arXiv subjects

Erik Skibsted

Publications and source records attributed to Erik Skibsted.

15 recordsLinked to original sources

arXiv2026

Finite energy subspace for time-periodic Schrödinger operators

For $N$-body Schrö\-dinger operators with time-periodic short-range pair-potentials we show by a time-dependent commutator method that all channel wave operators exist. For $N=2$ we prove asymptotic completeness by a simplified version of the method, recovering Yajima's completeness result \cite{Yaj1} proven by a stationary method. We propose a definition of a \emph{finite energy subspace}, intuitively consisting of states with `finite asymptotic energy'. This geometric notion is used to characterize the \emph{wave operator subspace} given as the direct sum of the ranges of the channel wave operators. Thus our main result states that the two subspaces coincide. In turn they coincide for $N= 2$ with the orthogonal subspace of the pure point subspace of the monodromy operator (by asymptotic completeness). For $N\geq 3$ asymptotic completeness for time-periodic short-range potentials remains an open problem. The main result of the paper may in this case potentially serve as an intermediate step for proving (or possibly disproving) asymptotic completeness. Another potential ingredient could be a key intermediate result of the paper, asserting that the states orthogonal to the pure point subspace obey a (sharp) \emph{minimal velocity bound}. Thus it remains an open problem possibly to derive a good maximal velocity bound for $N\geq 3$. Our results apply to $N$-body systems of particles in a time-periodic external electric field with time-mean equal to zero, for example the AC-Stark model.

math.SP↗

arXiv2024

Asymptotic completeness for short-range N-body systems revisited

We review Yafaev's approach to asymptotic completeness for systems of particles mutually interacting with short-range potentials. The theory is based on computation of commutators with time-independent (mostly bounded) observables yielding a sufficient supply of Kato smoothness bounds.

math.SP↗

arXiv2024

Green functions and completeness; the $3$-body problem revisited

Within the class of Derezi{ń}ski-Enss pair-potentials which includes Coulomb potentials a stationary scattering theory for $N$-body systems was recently developed \cite {Sk1}. In particular the wave and scattering matrices as well as the restricted wave operators are all defined at any non-threshold energy, and this holds without imposing any a priori decay condition on channel eigenstates. In this paper we improve for the case of $3$-body systems on the known \emph{weak continuity} properties in that we show that all non-threshold energies are \emph{stationary complete} in this case, resolving a conjecture from \cite {Sk1} in the special case $N=3$. A consequence is that the above scattering quantities depend \emph{strongly continuously} on the energy parameter at all non-threshold energies, hence not only almost everywhere as previously demonstrated (for an arbitrary $N$). Another consequence is that the scattering matrix is unitary at any such energy. As a side result we give an independent stationary proof of asymptotic completeness for $3$-body systems with long-range pair-potentials. This is an alternative to the known time-dependent proofs \cite{De, En}.

math-ph↗

arXiv2024

Stationary completeness: the $N$-body short-range case

For a general class of $N$-body Schrödinger operators with short-range pair-potentials the wave and scattering matrices as well as the restricted wave operators are all defined at any non-threshold energy. This holds without imposing any a priori decay condition on channel eigenstates and even for models including long-range potentials of Dereziński-Enss type. In this paper we improve for short-range models on the known weak continuity properties in that we show that all non-threshold energies are stationary complete, resolving in this case a conjecture from [Sk1]. A consequence is that the above scattering quantities depend strongly continuously on the energy parameter at all non-threshold energies (improving on previously almost everywhere proven properties). Another consequence is that the scattering matrix is unitary at any such energy. As a side result we obtain a new and purely stationary proof of asymptotic completeness for $N$-body short-range systems.

math-ph↗

arXiv2023

Spectral analysis of $N$-body Schrödinger operators at two-cluster thresholds

This book provides a systematic study of spectral and scattering theory for many-body Schrödinger operators at two-cluster thresholds. While the two-body problem (reduced after separation of the center of mass motion to a one-body problem at zero energy) is a well-studied subject, the literature on the many-body problem is sparse. However our analysis covers for example the system of three particles interacting by Coulomb potentials and restricted to a small energy region to the right of a fixed nonzero two-body eigenvalue $λ_0$. In general we address the question: How does scattering quantities for the many-body atomic and molecular models behave in the limit when the total energy approaches a fixed two-cluster threshold $λ_0$? This includes mapping properties and singularities of the limiting scattering matrix, asymptotics of the total scattering cross-section and absence of transmission from one channel to another in the small inter-cluster kinetic energy region. Our principal tools are the Feshbach-Grushin dimension reduction method and spectral analysis based on a certain Mourre estimate. Additional topics (of independent interest) are the limiting absorption principle, micro-local resolvent estimates, Rellich and Sommerfeld type theorems and asymptotics of the limiting resolvents at thresholds. While all of these features are fairly well-understood for two-body Schrödinger operators, they are poorly understood in the many-body case even for two-cluster thresholds. It is the goal of the book to remedy this point. The mathematical physics field under study is very rich, and there are many open problems, several of them stated explicitly in the book for the interested reader.

