute of Applied Math.and Mechanics National Academy of Sciences of Ukraine
Naboko, Sergey	St Petersburg State University
Nowaczyk, Marlena	AGH University of Science and Technology
Pyvovarchyk, Vyacheslav	South Ukraniaian National Pedagogical University
Rundell, Bill	Texas A&M
Rybkin, Alexei	University of Alaska Fairbanks
Sacks, Paul	Iowa State University
Simonov, Sergey	Vienna University of Technology
Tant, Katy	University of Strathclyde
Teschl, Gerald	Universitat Wien
Weikard, Rudi	University of Alabama at Birmingham
Wood, Ian	University of Kent
Woracek, Harald	Technische Universitatet Wien