Search

This work continues the study of the thermal Hamiltonian, initially proposed by J. M. Luttinger in 1964 as a model for the conduction of thermal currents in solids. The previous work [DL] contains a complete study of the "free" model in one spatial dimension along with a preliminary scattering result for convolution-type perturbations. This work complements the results obtained in [DL] by providing a detailed analysis of the perturbation theory for the one-dimensional thermal Hamiltonian. In more detail the following result are established: the regularity and decay properties for elements in the domain of the unperturbed thermal Hamiltonian; the determination of a class of self-adjoint and relatively compact perturbations of the thermal Hamiltonian; the proof of the existence and completeness of wave operators for a subclass of such potentials., 17 pages. Keywords: Thermal Hamiltonian, self-adjoint extensions, spectral theory, scattering theory

In the spectral and scattering theory for a Schrodinger operator with a time-periodic potential $$H(t)=p^2/2+V(t,x)$$ , the Floquet Hamiltonian $$K=-i\partial _t+H(t)$$ associated with H(t) plays an important role frequently, by virtue of the Howland–Yajima method. In this paper, we introduce a new conjugate operator for K in the standard Mourre theory, that is different from the one due to Yokoyama, in order to relax a certain smoothness condition on V.

We perform the scattering analysis of the evolution operator of quantum walks with an anisotropic coin, and we prove a weak limit theorem for their asymptotic velocity. The quantum walks that we consider include one-defect models, two-phase quantum walks, and topological phase quantum walks as special cases. Our analysis is based on an abstract framework for the scattering theory of unitary operators in a two-Hilbert spaces setting, which is of independent interest., 23 pages

In this work, we study inverse resonance problems for the Schr��dinger operator on the real line with the potential supported in $[0,1]$. In general, all eigenvalues and resonances can not uniquely determine the potential. (i) It is shown that if the potential is known a priori on $[0,1/2]$, then the unique recovery of the potential on the whole interval from all eigenvalues and resonances is valid. (ii) If the potential is known a priori on $[0,a]$, then for the case $a>1/2$, infinitely many eigenvalues and resonances can be missing for the unique determination of the potential, and for the case $a, 12 pages

We consider a Haag-Kastler net in a positive energy representation, admitting massive Wigner particles and asymptotic fields of massless bosons. We show that states of the massive particles are always vacua of the massless asymptotic fields. Our argument is based on the Mean Ergodic Theorem in a certain extended Hilbert space. As an application of this result we construct the outgoing isometric wave operator for Compton scattering in QED in a class of representations recently proposed by Buchholz and Roberts. In the course of this analysis we use our new technique to further simplify scattering theory of massless bosons in the vacuum sector. A general discussion of the status of the infrared problem in the setting of Buchholz and Roberts is given., Comment: 42 pages

Time-dependent scattering theory for a large class of translation invariant models, including the Nelson and Polaron models, restricted to the vacuum and one-particle sectors is studied. We formulate and prove asymptotic completeness for these models. The translation invariance imply that the Hamiltonians considered are fibered with respect to the total momentum. On the way to asymptotic completeness we determine the spectral structure of the fiber Hamiltonians, establish a Mourre estimate and derive a geometric asymptotic completeness statement as an intermediate step.

The probability of a quantum particle being detected in a given solid angle is determined by the $S$-matrix. The explanation of this fact in time dependent scattering theory is often linked to the quantum flux, since the quantum flux integrated against a (detector-) surface and over a time interval can be viewed as the probability that the particle crosses this surface within the given time interval. Regarding many particle scattering, however, this argument is no longer valid, as each particle arrives at the detector at its own random time. While various treatments of this problem can be envisaged, here we present a straightforward Bohmian analysis of many particle potential scattering from which the $S$-matrix probability emerges in the limit of large distances., Comment: some clarifications added in section 2.2; published version

We consider in this paper space-cutoff charged \({P(\varphi)_{2}}\) models arising from the quantization of the non-linear charged Klein–Gordon equation: $$(\partial_{t}+\i V(x))^{2}\phi(t, x)+ (-\Delta_{x}+ m^{2})\phi(t,x)+ g(x)\partial_{\overline{z}}P(\phi(t,x), \overline{\phi}(t,x))=0,$$ where V(x) is an electrostatic potential, g(x) ≥ 0 a space-cutoff, and \({P(\lambda, \overline{\lambda})}\) a real bounded below polynomial. We discuss various ways to quantize this equation, starting from different CCR representations. After describing the construction of the interacting Hamiltonian H we study its spectral and scattering theory. We describe the essential spectrum of H, prove the existence of asymptotic fields and of wave operators, and finally prove the asymptotic completeness of wave operators. These results are similar to the case when V = 0.