math-ph↗

arXiv2021

Stationary scattering theory, the $N$-body long-range case

Within the class of Derezi{ń}ski-Enss pair-potentials which includes Coulomb potentials and for which asymptotic completeness is known \cite{De}, we show that all entries of the $N$-body quantum scattering matrix have a well-defined meaning at any given non-threshold energy. As a function of the energy parameter the scattering matrix is weakly continuous. This result generalizes a similar one obtained previously by Yafaev for systems of particles interacting by short-range potentials \cite{Ya1}. As for Yafaev's paper we do not make any assumption on the decay of channel eigenstates. The main part of the proof consists in establishing a number of Kato-smoothness bounds needed for justifying a new formula for the scattering matrix. Similarly we construct and show strong continuity of channel wave matrices for all non-threshold energies. Away from a set of measure zero we show that the scattering and channel wave matrices constitute a well-defined `scattering theory', in particular at such energies the scattering matrix is unitary, strongly continuous and characterized by asymptotics of minimum generalized eigenfunctions.

math-ph↗

arXiv2019

Time-dependent scattering theory on manifolds

Based on our previous study [IS3] on the stationary scattering theory for the Schrodinger operator on a manifold possessing an escape function we complete our investigation by doing the time-dependent counterpart. A particular class of examples are manifolds with Euclidean and/or hyperbolic ends, possibly with unbounded and non-smooth obstacles. As an application we resolve a conjecture of [HPW] on cross-ends transmissions in its natural and strong form within the time-dependent framework.

math.DG↗

arXiv2016

Decay of eigenfunctions of elliptic PDEs, II

We study exponential decay rates of eigenfunctions of self-adjoint higher order elliptic operators on R^n. We are interested in decay rates as a function of direction. We show that the possible decay rates are to a large extent determined algebraically.

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arXiv2013

Decay of eigenfunctions of elliptic PDE's

We study exponential decay of eigenfunctions of self-adjoint higher order elliptic operators on $\R^d$. We show that the possible critical decay rates are determined algebraically. In addition we show absence of super-exponentially decaying eigenfunctions and a refined exponential upper bound.

math.SP↗

arXiv2012

Renormalized two-body low-energy scattering

For a class of long-range potentials, including ultra-strong perturbations of the attractive Coulomb potential in dimension $d\geq3$, we introduce a stationary scattering theory for Schrödinger operators which is regular at zero energy. In particular it is well defined at this energy, and we use it to establish a characterization there of the set of generalized eigenfunctions in an appropriately adapted Besov space generalizing parts of \cite{DS3}. Principal tools include global solutions to the eikonal equation and strong radiation condition bounds.

math-ph↗

arXiv2011

Sommerfeld radiation condition at threshold

We prove Besov space bounds of the resolvent at low energies in any dimension for a class of potentials that are negative and obey a virial condition with these conditions imposed at infinity only. We do not require spherical symmetry. The class of potentials includes in dimension $\geq3$ the attractive Coulomb potential. There are two boundary values of the resolvent at zero energy which we characterize by radiation conditions. These radiation conditions are zero energy versions of the well-known Sommerfeld radiation condition.

math-ph↗

arXiv2010

Two-body threshold spectral analysis, the critical case

We study in dimension $d\geq2$ low-energy spectral and scattering asymptotics for two-body $d$-dimensional Schrödinger operators with a radially symmetric potential falling off like $-γr^{-2},\;γ>0$. We consider angular momentum sectors, labelled by $l=0,1,\dots$, for which $γ>(l+d/2-1)^2$. In each such sector the reduced Schrödinger operator has infinitely many negative eigenvalues accumulating at zero. We show that the resolvent has a non-trivial oscillatory behaviour as the spectral parameter approaches zero in cones bounded away from the negative half-axis, and we derive an asymptotic formula for the phase shift.

math.SP↗

arXiv2009

Analyticity estimates for the Navier-Stokes equations

We study spatial analyticity properties of solutions of the Navier-Stokes equations and obtain new growth rate estimates for the analyticity radius. We also study stability properties of strong global solutions of the Navier-Stokes equations with data in H^r, r greater or equal 1/2, and prove a stability result for the analyticity radius.

math-ph↗

arXiv2007

Quantum scattering at low energies

For a class of negative slowly decaying potentials, including $V(x):=-γ|x|^{-μ}$ with $0<μ<2$, we study the quantum mechanical scattering theory in the low-energy regime. Using modifiers of the Isozaki-Kitada type we show that scattering theory is well behaved on the whole continuous spectrum of the Hamiltonian, including the energy 0. We show that the S-matrices are well-defined and strongly continuous down to the zero energy threshold. Similarly, we prove that the wave matrices and generalized eigenfunctions are norm continuous down to the zero energy if we use appropriate weighted spaces. These results are used to derive (oscillatory) asymptotics of the standard short-range and Dollard type S-matrices for the subclasses of potentials where both kinds of S-matrices are defined. For potentials whose leading part is $-γ|x|^{-μ}$ we show that the location of singularities of the kernel of $S(λ)$ experiences an abrupt change from passing from positive energies $λ$ to the limiting energy $λ=0$. This change corresponds to the behaviour of the classical orbits. Under stronger conditions we extract the leading term of the asymptotics of the kernel of $S(λ)$ at its singularities; this leading term defines a Fourier integral operator in the sense of Hörmander.

math-ph↗

arXiv2007

Classical and quantum dynamics for 2D-electromagnetic potentials asymptotically homogeneous of degree zero

We consider a charged particle moving in the plane subject to electromagnetic potentials with non-vanishing radial limits. We analyse the classical and the quantum dynamics for large time in the case the angular part of the (limiting) Lorentz force (defined for velocities that are purely radial) has a finite number of zeros at fixed energy. Any such zero defines a channel, and to the "stable" ones we associate quantum wave operators. Their completeness is studied in the case of zero as well as nonzero magnetic flux. In the latter case one needs possibly to incorporate a channel of spiraling states. These states are similar to those studied recently in the sign-definite case in \cite {CHS}.

math-ph↗