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We study and develop the stationary scattering theory for a class of one-body Stark Hamiltonians with short-range potentials, including the Coulomb potential, continuing our study in Adachi et al. (JDE 268: 5179–5206, 2020; Stationary scattering theory for 1-body Stark operators). The classical scattering orbits are parabolas parametrized by asymptotic orthogonal momenta, and the kernel of the (quantum) scattering matrix at a fixed energy is defined in these momenta. We show that the scattering matrix is a classical type pseudodifferential operator and compute the leading order singularities at the diagonal of its kernel. Our approach can be viewed as an adaption of the method of Isozaki-Kitada (Tokyo Univ. 35: 81–107, 1985) used for studying the scattering matrix for one-body Schrödinger operators without an external potential. It is more flexible and more informative than the more standard method used previously by Kvitsinsky-Kostrykin (Teoret. Mat. Fiz. 75(3): 416-430, 1988) for computing the leading order singularities of the kernel of the scattering matrix in the case of a constant external field (the Stark case). Our approach relies on Sommerfeld’s uniqueness result in Besov spaces, microlocal analysis as well as on classical phase space constructions.

We develop a new scheme of proofs for spectral theory of the [Formula: see text]-body Schrödinger operators, reproducing and extending a series of sharp results under minimum conditions. Our main results include Rellich’s theorem, limiting absorption principle bounds, microlocal resolvent bounds, Hölder continuity of the resolvent and a microlocal Sommerfeld uniqueness result. We present a new proof of Rellich’s theorem which is unified with exponential decay estimates studied previously only for [Formula: see text]-eigenfunctions. Each pair-potential is a sum of a long-range term with first-order derivatives, a short-range term without derivatives and a singular term of operator- or form-bounded type, and the setup includes hard-core interaction. Our proofs consist of a systematic use of commutators with ‘zeroth order’ operators. In particular, they do not rely on Mourre’s differential inequality technique.

We show an optimal version of the Rellich theorem for generalized many-body Schrodinger operators. It applies to singular potentials, in particular to a model for atoms and molecules with infinite mass and finite extent nuclei. Our proof relies on a Mourre estimate and a functional calculus localization technique., Comment: 11 pp

We compute up to a compact term the zero-energy scattering matrix for a class of potentials asymptotically behaving as −γ|x|−μ with 0 0. It turns out to be the propagator for the wave equation on the sphere at time .

We develop an extension of the abstract Mourre theory which consecutively is used to prove spectral properties of various systems coupled to a massless bosonic field. Our models include the spin-boson model and the standard model of quantum electrodynamics for a non-relativistic atom considered recently in [8] and [3] respectively.

We prove asymptotic completeness for short- and long-rangeN-body Stark Hamiltonians with local singularities of at most Coulomb type. Our results include the usual models for atoms and molecules.

We prove that the spectrum for a large class ofN-body Stark Hamiltonians is purely absolutely continuous. We need slow decay at infinity and local singularities of at most Coulomb type. In particular our results include the usual models for atoms and molecules.

We show absence of positive eigenvalues for generalized 2-body hard- core Schroedinger operators under the condition of bounded strictly convex obstacles. A scheme for showing absence of positive eigenvalues for generalized N -body hard-core Schroedinger operators, N \geq 2, is presented. This scheme involves high energy resolvent estimates, and for N = 2 it is implemented by a Mourre commutator type method. A particular example is the Helium atom with the assumption of infinite mass and finite extent nucleus.

We study regularity of bound states pertaining to embedded eigenvalues of a self-adjoint operator $H$, with respect to an auxiliary operator $A$ that is conjugate to $H$ in the sense of Mourre. We work within the framework of singular Mourre theory which enables us to deal with confined massless Pauli-Fierz models, our primary example, and many-body AC-Stark Hamiltonians. In the simpler context of regular Mourre theory our results boils down to an improvement of results obtained recently in \cite{CGH}., 70 pages

We study second order perturbation theory for embedded eigenvalues of an abstract class of self-adjoint operators. Using an extension of the Mourre theory, under assumptions on the regularity of bound states with respect to a conjugate operator, we prove upper semicontinuity of the point spectrum and establish the Fermi Golden Rule criterion. Our results apply to massless Pauli-Fierz Hamiltonians for arbitrary coupling., Comment: 30 pages, 2 figures

We prove smoothness of the 2-cluster-2-cluster and 2-cluster-N-cluster scattering amplitudes under a general short range condition on the potential and under a discreteness assumption on the channel energies.

We prove propagation estimates (of strong type) for long-rangeN-body Hamiltonians. Emphasis is on phase-space analysis in the free channel region.

We consider Hamiltonian systems of a certain class with unstable orbits moving to infinity. We prove a theorem showing that analogous quantum states do not exist. This theorem is applied to Schrodinger operators with potentials of degree zero which are Morse when restricted to the unit sphere.

This paper is the second in a series of two papers on asymptotic completeness for (generalized) three-body quantum systems with long-range interaction.

We discuss the perturbation of continuum eigenvalues without analyticity assumptions. Among our results, we show that generally a small perturbation removes these eigenvalues in accordance with Fermi's Golden Rule. Thus, generically (in a Baire category sense), the Schrodinger operator has no embedded non-threshold eigenvalues. I. Introduction It is well known (R-Sl) that a one-body Schrodinger operator — A + V(x), where V is sufficiently well behaved at infinity, cannot have eigenvalues λ embedded in the continuous spectrum (except possibly at threshold, λ = 0). The situation is quite different in the JV-body problem where continuum eigenvalues not only can exist, but do indeed exist in important physical situations: The operator Ho = -Δ1-Δ2-2/\x1\-2/\x2\ in L2{U6) (describing the Helium atom without electronic repulsion) has eigenvalues embedded in the continuous spectrum. While this example has an obvious symmetry, such symmetry is not necessary for the existence of embedded eigenvalues. An example in (F-H-HO-HO) can be modified to produce an embedded eigenvalue where no symmetry is apparent. In (Howl, 2) and (SI), analyticity assumptions are made which allow the treatment of embedded eigenvalues using the perturbation theory developed for use with isolated eigenvalues. The major idea in this theory is that when a small perturbation βW is added to the Schrodinger operator H, the continuum eigenvalue Eo turns into a "resonance," E0(β), which, while not necessarily an eigenvalue of H + βW, is a pole in the analytic continuation of certain matrix elements (φ,(H + βW — z)~ιφ) of the resolvent. The function E0(β) is analytic in β for \β\ small. E0(β) has an imaginary part which appears first to second order in β: τ d2E0(β)

For a quantum mechanical two-bodys-wave resonance we prove that the evolution of square integrable approximations of the Gamow function is outgoing and exponentially damped. An error estimate is given in terms of resonance energy and explicity. We obtain the Breit-Wigner form. The results are used in an α-decay model to prove general validity of the exponential decay law for periods of several lifetimes.