# 'Quantum integrability: old and new tools': first Volin Lecture tomorrow at 10:00.

Dear all, please find below a short summary of the first Volin's lecture, to be held tomorrow at 10:00 in IPhT.

The Organizers

Part 1 (before coffee break):

Bethe equations for the spectrum of anharmonic oscillator.

We will depart from the Schrodinger equation with polynomial potential,

discuss WKB approximation, Stokes phenomena, and derive the Bohr-Sommerfeld

quantization condition using very basic analytic continuation arguments.

Then we discuss the full quantum picture and show that the spectral

determinants for this system obey certain Wronskian relations which lead to

Bethe equations on the Hamiltonian's eigenvalues.

Part 2 (after coffee break)

Characters and integrability

We will recall some basic facts about Schur-Weyl duality and character

formulae. Then the discussion of SU(N) XXX spin chain from this perspective

is due, with the goal to introduce the concept of transfer matrices and

Hirota equations. We end up with presenting Wronskian solution of the

Hirota equations and Wronskian relations on Baxter Q-functions which, for

the SU(2) case, happen to be identical to those we observed when solving

the anharmonic oscillator.

Time permitting, generalization to non-compact supersymmetric cases will be

given, otherwise it is the first topic of the lecture next week.