Erik Balslev

Analytic scattering theory of two-body Schrödinger operators

We consider the self-adjoint analytic family of operators H ( z ) in L 2 ( R m ) defined for , associated with the operator H = H (1) = H 0 + V , where H 0 = − Δ and V is a dilation-analytic short-range potential. The analytic connection between the local wave and scattering operators associated with the operators H ( e iϑ ) is established. The scattering matrix S ( ϱ ) of H has a meromorphic continuation S ( z ) to S α with poles precisely at the resolvent resonances of H , and the local scattering operators of e −2 iϑ H ( e iϑ ) have representations in terms of the analytically continued scattering matrix S ( ϱe iϑ ).

Local distortion techniques and unitarity of the S-matrix for the 2-body problem

Abstract The two-body S-matrix for an interaction with exponential decay at infinity is defined in a time-independent way and its unitarity is proved directly by local distortion techniques. Complete sets of incoming and outgoing states, or delicate resolvent estimates are not needed for the proof.

Decomposition of many-body Schrödinger operators

The complex-dilated many-body Schrödinger operatorH(z) is decomposed on invariant subspaces associated with the “cuts” {μ+z−2R+}, whereμ is any threshold, and isolated spectral points. The interactions are dilation-analytic multiplicative two-body potentials, decaying asr−1+ε atr=0 and asr−1+ε atr=∞.

Resonances in three-body scattering theory

Resonances of the three-body Schrodinger operator with exponentially decaying two-body interactions are characterized for negative energies as (1) poles of an analytically continued resolvent acting between certain exponentially weighted spaces; (2) eigenvalues of the Schrodinger operator acting in a suitable space; (3) singular points of the Faddeev matrix; (4) singular points of the Lippman-Schwinger operator; (5) poles of the S-matrix; (6) poles of analytic families of exponentially growing eigenfunctions.

Local spectral deformation techniques for Schrödinger operators

Abstract Resonances are defined through local spectral deformation techniques for 2-body Schrodinger operators, where the potential satisfies certain analyticity conditions. The S -matrix is proved to be unitary and to have an analytic continuation with poles at resonances. The class of potentials includes a sum of a radial dilation-analytic potential and an exponentially decaying potential, generalising previous results on each of these two classes.

Spectral theory of Laplacians for Hecke groups with primitive character

It was proved by [Sel] tha t the Laplacian A(F) for congruence subgroups F of the modular group Fz has an infinite sequence of embedded eigenvalues {Ai} satisfying a Weyl law #{~i~<A}~(rFf/47r)A for A--+cx~. Here I l l is the area of the fundamental domain F of the group F, and the eigenvalues Ai are counted according to multiplicity. The same holds true for the Laplacian A(F; X), where X is a character on F and A(F; X) is associated with a congruence subgroup F1 of F. It is an important question whether this is a characteristic of congruence groups or it may hold also for some non-congruence subgroups of Fz. To investigate this problem Phillips and Sarnak studied per turbat ion theory for Laplacians A(F) with regular per turbat ions derived from modular forms of

Analyticity properties of eigenfunctions and scattering matrix

AbstractFor potentialsV=V(x)=O(|x|−2−ε) for |x|→∞,x∈ℝ3 we prove that if theS-matrix of (−Δ, −Δ+V) has an analytic extension

Resonances with a background potential

\tilde S(z)