We consider a one-dimensional Schrodinger operator with matrix-valued potential. For such an operator, the WKB construction breaks down, not only at the turning points, but also at the crossing points where the potential has multiple eigenvalues. We study the connection problem at the crossing points and compute the semiclassical asymptotics of what we call microlocal scattering matrix. We apply this result to the semiclassical resonance width arising from the molecule predissociation in quantum chemistry.
The overall goal of these lecture notes is twofold: (i) to provide an overview of the state of the art for boundary layer theory for stationary flows; (ii) to state the result, in simplified form, of a recent work, Reversal in stationary Prandtl equations by Sameer Iyer and Nader Masmoudi.
This survey is devoted to the main WKB aspects of the Isozaki-Kitada parametrix and to the applications thereof in scattering theory.
We study semiclassical 1D Schr & ouml;dinger operators of the form Pu =-h(2)u" + x(Y)W(x)u on a finite interval [0, b] for 0 < y is an element of R \ Q. We show that the WKB expansions of solution can be extended on [h(1-epsilon), b] for any e > 0. Using a different approximation near 0 and a matching procedure, we obtain the Cauchy data at 0 of such WKB solutions. This allows us to derive singular Bohr-Sommerfeld rules. We also pay special attention to uniformity in W for our expansions.
In this paper, we examine the relationship between WKB approximations and semiclassical parametrices. The fact that the two are related is a point of mathematical folk law in the field of semiclassical analysis. Here, we will set the relationship out precisely and illustrate it with a couple of examples. We will see that semiclassical parametrices for the Schr & ouml;dinger propagator can be constructed directly from WKB solutions and that this approach has the advantage of allowing us to consider parametrices accurate only to highest order (in h) rather than all orders. Such parametrices are often sufficient to suit the technical needs of semiclassical analysis and the WKB method of finding approximate solutions is particularly suitable for obtaining useful explicit expressions for the top-order terms.
These lectures are devoted to two integrable PDE on the line enjoying similar structures: the Benjamin-Ono equation and the Calogero-Moser derivative nonlinear Schr & ouml;dinger equation. For both equations, a Lax pair of operators is introduced on the Hardy space of the upper-half plane, and is used to prove conservation laws and explicit formulae, and to study soliton and multisoliton solutions. In the special case of the Benjamin-Ono equation, the small dispersion limit with general initial data is proved to exist and is identified. These lectures were presented at the 2024 PDE Days, Centre Paul Langevin, Aussois, France.
In this paper, we will prove Bohr-Sommerfeld quantization rules for the self-adjoint Zakharov-Shabat system and the Schr & ouml;dinger equation in the presence of two simple turning points bounding a classically allowed region. In particular, we use the method of comparison equations for 2x2 traceless first-order systems to provide a unified perspective that yields similar proofs in each setting. The use of a Weber model system gives results that are uniform in the eigenvalue parameter over the whole range from the bottom of the potential well up to finite values.
This survey focuses on the geometric problem of log-surfaces, which are pairs consisting of a smooth projective surface and a reduced non-empty boundary divisor. In the first part, we focus on the geography problem for complex log-surfaces associated with pairs of the form (P-2, C), where C is an arrangement of smooth plane curves admitting ordinary singularities. Specifically, we focus on the case in which C is an arrangement consisting of smooth rational curves as its irreducible components. In the second part, containing original new results, we study log-surfaces constructed as pairs consisting of a complex projective K3 surface and a rational curve arrangement. In particular, we provide some combinatorial conditions for such pairs to have the log-Chern slope equal to 3. Our survey is illustrated with many explicit examples of log-surfaces.
