Dedicated with great pleasure to Eduard R. Tsekanovskii on the occasion of his 65th birthday. Abstract. We revisit the computation of (2-modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels. The latter occur, for instance, in the form of Green’s functions associated with closed ordinary differential operators on arbitrary intervals on the real line. Our approach determines the (2-modified) Fredholm determinants in terms of solutions of closely associated Volterra integral equations, and as a result offers a natural way to compute such determinants. We illustrate our approach by identifying classical objects such as the Jost function for half-line Schrödinger operators and the inverse transmission coefficient for Schrödinger operators on the real line as Fredholm determinants, and rederiving the well-known expressions for them in due course. We also apply our formalism to Floquet theory of Schrödinger operators, and upon identifying the connection between the Floquet discriminant and underlying Fredholm determinants, we derive new representations of the Floquet discriminant. Finally, we rederive the explicit formula for the 2-modified Fredholm determinant corresponding to a convolution integral operator, whose kernel is associated with a symbol given by a rational function, in a straghtforward manner. This determinant formula represents a Wiener–Hopf analog of Day’s formula for the determinant associated with finite Toeplitz matrices generated by the Laurent expansion of a rational function. 1.

Abstract. A spectral mapping theorem is proved that resolves a key problem in applying invariant manifold theorems to nonlinear Schrödinger type equations. The theorem is applied to the operator that arises as the linearization of the equation around a standing wave solution. We cast the problem in the context of space-dependent nonlinearities that arise in optical waveguide problems. The result is, however, more generally applicable including to equations in higher dimensions and even systems. The consequence is that stable, unstable, and center manifolds exist in the neighborhood of a (stable or unstable) standing wave, such as a waveguide mode, under simple and commonly verifiable spectral conditions. 1. Main Results The local behavior near some distinguished solution, such as a steady state, of an evolution equation, can be determined through a decomposition into invariant manifolds, that is, stable, unstable and center manifolds. These (locally invariant) manifolds are characterized by decay estimates. While the flows on the stable and unstable manifolds are determined by exponential decay in forward and backward time respectively, that on the center manifold is ambiguous. Nevertheless, a determination of the flow on the center manifold can lead to a

In this paper we study the following nonlinear Schrodinger equation on the line, i @ @t u(t; x) = \Gamma d 2 dx 2 u(t; x) + V (x)u(t; x) + f(x; juj) u(t; x) ju(t; x)j ; u(0; x) = OE(x); where f is real-valued, and it satisfies suitable conditions on regularity, on grow as a function of u and on decay as x ! \Sigma1. The generic potential, V , is realvalued and it is chosen so that the spectrum of H := \Gamma d 2 dx 2 + V consists of one simple negative eigenvalue and absolutely-continuous spectrum filling [0; 1). The solutions to this equation have, in general, a localized and a dispersive component. The nonlinear bound states, that bifurcate from the zero solution at the energy of the eigenvalue of H, define an invariant center manifold that consists of the orbits of time-periodic localized solutions . We prove that all small solutions approach a particular periodic orbit in the center manifold as t ! \Sigma1. In general, the periodic orbits are different for t ! \Sigma1. O...

We sketch a derivation of abstract scattering theory from the microscopic first principles defined by Bohmian mechanics. We emphasize the importance of the fluxacross -surfaces theorem for the derivation, and of randomness in the impact parameter of the initial wave function---even for an, inevitably inadequate, orthodox derivation. 1 Dedicated to Joel Lebowitz, with love and admiration, for his 70th birthday. Supported in part by the DFG, by NSF Grant No. DMS95--04556, and by the INFN. Preprint submitted to Elsevier Preprint 8 November 1999 1

