谷島 賢二

We report our recent results on the existence and uniqueness of unitary propagators for [Formula: see text]-particle Schrödinger equations which may be applied to most interesting problems in physics.

We study the threshold behavior of two-dimensional Schrödinger operators with finitely many local point interactions. We show that the resolvent can either be continuously extended up to the threshold, in which case we say that the operator is of regular type, or it has singularities associated with [Formula: see text] or [Formula: see text]-wave resonances or even with an embedded eigenvalue at zero, for whose existence we give necessary and sufficient conditions. An embedded eigenvalue at zero may appear only if we have at least three centers. When the operator is of regular type, we prove that the wave operators are bounded in [Formula: see text] for all [Formula: see text]. With a single center, we always are in the regular type case.

We prove that, for arbitrary centres and strengths, the wave operators for three-dimensional Schrödinger operators with multi-centre local point interactions are bounded in Lp(R3) for 1 &lt p&lt 3 and unbounded otherwise.

We consider Schrodinger equations for N number of particles in (classical) electro-magnetic fields that are interacting with each other via time dependent inter-particle potentials. We prove that they uniquely generate unitary propagators on the state space under the conditions that fields are spatially smooth and do not grow too rapidly at infinity so that propagators for single particles satisfy Strichartz estimates locally in time, and that local singularities of inter-particle potentials are not too strong that time frozen Hamiltonians define natural selfadjoint realizations in . We also show, under very mild additional assumptions on the time derivative of inter-particle potentials, that propagators possess the domain of definition of the quantum harmonic oscillator as an invariant subspace such that, for initial states in , solutions are C (1) functions of the time variable with values in . New estimates of Strichartz type for propagators for N independent particles in the field will be proved and used in the proof.

We consider the continuity property in Lebesgue spaces L-P(R-m) of the wave operators W-+/- of scattering theory for Schrodinger operators H = -Delta + V on R-m, vertical bar V(x)vertical bar <= C < x >(-delta) for some delta > 2 when H is of exceptional type, i.e. N = {u is an element of < x >(s) L-2(R-m): (1 + (-Delta)V-1)u = 0} not equal {0} for some 1/2 < s < delta - 1/2. It has recently been proved by Goldberg and Green for m >= 5 that W-+/- are in general bounded in L-P(R-m) for 1 <= p < m/2, for 1 <= p < m if all phi is an element of N satisfy integral R-m V phi dx = 0 and, for 1 <= p < infinity if integral R-m x(i)V phi dx = 0, i = 1,...,m in addition. We make the results for p > m/2 more precise and prove in particular that these conditions are also necessary for the stated properties of W-+/-. We also prove that, for m = 3, W-+/- are bounded in L-P(R-3) for 1 < p < 3 and that the same holds for 1 < p < infinity if and only if all phi is an element of N satisfy integral R-3 V phi dx = 0 and integral R-3 x(i)Vdx = 0, i = 1, 2, 3, simultaneously.

We consider the fundamental solution for the Cauchy problem for perturbations of the harmonic oscillator by time dependent potentials which grow at spatial infinity slower than quadratic but faster than linear functions and whose Hessian matrices have a fixed sign. We prove that the fundamental solution at resonant times grows indefinitely at spatial infinity with an algebraic growth rate, which increases indefinitely when the growth rate of perturbations at infinity decreases from the near quadratic to the near linear ones.

Let H = -Delta + V(x) be a Schrodinger operator on L-2(R-4), H-0 = -Delta. Assume that vertical bar V(x)vertical bar + vertical bar del V(x)vertical bar <= C < x >(-delta) for some delta > 8. Let W-+/- = s-lim(t ->+/-infinity) e(itH) e-(itH0) be the wave operators. It is known that W-+/- extend to bounded operators in L-p(R-4) for all 1 <= p <= infinity, if 0 is neither an eigenvalue nor a resonance of H. We show that if 0 is an eigenvalue, but not a resonance of H, then the W-+/- are still bounded in L-p(R-4) for all p such that 4/3 < p < 4.

Let H = -Delta + V (x) be a three dimensional Schrodinger operator. We study the time decay in L-p spaces of scattering solutions e(-itH)P(c)u, where P-c is the orthogonal projection onto the continuous spectral subspace of L-2(R-3) for H. Under suitable decay assumptions on V (x) it is shown that they satisfy the so-called L-p- L-q estimates parallel to e(-itH)P(c)u parallel to(p) <= (4 pi vertical bar t vertical bar)(-3(1/2-1/p)) parallel to u parallel to(q) for all 1 <= q <= 2 <= p <= infinity with 1/p + 1/q = 1 if H has no threshold resonance and eigenvalue; and for all 3/ 2 < q <= 2 <= p < 3 if otherwise.

Let H( t)=- Delta+ V( t, x) be a time- dependent Schrodinger operator on L-2(R-3). We assume that V(t, x) is 2pi-periodic in time and decays sufficiently rapidly in space. Let U(t, 0) be the associated propagator. For u(0) belonging to the continuous spectral subspace of L-2(R-3) for the Floquet operator U(2pi, 0), we study the behavior of U(t, 0) u(0) as t --> infinity. in the topology of x-weighted spaces, in the form of asymptotic expansions. Generically the leadingterm is t(-3/2)B(1)u(0). Here B-1 is a finite rank operator mapping functions of x to functions of t and x, periodic in t. If n is an element of Z is a threshold resonance of the corresponding Floquet Hamiltonian - ipartial derivative(t)+ H(t), the leading behavior is t(-1/2)B(0)u(0). The point spectral subspace for U(2pi, 0) is finite dimensional. If U(2pi, 0) phi(j) = e(-i2pilambdaj)phi(j), then U(t, 0) phi(j) represents a quasi-periodic solution.

We study smoothing properties for time-dependent Schrodinger equations ipartial derivativeu/partial derivativel = -(1 /2) Deltau + V(x)u, u(0) = phi is an element of L-2 (R-n), with potentials which satisfy V(x) = O(\x\(m)) at infinity, m greater than or equal to 2. We show that the solution u(t, x) is 1/m times differentiable with respect to x at almost all t is an element of R, and explain that this is the result of the fact that the sojourn time of classical particles with energy lambda in arbitrary compact set is less than CTlambda(-1/m) during [0, T] when lambda is very large. We also show Strichartz's inequality with derivative loss for such potentials and give its application to nonlinear Schrodinger equations. (C) 2004 Elsevier Inc. All rights reserved.

We study the spectral and scattering theory of the Nelson model for an atom interacting with a photon field in the subspace with less than two photons. For the free electron-photon system, the spectral property of the reduced Hamiltonian in the center of mass coordinates and the large time dynamics are determined. If the electron is under the influence of the nucleus via spatially decaying potentials, we locate the essential spectrum, prove the absence of singular continuous spectrum and the existence of the ground state, and construct wave operators giving the asymptotic dynamics.

We prove a smoothing property for one dimensional time dependent Schrodinger equations with potentials which satisfy V(x) similar to C\x\(k) at infinity, k > 2. As an application, we show that the initial value problem for certain nonlinear Schrodinger equations with such potentials is L-2 well-posed. We also prove a sharp asymptotic estimate of the L-p-norm of the normalized eigenfunctions of H = -Delta + V for large energy.

We show that the fundamental solution of the initial value problem for the time dependent SchrÖdinger equation is bounded and continuous for a class of non-smooth potentials. The class is large enough to accomodate Coulomb potentials if the spatial dimension is three.