Peter Rejto

A limiting absorption principle for Schrödinger operators with Von Neumann-Wigner type potentials

In this paper we prove the main step in establishing a limiting absorption principle for von Neumann-Wigner type Schrödinger Hamiltonians of the form −Δ+csinb|x|/|x|+V(x), whereV(x) is a short range potential. The first fundamental step is to obtain a limiting absorption principal for the “free” operator −Δ+csinb.|x|/|x|. The free operator is unitarily equivalent to a direct sum of ordinary differential operators. We obtain uniform estimates for the resolvents of these ordinary differential operators. by obtaining uniform estimates for the Weyl-Green kernels of these resolvents. In turn, these latter estimates require uniform estimates on the Wronskians of certain generalized eigen-solutions of these differential operators.

Unique solvability of nonlinear Volterra equations in weighted spaces

Abstract We investigate integral equations of the form x(t) = g(t) + ∫−∞t F(t, s, x(s)) ds. (∗) In general, this equation is history-dependent, so one needs to give an initial condition on (−∞, 0] in order to obtain a unique solution. By introducing a weight function on R , we can single out a class of admissible solutions, and give conditions for the unique solvability of (∗) in this restricted class. We also study some Fredholm equations on these weighted spaces. In addition, we also treat a class of equations of the first kind for which similar conclusions can be drawn.

Weighted resolvent estimates for Volterra operators on unbounded intervals

The interval may be bounded or unbounded, including the case of the whole real line, and the kernel V(r, q) is supposed to be such that V maps X into X (we shall make precise assumptions later). The case of a bounded interval f with X = W(Y) and V( q is classical. In this case, the operator V is bounded and its spectrum consists of (0). In other words, the spectral radius of V is zero. A standard proof of these two facts is to prove that for 13 # 0 the Neumann series for the inverse of (AZV) converges with respect to the operator norm. The purpose of this note is to study the case of a possibly unbounded

Schrödinger Operators with Oscillating Potentials

Dae to the pioneering contributions of Kato-Kuroda [10], [11] and to the more recent works of Agmon [l] and Enss [6], a spectral and scattering theory for Schrodinger operators with short range potentials is now well established. An interesting example of a potential which does not belong to this class is the Wigner-von- Neumann [17] potential. This potential is the sum of a short range potential and of an oscillating one which is of the form,

A Schrödinger Operator with an Oscillating Potential

{P_o}\left( x \right) = c\frac{{\sin b{{\left| x \right|}^\alpha }}}{{{{\left| x \right|}^\beta }}},\alpha ,\beta > 0

On partly gentle perturbations III

(1.1) where c = -8, b = 2 and α = β = 1.

Limiting absorption principle for some Schrödinger operators with exploding potentials, II

Several authors formulated a limiting absorption principle for several classes of operators. For the physical significance of this principle we refer to the paper of Eidus [Ei] and for brevity, for additional references we only refer to the AMS Memoir of Ben-Artzi and Devinatz [BD].

A limiting absorption principle for Schrödinger operators with oscillating potentials. Part II: -Δ + (c sin b |x|/|x|) + V(x) for short range V and small coupling constant c

A relatively large amount of excellent work has been done during the past decade on general spectral and scattering theories for Schrodinger operators with long range oscillating potentials. However, most of these works do not include operators of the form H = - Δ + c sin b r r + V ( x ) , where V is a short range potential. When V is radially symmetric, the problem has been successfully dealt with in recent years. On the other hand when V is not radially symmetric only one recent paper deals with these operators, but only for high energy values. In this paper we shall consider the spectral theory for this specific operator and compare our results with the previously mentioned paper.