Narrow Results

We present the spectral and scattering theory of the Casimir operator acting on radial functions in $L^2(\mathrm{\text{SL}}(2,ℝ))$. After a suitable decomposition, these investigations consist in studying a family of differential operators acting on the half-line. For these operators, explicit expressions can be found for the resolvent, for the spectral density, and for the Møller wave operators, in terms of the Gauss hypergeometric function. An index theorem is also introduced and discussed. The resulting equality, generically called Levinson’s theorem, links various asymptotic behaviors of the hypergeometric function. This work is a first attempt to connect group theory, special functions, scattering theory, $C^{\ast }$-algebras, and Levinson’s theorem.

In this article we continue our analysis of Schrödinger operators with a random potential using scattering theory. In particular the theory of Krein's spectral shift function leads to an alternative construction of the density of states in arbitrary dimensions. For arbitrary dimension we show existence of the spectral shift density, which is defined as the bulk limit of the spectral shift function per unit interaction volume. This density equals the difference of the density of states for the free and the interaction theory. This extends the results previously obtained by the authors in one dimension. Also we consider the case where the interaction is concentrated near a hyperplane.

We study the scattering theory for a class of non-relativistic quantum field theory models describing a confined non-relativistic atom interacting with a massless relativistic bosonic field. We construct invariant spaces which are defined in terms of propagation properties for large times and which consist of states containing a finite number of bosons in the region {|x| ≥ ct} for t → ±∞. We show the existence of asymptotic fields and we prove that the associated asymptotic CCR representations preserve the spaces and induce on these spaces representations of Fock type. For these induced representations, we prove the property of geometric asymptotic completeness, which gives a characterization of the vacuum states in terms of propagation properties. Finally we show that a positive commutator estimate imply the asymptotic completeness property, i.e. the fact that the vacuum states of the induced representations coincide with the bound states of the Hamiltonian.

For the massless Dirac equation outside a slow Kerr black hole, we prove asymptotic completeness. We introduce a new Newman–Penrose tetrad in which the expression of the equation contains no artificial long-range perturbations. The main technique used is then a Mourre estimate. The geometry near the horizon requires us to apply a unitary transformation before we find ourselves in a situation where the generator of dilations is a good conjugate operator. The results are eventually re-interpreted geometrically to provide the solution to a Goursat problem on the Penrose compactified exterior.

A Gelfand triplet for the Hamiltonian H of the Friedrichs model on ℝ with multiplicity space , , is constructed such that exactly the resonances (poles of the inverse of the Livšic-matrix) are (generalized) eigenvalues of H. The corresponding eigen(anti)linear forms are calculated explicitly. Using the wave matrices for the wave (Möller) operators the corresponding eigen(anti)linear forms on the Schwartz space for the unperturbed Hamiltonian H₀ are also calculated. It turns out that they are of pure Dirac type and can be characterized by their corresponding Gamov vector λ → k/(ζ₀ - λ)^-1, ζ₀ resonance, , which is uniquely determined by restriction of to , where denotes the Hardy space of the upper half-plane. Simultaneously this restriction yields a truncation of the generalized evolution to the well-known decay semigroup for t ≥ 0 of the Toeplitz type on . That is: Exactly those pre-Gamov vectors λ → k/(ζ - λ)^-1, ζ from the lower half-plane, , have an extension to a generalized eigenvector of H if ζ is a resonance and if k is from that subspace of which is uniquely determined by its corresponding Dirac type antilinear form.

The addendum refers mainly to Sec. 5 of the paper (Friedrichs model on the positive half line). The "Schwartz space framework" is omitted because it is dispensable for the results. Improvements of the proofs are indicated. A supplement presents in the special case G₀ := ℂ\(-∞,0] a surprising implication: the scattering matrix has only simple poles and its "main part" is a linear combination of all Gamov vectors.

