**November 26, 2018 at 16:00 - November 30, 2018**
- BCAM

**Aingeru Fernández Bertolin (UPV/EHU) and Diana Stan (University of Cantabria)**

**DATES:** 26-30 November 2018 (5 sessions)

**TIME:** 16:00 - 18:00 (a total of 10 hours)

**LOCATION:** BCAM Seminar room

This course is an introduction to the Unique Continuation (UC) Property in the sense of Hardy Un-certainty Principle: consider u(t,x) a solution to an evolution equation at time t and space variable x. Assuming that u(0,x) and u(1,x) decay sufficiently fast for large |x|, then we derive that the only possibility is u=0.

We will prove the UC property for the Schrödinger equation with potential, both in the continuous and discrete settings.

**PROGRAMME:**
**1.** Intro and motivation: Hardy’s theorem and the relation between the theorem and UC properties for the free Schrödinger equation.

**2.** Unique Continuation for the Schrödinger equation with potential. Persistence properties for solutions with fast decay at two different times.

**3.** Monotonicity formulas (log-convexity) and Carleman estimates using real variable methods.

**4.** If time permits, some words on the sharp result.

**5.** Unique Continuation for Discrete Schrödinger Equation.

**OBJECTIVES:**
Get familiar with the techniques of Carleman estimates and monotonicity formulas.

**PREREQUISITES: **
Basic functional analysis and PDEs.

***Registration is free, but mandatory before November 21st: **So as to inscribe go to

https://bit.ly/2pVNl4M and fill the registration form. Student grants are available. Please, let us know if you need support for travel and accommodation expenses when you fill the form.