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Register a new userRefined Weyl law for homogeneous perturbations of the harmonic oscillator
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Let $H$ denote the harmonic oscillator Hamiltonian on $mathbb{R}^d,$ perturbed by an isotropic pseudodifferential operator of order $1.$ We consider the Schrodinger propagator $U(t)=e^{-itH},$ and find that while $operatorname{singsupp} operatorname{Tr} U(t) subset 2 pi mathbb{Z}$ as in the unperturbed case, there exists a large class of perturbations in dimension $d geq 2$ for which the singularities of $operatorname{Tr} U(t)$ at nonzero multiples of $2 pi$ are weaker than the singularity at $t=0$. The remainder term in the Weyl law is of order $o(lambda^{d-1})$, improving in these cases the $O(lambda^{d-1})$ remainder previously established by Helffer--Robert.
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Let $T$ be the generator of a $C_0$-semigroup $e^{-Tt}$ which is of finite trace for all $t>0$ (a Gibbs semigroup). Let $A$ be another closed operator, $T$-bounded with $T$-bound equal to zero. In general $T+A$ might not be the generator of a Gibbs semigroup. In the first half of this paper we give sufficient conditions on $A$ so that $T+A$ is the generator of a Gibbs semigroup. We determine these conditions in terms of the convergence of the Dyson-Phillips expansion corresponding to the perturbed semigroup in suitable Schatten-von Neumann norms. In the second half of the paper we consider $T=H_vartheta=-e^{-ivartheta}partial_x^2+e^{ivartheta}x^2$, the non-selfadjoint harmonic oscillator, on $L^2(mathbb{R})$ and $A=V$, a locally integrable potential growing like $|x|^{alpha}$ for $0leq alpha<2$ at infinity. We establish that the Dyson-Phillips expansion converges in this case in an $r$ Schatten-von Neumann norm for $r>frac{4}{2-alpha}$ and show that $H_vartheta+V$ is the generator of a Gibbs semigroup $mathrm{e}^{-(H_vartheta+V)tau}$ for $|arg{tau}|leq frac{pi}{2}-|vartheta|$. From this we determine asymptotics for the eigenvalues and for the resolvent norm of $H_vartheta+V$.
The one dimensional wave equation serves as a basic model for imaging modalities such as seismic which utilize acoustic data reflected back from a layered medium. In 1955 Peterson et al. described a single scattering approximation for the one dimensional wave equation that relates the reflection Greens function to acoustic impedance. The approximation is simple, fast to compute and has become a standard part of seismic theory. The present paper re-examines this classical approximation in light of new results concerning the (exact) measurement operator for reflection imaging of layered media, and shows that the classical approximation can be substantially improved. We derive an alternate formula, called the refined impedance transform, that retains the simplicity and speed of computation of the classical estimate, but which is qualitatively more accurate and applicable to a wider range of recorded data. The refined impedance transform can be applied to recorded data directly (without the need to deconvolve the source wavelet), and solves exactly the inverse problem of determining the value of acoustic impedance on the far side of an arbitrary slab of unknown structure. The results are illustrated with numerical examples.
We return to the description of the damped harmonic oscillator by means of a closed quantum theory with a general assessment of previous works, in particular the Bateman-Caldirola-Kanai model and a new model recently proposed by one of the authors. We show the local equivalence between the two models and argue that latter has better high energy behavior and is naturally connected to existing open-quantum-systems approaches.
In this paper we give a new and simplified proof of the theorem on selection of standing waves for small energy solutions of the nonlinear Schrodinger equations (NLS) that we gave in cite{CM15APDE}. We consider a NLS with a Schrodinger operator with several eigenvalues, with corresponding families of small standing waves, and we show that any small energy solution converges to the orbit of a time periodic solution plus a scattering term. The novel idea is to consider the refined profile, a quasi--periodic function in time which almost solves the NLS and encodes the discrete modes of a solution. The refined profile, obtained by elementary means, gives us directly an optimal coordinate system, avoiding the normal form arguments in cite{CM15APDE}, giving us also a better understanding of the Fermi Golden Rule.
In this paper we prove uniqueness for an inverse boundary value problem (IBVP) arising in electrodynamics. We assume that the electromagnetic properties of the medium, namely the magnetic permeability, the electric permittivity and the conductivity, are described by continuously differentiable functions.
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