In this paper, extending some ideas of Fano in [Fano, 1949] and of the first and last authors in [Andreatta-Pignatelli, 2023], we study the birational geometry of the Hilbert scheme of 0-dimensional subschemes of length 2 of a rational normal scroll F-n. This fourfold has three elementary contractions associated to the three faces of its nef cone. We study natural projective realizations of these contractions. In particular, given a smooth rational normal scroll S(a,b )of degree r in Pr+1 with 1 <= a <= b and a+b=r, i.e., S-a,S-b=P(O(P)1(a)circle plus O(P)1(b)) embedded in Pr+1 with its O(1) line bundle (from an abstract viewpoint S-a,S-b congruent to Fb-a), we consider the variety X-a,X-b subset of G(1,r+1) described by all lines that are secant or tangent to S-a,S-b. The variety X-a,X-b is the image of some of the aforementioned contractions; it is smooth if a>1, and it is singular at a unique point if a=1. We compute the degree of X-a,X-b and the local structure of the singularity of X-a,X-b when a=1. Finally, we discuss in some detail the case r=4, originally considered by Fano in [Fano, 1949], because the smooth hyperplane sections of X-2,X-2 and X-1,X-3 are the Fano 3-folds that appear as number 16 in the Mori-Mukai list of Fano 3-folds with Picard number 2. We prove that any smooth hyperplane section of X-2,X-2 is also a hyperplane section of X-1,X-3,X- and we discuss the GIT-stability of the smooth hyperplane sections of X-1,X-3, where G is the subgroup of the projective automorphisms of X-1,X-3 coming from the ones of S-1,S-3
This article surveys and develops the use of simultaneous uniformization for the study of chord-arc curves in the BMO Teichmüller space. The method of simultaneous uniformization provides a unified complex-analytic framework in which chord-arc curves are parametrized by their BMO embeddings and the logarithm of derivatives of these embeddings forms a biholomorphic image in the Banach space of BMO functions. We review this correspondence and its consequences, such as the relation to reparametrizations by strongly quasisymmetric homeomorphisms, in a rather self-contained manner in order to highlight the coherence of the approach.The main new contribution of this exposition concerns the Cauchy transform of BMO functions on a chord-arc curve. We show that the Cauchy transform is expressed through the derivative of the biholomorphic map arising from simultaneous uniformization and consequently depends holomorphically on the variation of the chord-arc curve. This result connects classical singular integral operators with the complex structure of Teichmüller spaces and illustrates the effectiveness of the method. We also outline the parallel theory in the VMO Teichmüller space, which exhibits further structural properties.
In this article, we give an introduction to asymptotic stability in two dimensional incompressible flows, and a non-technical overview of the recent proof of uniform-in-viscosity inviscid damping and vorticity depletion near periodic shear flows on a non-square torus.
This is a survey on what is known, up to date, on normal subgroups of Cremona groups. There are several different approaches to showing that they exist, and we will take a look at each of them, more or less in chronological order.
We study the intersection form F_{X} on the second cohomology group H^{2}(X, \mathbb{Z}) of a compact Kähler manifold X of dimension n . Although the structure of F_{X} is relatively well understood in dimensions two and three, much less is known for n \geq 4 . We investigate the fundamental properties of F_{X} in higher dimensions and discuss several applications to birational geometry. Finally, we present a number of open problems concerning the relationship between birational invariants and topological invariants of Kähler manifolds.
We formulate an effective variant of the Yau–Tian–Donaldson conjecture and then review effective results on K-stability of spherical varieties, that is, K-stability criteria which can be effectively computed given the combinatorial data associated with the variety. We focus on the standard notion of K-stability as defined by Donaldson for constant scalar curvature Kähler metrics.
We construct a non-commutative version of the Grassmann variety G(2,4) as a non-commutative moduli space of linear subspaces in a projective space.
For a locally free sheaf $\mathcal{E}$ on a smooth projective curve, we can define the punctual Quot scheme which parametrizes torsion quotients of $\mathcal{E}$ of length $n$ supported at a fixed point. It is known that the punctual Quot scheme is a normal projective variety with canonical Gorenstein singularities. In this note, we show that the punctual Quot scheme is a $\mathbb{Q}$-factorial Fano variety of Picard number one.
In this note, we report some recent progress on the Jordan property for (birational) automorphism groups of projective varieties and compact complex varieties.
WS: it is difficult to write about Thomas Kappeler in the past tense. He was a brilliant mathematician, but more importantly he was a wonderfully open, generous, and friendly person. I was fortunate that we had many opportunities to spend time together and discuss mathematics. I greatly miss him.A Stokes wave is a traveling free-surface periodic water wave that is constant in the direction transverse to the direction of propagation. Even Stokes waves of very small amplitude are unstable when subjected to various perturbations. We present a brief survey of this phenomenon with emphasis on transverse perturbations.
We give an essentially self-contained treatment of the fundamental analytic and algebraic features of regularity structures and their applications to the study of singular stochastic PDEs.
