# Analysis and PDE Seminar Homepage, Fall 2013

Location: 383N

Time: Thursday, 2:15-3:15pm.

### Stability estimates for the inverse boundary value problem for the Helmholtz equation and iterative reconstruction''

Abstract: We consider time-harmonic waves described by the Helmholtz equation and view the Dirichlet-to-Neumann map as the data. We establish a relationship between this map and seismic data, and include attenuation. We obtain conditional Lipschitz stability for the associated inverse boundary value problem with partial boundary data. The stability is obtained for wavespeed models of the form of finite linear combinations of piecewise constant functions, naturally including the presence of conormal singularities. The dimension of the corresponding space is determined by the number of linearly independent combinations. We give an estimate of the stability constant in terms of the dimension of this model space. Using multi-frequency data, we then design a multi-level iterative method with associated conditions expressed in terms of compression for convergence to the unique solution derived from the mentioned stability estimates.

Joint research with E. Beretta, L. Qiu and O. Scherzer.

### Second microlocalization at coisotropic submanifolds''

Abstract: Solutions to linear hyperbolic equations such as the classical wave equation often enjoy what Hörmander (and later Melrose) termed Lagrangian regularity''. Indeed, the fundamental solution to the wave equation is a Lagrangian distribution associated to the graph of geodesic flow on the cotangent bundle. In this talk, we will describe a generalization of this type of regularity where the Lagrangian submanifold of the cotangent bundle of the ambient space is replaced by a coisotropic submanifold, i.e., coisotropic regularity''. We will also describe a second microlocal'' pseudodifferential calculus associated to such a coisotropic, i.e., a calculus which will test for coisotropic regularity in an analogous fashion to how the usual pseudodifferential calculus tests for smooth (or Sobolev) regularity (via a second wavefront set). Time permitting, we'll discuss the propagation of second microlocal singularities in analogy with Hörmander's theorem on propagation of singularities in the smooth regularity setting.

### Resonances in Scattering by Two Magnetic Fields at Large Separation and a New Complex Scaling Method''

Abstract: We study the quantum resonances in magnetic scattering in two dimensions. The scattering system consists of two obstacles by which the magnetic fields are completely shielded. The trajectories trapped between the two obstacles are shown to generate the resonances near the positive real axis, when the distance between the obstacles goes to infinity. The location is described in terms of the backward amplitudes for scattering by each obstacle. A difficulty arises from the fact that even if the supports of the magnetic fields are largely separated from each other, the corresponding vector potentials are not expected to be well separated. To overcome this, we make use of a gauge transformation and develop a new type of complex scaling method. We can cover the scattering by two solenoids at large separation as a special case. The obtained result heavily depends on the magnetic fluxes of the solenoids. This indicates that the Aharonov–Bohm effect influences the location of resonances. This is joint work with Hideo Tamura.

### Semilinear wave equations on asymptotically de Sitter, Kerr-de Sitter and Minkowski spacetimes''

Abstract: I will discuss the small data solvability of suitable semilinear wave and Klein-Gordon equations on geometric classes of spaces, which include asymptotically de Sitter, Kerr-de Sitter and Minkowski spacetimes. Our results are obtained by showing the global Fredholm property, and indeed invertibility, of the underlying linear operator on suitable $L^2$-based function spaces, which also possess appropriate algebra or more complicated multiplicative properties. The linear framework is based on b-analysis, introduced in this context by Vasy to describe the asymptotic behavior of solutions of linear equations. An interesting feature of the analysis is that resonances, namely poles of the inverse of the Mellin transformed b-normal operator, play an important role. Joint work with Andras Vasy.