The problem of constructing generalized point interactions of the second deriv- ative operator in L 2 (R) leading to the same scattering data as for reflectionless potentials is considered. It is proved that this problem has a solution only if extensions in Pontrya- gin spaces are involved. The solution of the inverse scattering problem is not unique, this is illustrated by considering the scattering data for soliton of the Korteweg-de Vries equa- tion. Mathematics Subject Classifications (2000). 34B25, 47E05, 47B50, 81U40.

Within the framework of local quantum physics we construct a scattering theory of stable, massive particles without assuming mass gaps. This extension of the Haag-Ruelle theory is based on advances in the harmonic analysis of local operators. Our construction is restricted to theories complying with a regularity property introduced by Herbst. The paper concludes with a brief discussion of the status of this assumption., As appeared in Letters in Mathematical Physics

A new approach to the inverse scattering problem proposed by Schroer, is applied to two-dimensional integrable quantum field theories. For any two-particle S-matrix S_2 which is analytic in the physical sheet, quantum fields are constructed which are localizable in wedge-shaped regions of Minkowski space and whose two-particle scattering is described by the given S_2. These fields are polarization-free in the sense that they create one-particle states from the vacuum without polarization clouds. Thus they provide examples of temperate polarization-free generators in the presence of non-trivial interaction.

In this Letter we show how the scattering amplitudes of nonrelativistic one-particle Schrodinger operators with a scalar (not necessarily rotation invariant) potential may be obtained from the scattering cross-sections for the system where a scalar potential is added and whose scattering amplitudes are known explicitly.

We consider perturbations of a model quantum system consisting of a single bound state and continuum radiation modes. In many problems involving the interaction of matter and radiation, one is interested in the effect of time-dependent perturbations. A time-dependent perturbation will couple the bound and continuum modes causing ‘radiative transitions’. Using techniques of time-dependent resonance theory, developed in earlier work on resonances in linear and nonlinear Hamiltonian dispersive systems, we develop the scattering theory of short-lived (\( (\mathcal{O}(t^{ - 1 - {\varepsilon }} ))\)(t−1−e)) spatially localized perturbations. For weak pertubations, we compute (to second order) the ionization probability, the probability of transition from the bound state to the continuum states. These results can also be interpreted as a calculation, in the paraxial approximation, of the energy loss resulting from wave propagation in a waveguide in the presence of a localized defect.

Quantum scattering in the presence of a constant electric field (‘Stark effect’) is considered. It is shown that the scattering matrix has a meromorphic continuation in the energy variable to the entire complex plane as an operator on L2(R n-1). The allowed potentials V form a general subclass of potentials that are short-range relative to the free Stark Hamiltonian: Roughly, the potential vanishes at infinity, and admits a decomposition $$V = V_\mathcal{A} + V_e$$ , where $$V_\mathcal{A}$$ is analytic in a sector with $$V_\mathcal{A} (x) = O(\left\langle {x_{} } \right\rangle ^{ - 1/2 - \varepsilon } )$$ , and $$V_e (x) = O({\text{e}}^{\mu x_1 } )$$ , for x1 0. These potentials include the Coulomb potential. The wave operators used to define the scattering matrix are the two Hilbert space wave operators.

We study the direct spectral transform for the heat equation associated with the Kadomtsev-Petviashvili. We show that, for real nonsingular exponentially decaying-at-infinity potential, the directproblem is nonsingular for arbitrarily large potentials. Earlier, thisstatement was proved only for potentials satisfying the ‘small norm’assumption.

The scattering of a quantum charged particle on the hyperbolic plane in the Abelian Aharonov-Bohm field is considered. S-matrix, scattering amplitude and its flat space limit are constructed in a closed form by the Sommerfeld integral approach.

A four-parameter family of all self-adjoint operators corresponding to the one-dimensional Dirac Hamiltonian with point interaction is characterized in terms of boundary conditions. The spectrum and eigenvectors, and the scattering parameters are calculated. It is shown that the nonrelativistic limit reproduces (in the norm resolvent sense) the four-parameter family of Schrodinger operators with point interaction, their eigenvalues and scattering parameters.

When the potentialq(x) ∈L 1 1 with a singular term, the continuities of the scattering matrix of the Schrodinger equation are investigated. By means of the transformation approach, we arrive at the conclusion that the scattering matrix S(k) of such a potential is continuous for the wholek,- ∞

In this Letter, we give results on precise microlocalized time-decay estimates in three-body long-range scattering problems. We prove the asymptotic completeness of wave operators in three-body long-range scattering for a class of long-range interactions of the form V 1(x)+V 2(x), where V 1 is nonnegative and decays like O(|x|−e0), for some e0 > 1/2 and V 2 decays like O(|x|-y) for some γ > 2(1−e0)/e0.

We compute the scattering operator for a simplified model of an atom interacting a with a quantized field. The field is restricted to the vacuum and one-particle sectors, and the atom has only two states. We also solve the inverse scattering problem for the same model. The methods used rely on the particular form of the interaction, which is chosen to mimic the interaction between radiation and matter.