In this paper, we explain the photoelectric effect in a variant of the standard model of non relativistic quantum electrodynamics, which is in some aspects more closely related to the physical picture, than the one studied in [5]. Now, we can apply our results to an electron with more than one bound state and to a larger class of electron-photon interactions. We will specify a situation, where the second order of ionization probability is a weighted sum of single photon terms. Furthermore, we will see that Einstein's equality

We introduce an abstract class of bosonic QFT Hamiltonians and study their spectral and scattering theories. These Hamiltonians are of the form H = dΓ(ω) + V acting on the bosonic Fock space Γ(𝔥), where ω is a massive one-particle Hamiltonian acting on 𝔥 and V is a Wick polynomial Wick(w) for a kernel w satisfying some decay properties at infinity.

We describe the essential spectrum of H, prove a Mourre estimate outside a set of thresholds and prove the existence of asymptotic fields. Our main result is the asymptotic completeness of the scattering theory, which means that the CCR representations given by the asymptotic fields are of Fock type, with the asymptotic vacua equal to the bound states of H. As a consequence, H is unitarily equivalent to a collection of second quantized Hamiltonians.

We consider time delay and symmetrized time delay (defined in terms of sojourn times) for quantum scattering pairs {H₀ = h(P), H}, where h(P) is a dispersive operator of hypoelliptic-type. For instance, h(P) can be one of the usual elliptic operators such as the Schrödinger operator h(P) = P² or the square-root Klein–Gordon operator . We show under general conditions that the symmetrized time delay exists for all smooth even localization functions. It is equal to the Eisenbud–Wigner time delay plus a contribution due to the non-radial component of the localization function. If the scattering operator S commutes with some function of the velocity operator ∇h(P), then the time delay also exists and is equal to the symmetrized time delay. As an illustration of our results, we consider the case of a one-dimensional Friedrichs Hamiltonian perturbed by a finite rank potential.

Our study puts into evidence an integral formula relating the operator of differentiation with respect to the kinetic energy h(P) to the time evolution of localization operators.

We prove a limiting absorption principle for the standard model of non-relativistic quantum electrodynamics (QED) and for Nelson's model describing interactions of electrons with phonons. To this end, we use the spectral renormalization group technique on the continuous spectrum in conjunction with Mourre theory.

This paper analyzes the scattering theory for periodic tight-binding Hamiltonians perturbed by a finite range impurity. The classical energy gradient flow is used to construct a conjugate (or dilation) operator to the unperturbed Hamiltonian. For dimension d ≥ 3, the wave operator is given by an explicit formula in terms of this dilation operator, the free resolvent and the perturbation. From this formula, the scattering and time delay operators can be read off. Using the index theorem approach, a Levinson theorem is proved which also holds in the presence of embedded eigenvalues and threshold singularities.

We define, prove the existence and obtain explicit expressions for classical time delay defined in terms of sojourn times for abstract scattering pairs (H₀, H) on a symplectic manifold. As a by-product, we establish a classical version of the Eisenbud–Wigner formula of quantum mechanics. Using recent results of Buslaev and Pushnitski on the scattering matrix in Hamiltonian mechanics, we also obtain an explicit expression for the derivative of the Calabi invariant of the Poincaré scattering map. Our results are applied to dispersive Hamiltonians, to a classical particle in a tube and to Hamiltonians on the Poincaré ball.

We review the spectral analysis and the time-dependent approach of scattering theory for manifolds with asymptotically cylindrical ends. For the spectral analysis, higher order resolvent estimates are obtained via Mourre theory for both short-range and long-range behaviors of the metric and the perturbation at infinity. For the scattering theory, the existence and asymptotic completeness of the wave operators is proved in a two-Hilbert spaces setting. A stationary formula as well as mapping properties for the scattering operator are derived. The existence of time delay and its equality with the Eisenbud–Wigner time delay is finally presented. Our analysis mainly differs from the existing literature on the choice of a simpler comparison dynamics as well as on the complementary use of time-dependent and stationary scattering theories.

The question of self-adjoint realizations of sign-indefinite second-order differential operators is discussed in terms of a model problem. Operators of the type are generalized to finite, not necessarily compact, metric graphs. All self-adjoint realizations are parametrized using methods from extension theory. The spectral and scattering theories of the self-adjoint realizations are studied in detail.