### Quantum Fisher Information for Noisy Dynamics: A semi-classical approach''

Abstract: Parameter estimation of probability distributions is one of the most basic tasks in information theory, and has been generalized to the quantum regime since the description of quantum measurement is essentially probabilistic. The "quantum metrology" prescription is straightforward for closed systems evolving unitarily,  becoming more challenging when the system is coupled (more realistically) to an external environment. This produces "noisy" non-unitary dynamics. This noise, or decoherence, degrades the precision in any parameter estimation -- precision that is quantified by Quantum Fisher Information. For physically relevant noise models including both phase diffusion and dissipation we investigate the scaling of single parameter precision with the noise amplitude and "resource" N, the number of system dimensions. Using a novel operator approach (rather than WKB), we find new saturable precision bounds in the asymptotic limit of large N in tandem with those quantum states uniquely capable of reaching these bounds. Convergence to asymptotic behaviour can set in quickly for modest N<100, and as such our analysis is relevant to sensing and metrology experiments incorporating ensembles of 10  to 100 particles; potentially atoms, magnetic spins, flux-qubits or photons.

### Boundary values of resolvents of self-adjoint operators in Krein spaces and applications to the Klein-Gordon equation''

Abstract : 30 years ago, E. Mourre showed that a local in energy positive commutator estimate for a selfadjoint operator H entails a limiting absorption principle for this operator and thus the absence of singular continuous spectrum. This result had a very deep impact in scattering theory leading in particular to asymptotic completeness results for quantum N-particle systems. A lot of efforts had been made to weaken the original hypotheses in the work of Mourre. A central requirement remained however that the hamiltonian H is a selfadjoint operator on a Hilbert space. Whereas this is a very natural requirement for the Schrödinger equation, it turns out that it is in general not fulfilled for the Klein-Gordon equation when this equation is coupled to an electric field or associated to a lorentzian metric which is not stationary (in the sense that there exists no global timelike Killing vector field). The natural setting in this situation seems to be the one of a selfadjoint operator on a so called Krein space. A Krein scalar product is a non degenerate hermitian form which is not necessarily positive. Krein spaces are generalizations of Hilbert spaces.

The talk is devoted to the proof of resolvent estimates for the boundary values of resolvents of selfadjoint operators on a Krein space. Our result generalizes the result of Mourre to the Krein space setting. We will also discuss applications to the Klein-Gordon operator coupled to an electromagnetic field. This is joint work with Vladimir Georgescu and Chrsitian Gérard.

### Critical Self-Gravitating Wave Maps''

Abstract: Wave maps are maps from a Lorentzian manifold to a Riemannian manifold which are critical points of a Lagrangian which is a natural geometrical generalization of the free wave Lagrangian. Self-gravitating wave maps are those from an asymptotically flat Lorentzian manifold which evolves according to Einstein's equations of general relativity with the wave map itself as the source. The energy of wave maps is scale invariant if the domain manifold is 2+1 dimensional, hence it is referred to as the critical dimension.

Apart from a purely mathematical interest, the motivation to study critical self-gravitating wave maps is that they occur naturally in 3+1 Einstein's equations of general relativity. Therefore, studying critical self-gravitating wave maps could be a fruitful way of understanding the ever elusive global behavior of Einstein's equations. A few central questions concerning the study of critical self-gravitating wave maps are local and global existence, blow up criterion, profile, compactness and bubbling.

In this talk, after a brief discussion on the background and formulation of the Cauchy problem of critical self-gravitating wave maps, we shall present a recent proof of the non-concentration of energy of critical equivariant self-gravitating wave maps before pointing out potential generalizations and applicable methods therein.

### The trace formula for diffractive closed geodesics on a conic manifold''

Abstract: We consider the trace of the wave group on a compact manifold with conic singularities. The trace of the wave group, which on the one hand equals $\sum e^{-it\lamba_j}$ where $\lambda_j^2$ are the eigenvalues of the Laplacian, is on the other hand a distribution in $t,$ singular at lengths of closed geodesics. Those closed geodesics that interact with the cone points generically do so `diffractively,' carrying singularities into regions of phase space inaccessible to ordinary geodesic flow. We describe a formula for the leading order singularity of the wave trace at lengths of closed diffractive geodesics, generalizing the formula due to Duistermaat-Guillemin in the smooth setting and to Hillairet in the setting of flat surfaces with conic singularities. This is joint work with G. Austin Ford.