For scattering by a finite range potential, it is shown that the scattering length is convergent to the capacity of the support of the finite range potential in the semi-classical limit. This result has been already proved by use of probabilistic methods. Here we give an alternative proof without using any probabilistic idea. The proof is purely analytical and very simple.

The three-body quantum scattering problem reduced by the expansion of the wavefunction over the specially constructed basis to a two-body problem is considered. The asymptotics of this basis, as well as the solutions of the effective two-body equations are derived. A total S-matrix for 2 → (2, 3) processes is expressed in terms of adiabatic amplitudes and vice versa.

The quantum nonlinear Schrodinger equation with attractive coupling is considered through the quantum inverse scattering method. The asymptotic fields arising in this model are characterized in terms of scattering data operators.

We study completeness of the wave operators between pairs of spaces not using the subspace of absolute continuity. Rather we work with the orthogonal complement of the eigenvectors, a subspace which is more natural from the viewpoint of physics. Moreover, we also give sufficient conditions that the singular continuous spectrum be absent. In this case the subspaces mentioned above coincide.

The Schrodinger operator H=−Δ+V is considered when the potential V is central, oscillating and possibly unbounded. An eigenfunction expansion for H is derived. Also, the phase shift of the generalized eigenfunctions of H is computed. The expansion and shift are, together, applied to the study of the wave operators of scattering theory. The modified wave operators, for H and H0=−Δ are shown to exist and be complete and a condition on V is derived which is necessary and sufficient for the Moller wave operators to exist and be complete. Spectral properties of H are also discussed.

For a spherically symmetric potential such that rV∈L1(a, ∞), ∀a>0, and is such that, if we define W=−∫r∞V(t) d(t), W belongs to L1 (0, ∞) and rW→0 as r→0, we show that the number of bound states in any partial-wave satisfies the bound n⩽2 ∫0∞r W2 dr. It was shown in a previous paper [1] that this class of potentials is regular from the point of view of abstract scattering theory as well as from the time-independent theory and the Jost function approach. We show also that, for large values of the coupling constant, n(gV) has the asymptotic behaviour C±∣g∣∫0∞∣W(r) dr as g→±∞.

We give results on the behaviour at infinity of commutators of the form [ϕ(H), f(Q)], where H is a Schrodinger operator and Q denotes the position operator in [ϕ(H),f(Q)]. These results are applied to obtain propagation properties and asymptotic completeness below the three-body threshold for N-body systems.

In the space-time with signature (2,2) the self-duality equations in the specific case of potentials independent of one of the coordinates are reduced to a relativistic-invariant system in the (2-1)-dimensional space-time. A general solution of this system is constructed by means of IST. A soliton solution, finite in all directions, is discussed. It is found that there is no classical scattering of both solitons and continuous spectrum waves.

Let \(H = - \tfrac{1}{2}\vartriangle + \sum\nolimits_{j = 1}^n {W_j (Q)P_j + W_0 (Q)} \) be the selfadjoint operator for the static electromagnetic field where Wj for 0, 1, 2, ..., n is a sum of (i) a short-range potential and (ii) a smooth long-range potential decreasing at ∞ as |x|-δ with δ in (0, 1]. Then for δ>1/2, asymptotic completeness holds for the scattering system (H, H0).

We define a modified free-time evolution for the Dirac equation with long-range potentials β(1/|x|), where β is the Dirac matrix, and prove a strong asymptotic completeness of the corresponding wave operators. Our methods also work for the magnetic fields α·A(x).

A manifestly relativistic-invariant formulation of the method of inverse scattering transform for relativistic-invariant equations is proposed. The sine-Gordon model and the massive Thirring model are considered.

The generalizations of the nonlinear Schrodinger equation (NS) associated with the orthosymplectic superalgebras are formulated. The simplest osp(1/2)-NS model is solved by the quantum inverse scattering method on a finite interval under periodic boundary conditions as well as on the whole line in the case of a finite number of excitations.

Scattering of a quantum mechanical particle by a long-range potential is studied using Enss's time-dependent method. More precisely, a simple and natural extension of the Enss method to a two Hilbert space setting is established. Applied to the consideration of long-range scattering, this extended Enss method reduces the problem of proving existence and completeness of the wave operators to the problem of solving an eikonal equation on a cone in phase space. Relying on a solution of the eikonal equation constructed by Isozaki and Kitada, it is shown that the wave operators exist and are complete whenever the potential is in the long-range class introduced by Hormander.

It is shown that in scattering theory, the Heisenberg relation has the form\(\left\langle {\psi , H\psi } \right\rangle \Delta Q_{\Omega _ \pm ^ * \psi }^2 \geqslant \frac{9}{4}h^2\) for a wide class of potentials.H is the Hamiltonian of scattered particles, Ψ is a scattering state, and Ω± are wave operators. We discuss the interpretation of the obtained inequality and its entropic formulation.