# Seminar

 Title : Denis Grebenkov (École Polytechnique). The Bloch-Torrey operator: eigenmodes localization, branching spectrum, asymptotic analysis, and applications. Contact : Maryna Kachanovska Date : 03/02/2022 Place : 14h, amphi 2234
The Bloch-Torrey operator A_q = −\Delta + iqx, where \Delta is the Laplace operator, is a non-self-adjoint differential operator that naturally appears in the context of magnetic resonance imaging [1,2]. In 1D, it reduces to the complex Airy operator −\frac{d^2}{dx^2}+iqx. If q \neq 0, the latter has an empty spectrum on \mathbb{R} but a discrete spectrum on \mathbb{R}^+ (with Dirichlet, Neumann, or Robin boundary conditions), in contrast with the well-known spectral properties of the Laplace operator (i.e. when q=0). Moreover, the eigenmodes are localized at x=0 [1,3]. These unconventional properties suggest a conjecture that the spectrum of the Bloch-Torrey operator is discrete in any domain, in particular unbounded domains, except for trivial cases (such as free space \mathbb{R}^n with no obstacle). This conjecture was proved in [4,5] for the 2D case. Additionally, an approximate eigenmode construction performed at high q indicates that the eigenmodes are localized near the boundary of the obstacles in the domain. The Bloch-Torrey operator A_q has the length scale l_q =q^{-1/3}, which controls the spread of the localized eigenmodes. Thus, localization typically happens when l_q is much smaller than any other relevant geometric length scale in the domain. This is illustrated on Fig. 1 which shows the evolution of the spectrum of A_q inside a disk of radius R with Neumann boundary condition as a function of R/l_g =(qR)^{1/3}, along with some plots of the corresponding eigenmodes (obtained with a finite elements scheme) [2]. Quite remarkably, the transition from delocalized to localized eigenmodes is associated with branching points in the spectrum, where two real eigenvalues coalesce and become a conjugate pair. After the branching point, in the localization (high-q) regime, the eigenvalues \lambda_{k,l} (with positive integer indices k, l) obey the asymptotic expansion [5]. This asymptotic formula may be generalized to other boundary geometry, boundary condition, and higher dimension [2,5]. In this talk, we review the above properties and discuss their implications for magnetic resonance imaging. References [1] S. D. Stoller, W. Happer, and F. J. Dyson, Phys. Rev. A, 44, 7459—7477 (1991) [2] N. Moutal, D. S. Grebenkov, K. Demberg, T. A. Kuder, J. Magn. Res. 305, 162-174 (2019) [3] D. S. Grebenkov, B. Helffer, and R. Henry, SIAM J. Math. Anal.49, 1844—1894 (2017) [4] Y. Almog, D. S. Grebenkov and B. Helffer, J. Math. Phys. 59, 041501 (2018) [5] D. S. Grebenkov and B. Helffer, SIAM J. Math. Anal.50, 622—676 (2018)