this paper we prove existence of modi ed wave operators, with probability one, for the family of random operators on L ), d 2, H = 1 4 V (1.1) where x n jnj (1.2) with uniformly bounded independent ! n with mean 0, and > 2 . The most important example are Bernoulli variables ! n = 1, although we would like to point out that no assumption is made about identical distribution of the ! n . Here is a standard C bump function with small support and N R is a set of points with the property that fR < jxj < 2Rg\N is a maximally R separated set of points so that the summands in (1.2) have disjoint supports. We refer to (1.1) with V as in (1.2) as 4 -model. Our methods also apply to more general potentials than (1.2) for which the individual bumps are not rescaled versions of a single bump function . More precisely, consider n (x n) (1.3) where n is a C function supported in a ball B(0; jnj ) satisfying the derivative bounds x2R n (x)j C jnj jj and all n 2 N . The net N is just as above, C is uniform in N , and the functions f n ( n)g n2N have disjoint supports. For simplicity, we shall restrict ourselves to the model (1.2), but all arguments apply just as well to the case (1.3). Recall that wave operators of (1.1) are de ned as the strong L limit itH 0 (1.4) provided it exists. Here H 0 = 4. These are the quantum mechanical analogues of classical wave operators, which allow one to parameterize the trajectories with positive energy in the force eld rV by means of the free trajectories. The most basic result is that jV (x)j Cjxj guarantees the existence of the limit (1.4). Dollard [8] showed that wave operators do not exist for the Coulomb potential V (x) = jxj in R . Alsholm, Kato [1],...

In the paper [25]... In this paper we consider the quantum mechanical scattering in a hedgehog-shaped space which is constructed by gluing a finite number of half-lines to distinct points of a compact Riemannian manifold of dimension less than four. The Hamiltonian of a quantum particle in such a system coincides with a Schrödinger operator on the punctured manifold (the points of gluing are removed) and with the free Schrödinger operator on each half-line. At the gluing points, some boundary conditions are imposed. In particular, the Schröodinger operator in a magnetic field is included in our scheme. The approach we use is based on the Krein resolvent formula from operator extension theory [50], therefore in Sec. 1 we give a very brief sketch of results needed from this theory. Sec. 2 is devoted to the construction of Schrödinger operators on the hedgehog-shaped space; we use the theory of boundary value spaces [35] to describe all possible kinds of boundary conditions defining the Schrödinger operators. We distinguish among them operators of "Dirichlet" and of "Neumann" type. It is worth noting that the results of Sec. 2 are valid for all Riemannian manifolds of dimension less than four, not only for the compact ones. In principle, the definition of the Schrödinger operator on a hedgehog-shaped space may be given in the framework of pseudo-differential operator theory on such a space [66], but our approach is more convenient for investigating the scattering parameters and connected with the approach to spectral problems for point perturbations on Riemannian manifolds [8], [9]...

We survey recent results on spectral properties of random Schrodinger operators. The focus is set on the integrated density of states (IDS).

. We consider an electron bound by some anharmonic external potential and coupled to the quantized radiation field in the dipole approximation. We prove asymptotic completeness for the photon scattering. This means that an arbitrary initial state has a long time asymptotic, which consists of electron plus radiation field in their coupled ground state and finitely many outgoing photons. 1 Introduction and main results Rayleigh scattering refers to scattering of photons from a bound electron. In its simplest version one imagines an electron confined by a prescribed external potential and coupled to the radiation field. In the remote past electron and field are in their coupled ground state and there are a few far away incoming photons. As travelling inwards these excite the atom, scatter off, and in the far future there will be some outgoing photons leaving the relaxed atom behind. As first noted by van Kampen [1], in the dipole approximation the joint Hamiltonian becomes quadratic p...

Abstract. In this paper we develop a quantitative version of Enss ’ method to establish global-in-time decay estimates for solutions to Schrödinger equations on manifolds. To simplify the exposition we shall only consider Hamiltonians of the form H: = − 1 2 ∆M, where ∆M is the Laplace-Beltrami operator on a manifold M which is a smooth compact perturbation of three-dimensional Euclidean space R3 which obeys the non-trapping condition. We establish a global-in-time local smoothing estimate for the Schrödinger equation ut = −iHu. The main novelty here is the global-in-time aspect of the estimates, which forces a more detailed analysis on the low and medium frequencies of the evolution than in the local-in-time theory. In particular, to handle the medium frequencies we require the RAGE theorem (which reflects the fact that H has no embedded eigenvalues), together with a quantitative version of Enss ’ method decomposing the solution asymptotically into incoming and outgoing components, while to handle the low frequencies we need a Bernstein inequality (which reflects the fact that H has no eigenfunctions or resonances at zero). 1.

We give a number of results concerning dierent possible spectral types for Schrodinger operators with sparse potentials. These potentials are in between stationary (e.g., random) potentials and the short range potentials familiar from scattering theory. They decay at in nity in some averaged sense, however in such a way that there is enough \space" for surprising spectral properties.