# Inverse Scattering - The Time Dependent Approach, fall 2007

### Lecturer

### Scope

5 sp.

### Type

Advance studies.

### Course notes

See this page for the course notes in pdf form.

### Lectures

The lectures will take place on Mondays at 14-16, alternating between Exactum (room B120) and TKK (room U345, from 5.11. on U262). The Inverse problems seminar this fall will therefore consist of Professor Weder's lectures. The first lecture is on Monday, 17 Sep, at 14-16 in Exactum B120. The lectures will last until the end of the semester.

### Contents

1. A brief introduction to scattering theory.

The wave and scattering operators. Definition and main properties. The definition of the scattering matrix from the scattering operator. The relation to the stationary (time-independent) method. The limiting absorption principle. The stationary formula for the scattering matrix. The scattering amplitude and the scattering solutions.

2. Propagation properties of free wave packets that evolve under the Schrödinger equation without potential. The method of stationary phase.

3. Two-body short-range potentials.

High-energy estimates for the wave and scattering operators. Reconstruction of the potential from the high-energy limit inverting the Radon transform.

4. Inverse scattering by magnetic fields.

The Aharonov-Bohm effect. Reconstruction of the regular magnetic potential and the hidden magnetic flux.

5. Inverse Scattering for the non-linear Schrödinger equation with a potential.

Definition of the non-linear scattering operator. Reconstruction of the scattering operator of the linearized equation from the low-energy limit of the non-linear scattering operator and reconstruction of the potential. Reconstruction of the non-linearity from the non-linear scattering operator with reference dynamics containing the potential. Application to the reconstruction of the parameters of a quantum capacitor.

6. Depending on the time available and on the interest of the students other topics will be discussed. For example. Inverse scattering by two-body long-range potentials. Inverse scattering for N-body systems with short- and long-range potentials. Inverse scattering of two clusters (atom-atom scattering ), reconstruction of the effective potential. Inverse scattering for the non-linear Klein-Gordon equation.

### Bibliography

References

1. M. Reed and B. Simon, Methods of Modern Mathematical Physics III Scattering Theory. Academic Press, New York, 1979.

2. V. Enss and R. Weder, The Geometrical Approach to Multidimensional Inverse Scattering, J. Math. Phys. 36, (1995), 3902-3921.

3. R. Weder, Multidimensional Inverse Scattering in an Electric Field, J. Functional . Analysis, 139, (1996), 441-465.

4.V. Enss and R. Weder, Inverse Two Cluster Scattering, Inverse Problems, 12, (1996), 409-418.

5. R. Weder, The W(kp)-Continuity of the Schrödinger Wave Operators on the Line, Comm. Math. Phys., 208 (1999), 507-520.

6. R. Weder, Lp-Lp´ Estimates for the Schrödinger Equation on the Line and Inverse Scattering for the Non linear Schrödinger Equation with a Potential, J. Funct. Analysis, 170 (2000), 37-68.

7. R. Weder, Inverse Scattering on the Line for the Nonlinear Klein-Gordon Equation with a Potential, J. Math. Anal. Applications, 252 (2000), 102-123.

8. R. Weder, Inverse Scattering for the Nonlinear Schrödinger Equation. Reconstruction of the Potential and the Nonlinearity, Mathematical Methods in the Applied Sciences, 24 (2001), 245-254.

9. R. Weder, Multidimensional Inverse Scattering for the Nonlinear Klein--Gordon Equation with a Potential, Journal of Differential Equations, 184 (2002), 62-77.

10. R. Weder, The Aharonov-Bohm Effect and Time-Dependent Inverse Scattering Theory, Inverse Problems, 18 (2002), 1041-1056.

11. R. Weder, The Lp-Lp´ Estimate for the Schrödinger Equation on the Half-Line, Journal of Mathematical Analysis and Applications, 281 (2003), 233-243.

12. R. Weder, Inverse Scattering for the Forced Non-Linear Schrödinger Equation with a Potential on the Half-Line, Mathematical Methods in the Applied Sciences, 28 (2005), 1219-1236.

13. M. A. Sandoval Romero and R. Weder, The Initial Value Problem, Scattering and Inverse Scattering, for Schrödinger Equations with a Potential and a Non-Local non-Linearity, Journal of Physics A. Mathematical and General, 39 (2006), 11461-11478.