Quantum systems at the Brink
For quantum systems one usually distinguishes between bound states energies and the continuous spectrum. The continuous spectrum is where transport happens, the bound states energies are eigenvalues of the hamiltonian belonging to the discrete spectrum. Usually the bound state eigenvalues are well below the continuous spectrum, with a safety distance to the continuous (or. more precisely, the essential) spectrum. In this case usual perturbation theory works and there are simple criteria for the existence or non-existence of bound states.
Now imagine you have a quantum system which depends on a parameter and you can tune this parameter such that the ground state energy hits the edge of the continuous spectrum at some critical value. Does the ground state survive or does it disappear at the critical parameter? I.e., does the ground state gets wider and wider and it disappears in the limit or does it stay finite and then explodes, if one varies the parameter just a tiny bit past its critical value?
How can one decide between these scenarios? How does one prove the existence of a ground state at the critical parameter value, when the ground state energy is at the edge of the continuous spectrum? This is obviously a highly unstable situation where perturbation theory does not apply since there is no safety distance of the ground state energy to the continuous spectrum anymore when the parameter is critical.
What is the physical mechanism which stabilizes a quantum system so that its ground state is still bound at the critical parameter?
Can one get decay estimates for the ground state even though the usual approaches to this type of questions usually need a safety distance of the ground state energy to the continuous spectrum?
In this lecture series, we develop the necessary tools to study these types of questions.
schedule:
Monday, March 31st, 14.00 - 16.00
Thursday, April 3rd, 10:00 - 12:00
Friday, April 4th, 14:00 - 16:00
Tuesday, April 8th, 14:00 - 16:00
Thursday, April 10th, 10:00 - 12:00