On the Spectral Properties of the Perturbed Landau Hamiltonian

The Landau Hamiltonian governing the behavior of a quantum particle in dimension 2 in a constant magnetic field is perturbed by a compactly supported magnetic field and a similar electric field. We describe how the spectral subspaces change and how the Landau levels split under this perturbation.     

The Spectral Shift Function and the Friedel Sum Rule

We study the relationship between the spectral shift function and the excess charge in potential scattering theory. Although these quantities are closely related to each other, they have been often formulated in different settings so far. Here, we first give an alternative construction of the spectral shift function, and then we prove that the spectral shift function thus constructed yields the Friedel sum rule.     

Spectral Shift Function in the Large Coupling Constant Limit

In the large-coupling constant limit we obtain an asymptotic expansion in powers of of the derivative of the spectral shift function corresponding to . Here the potential W(x) is positive and near infinity for some δ> n and ω₀ ∈C^∞ (S^n-1). submitted 18/05/05, accepted 19/10/05     

The Spectral Shift Function for Certain Block Operator Matrices

Let L₀ be a 2 × 2 diagonal self‐adjoint block operator matrix with entries A and D. If operators B and B* are added to the off diagonal zeros, certain parts of the spectrum of L₀ move to the right and other parts move to the left. In this paper it is shown that, correspondingly, if B is a trace class operator M. G. Krein's spectral shift function is of constant sign on certain intervals.     

The Krein Spectral Shift Function in Semifinite von Neumann Algebras

We show the existence of a spectral shift function in the sense of Krein for bounded trace class perturbations of a self-adjoint operator affiliated with a semifinite von Neumann algebra     

Representation of the Singularities of a Function

We show that a functionfhaving a little uniform smoothness can be locally represented at any pointx₀as [equation] wheregis an indefinitely oscillatin function; the value of β of the regularity ofrare related by the 2-microlocal regularity iff.     

Spectral Shift Function of the Schrödinger Operator in the Large Coupling Constant Limit

The spectral shift function of a Schrödinger operator with a perturbation of definite sign is considered. The asymptotics of the spectral shift function for large coupling constant is studied, and results concerning positive and negative perturbations are compared. A more general case of unperturbed operator given by a function of the Laplacian is discussed. This case explains the dependence of the asymptotics of the spectral shift function on the perturbation potential on the one hand and on the order of the unperturbed operator on the other hand.     

关于Landau不等式

本文给出Landau不等式的一种改进。     

Jet cohomology of Isolated Hypersurface Singularities and Spectral Sequences

We study jet cohomology of isolated hypersurface singularities defined by partial differential forms and prove formulas to compute jet cohomology groups by linear algebra.

     

A Time-Independent Approach for the Study of the Spectral Shift Function and an Application to Stark Hamiltonians

In this paper we develop a time-independent approach for the study of the spectral shift function (SSF for short). We apply this method for the perturbed Stark Hamiltonian. We obtain a weak and a Weyl-type asymptotics with optimal remainder estimate of the SSF of the operator pair (P = P₀ + V(x), P₀ = ? h²Δ +x₁), x = (x₁,…, x_n) where V(x) ∈ 𝒞^∞(?ⁿ, ?) decays sufficiently fast at infinity, and h is a small positive parameter. Near a non-trapping energy λ, we give a pointwise asymptotic expansions in powers of h of the derivative of the SSF, and we compute explicitly the two leading terms.     

On the Singularities of Fractionalized Potential Functions

The singularities of harmonic functions generated under the action of an integral transform with a fractionalized kernel are studied. A simple relationship arises connecting pairs of singularities.     

The Singularities of the Distance Function Near Convex Boundary Points

In an open bounded set Ω, we consider the distance function from ∂Ω associated to a Riemannian metric with C ^1,1 coefficients. Assuming that Ω is convex near a boundary point x ₀, we show that the distance function is differentiable at x ₀ if and only if there exists the tangent space to ∂Ω at x ₀. Furthermore, if the distance function is not differentiable at x ₀ then there exists a Lipschitz continuous curve, with initial point at x ₀, such that the distance function is not differentiable along such a curve.      

On the Spectral Function of Some Higher Order Elliptic or Degenerate-elliptic Operators

^m σ/|α|=|β|=m D^α(a _αβD^β) is a higher order degenerate-elliptic operator on L²(R^N) and V ∈ L¹(R^N) we show in particular that s(λ) > 0 for all λ > 0 provided that ∫_RN V(x)dx≥ 0, V not equivalent to 0 and N ≤ 2m.     

On the Singularities of Surfaces Ruled by Conics

We classify the singularities of a surface ruled by conics: they are rational double points of type A _n or D _n . This is proved by showing that they arise from a precise series of blow-ups of a suitable surface geometrically ruled by conics. We determine also the family of such surfaces which are birational models of a given surface ruled by conics and obtained in a “minimal way” from it.     

On the equilibrium position of Ginzburg Landau vortices

We study a few problems related to superconducting vortices. Our main concern is the stable equilibrium distribution of them. Our starting point is the asymptotic form of the Ginzburg Landau energy functional with a large Ginzburg Landau parameter. We consider in particular the interaction of the vortices with an applied magnetic field, and the effects of impurities on the vortex locations.     

Spectral Invariance of Pseudodifferential Boundary Value Problems on Manifolds with Conical Singularities

Journal of Fourier Analysis and Applications - We prove the spectral invariance of the algebra of classical pseudodifferential boundary value problems on manifolds with conical singularities in the...