We analyze scattering in a system of two (distinguishable) particles moving on the half-line ${\overline{ℝ}}_{+}$ under the influence of singular two-particle interactions. Most importantly, due to the spatial localization of the interactions, the two-body problem is of a non-separable nature. We will discuss the presence of embedded eigenvalues and using the obtained knowledge about the kernel of the resolvent, we prove a version of the limiting absorption principle. Furthermore, by an appropriate adaptation of the Lippmann–Schwinger approach, we are able to construct generalized eigenfunctions which consequently allow us to establish an explicit expression for the (on-shell) scattering amplitude. An approximation of the scattering amplitude in the weak-coupling limit is also derived.

We present a suitable framework for the definition of quantum time delay in terms of sojourn times for unitary operators in a two-Hilbert spaces setting. We prove that this time delay defined in terms of sojourn times (time-dependent definition) exists and coincides with the expectation value of a unitary analogue of the Eisenbud–Wigner time delay operator (time-independent definition). Our proofs rely on a new summation formula relating localization operators to time operators and on various tools from functional analysis such as Mackey’s imprimitivity theorem, Trotter–Kato formula and commutator methods for unitary operators. Our approach is general and model-independent.

We consider mathematical models of the weak decay of the vector bosons $W^{\pm }$ into leptons. The free quantum field Hamiltonian is perturbed by an interaction term from the standard model of particle physics. After the introduction of high energy and spatial cut-offs, the total quantum Hamiltonian defines a self-adjoint operator on a tensor product of Fock spaces. We study the scattering theory for such models. First, the masses of the neutrinos are supposed to be positive: for all values of the coupling constant, we prove asymptotic completeness of the wave operators. In a second model, neutrinos are treated as massless particles and we consider a simpler interaction Hamiltonian: for small enough values of the coupling constant, we prove again asymptotic completeness, using singular Mourre’s theory, suitable propagation estimates and the conservation of the difference of some number operators.

We consider the cubic-quintic nonlinear Schrödinger equation of up to three space dimensions. The cubic nonlinearity is thereby focusing while the quintic one is defocusing, ensuring global well-posedness of the Cauchy problem in the energy space. The main goal of this paper is to investigate the interplay between dispersion and orbital (in-)stability of solitary waves. In space dimension one, it is already known that all solitons are orbitally stable. In dimension two, we show that if the initial data belong to the conformal space, and have at most the mass of the ground state of the cubic two-dimensional Schrödinger equation, then the solution is asymptotically linear. For larger mass, solitary wave solutions exist, and we review several results on their stability. Finally, in dimension three, relying on previous results from other authors, we show that solitons may or may not be orbitally stable.

We study the spectral and scattering theory of light transmission in a system consisting of two asymptotically periodic waveguides, also known as one-dimensional photonic crystals, coupled by a junction. Using analyticity techniques and commutator methods in a two-Hilbert spaces setting, we determine the nature of the spectrum and prove the existence and completeness of the wave operators of the system.

We propose a mathematically rigorous construction of the scattering matrix and the interacting fields in models of relativistic perturbative quantum field theory with massless fields and long-range interactions. We consider quantum electrodynamics and a certain model of interacting scalar fields in which the standard definition of the scattering matrix is not applicable because of the infrared problem. We modify the Bogoliubov construction using the ideas of Dollard, Kulish and Faddeev. Our modified scattering matrix and modified interacting fields are constructed with the use of the adiabatic limit which is expected to exist in arbitrary order of perturbation theory. In the paper, we prove this assertion in the case of the first- and the second-order corrections to the modified scattering matrix and the first-order corrections to the modified interacting fields. We study the physical properties of our construction. We conclude that the electrons and positrons are always surrounded by irremovable clouds of photons. Moreover, the physical energy-momentum operators do not coincide with the standard ones and their joint spectrum does not contain the mass hyperboloid.