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Search: MSC category 47A40 ( Scattering theory [See also 34L25, 35P25, 37K15, 58J50, 81Uxx] )

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1. CJM 2003 (vol 55 pp. 449)

Albeverio, Sergio; Makarov, Konstantin A.; Motovilov, Alexander K.
Graph Subspaces and the Spectral Shift Function
We obtain a new representation for the solution to the operator Sylvester equation in the form of a Stieltjes operator integral. We also formulate new sufficient conditions for the strong solvability of the operator Riccati equation that ensures the existence of reducing graph subspaces for block operator matrices. Next, we extend the concept of the Lifshits-Krein spectral shift function associated with a pair of self-adjoint operators to the case of pairs of admissible operators that are similar to self-adjoint operators. Based on this new concept we express the spectral shift function arising in a perturbation problem for block operator matrices in terms of the angular operators associated with the corresponding perturbed and unperturbed eigenspaces.

Categories:47B44, 47A10, 47A20, 47A40

2. CJM 2002 (vol 54 pp. 998)

Dimassi, Mouez
Resonances for Slowly Varying Perturbations of a Periodic Schrödinger Operator
We study the resonances of the operator $P(h) = -\Delta_x + V(x) + \varphi(hx)$. Here $V$ is a periodic potential, $\varphi$ a decreasing perturbation and $h$ a small positive constant. We prove the existence of shape resonances near the edges of the spectral bands of $P_0 = -\Delta_x + V(x)$, and we give its asymptotic expansions in powers of $h^{\frac12}$.

Categories:35P99, 47A60, 47A40

3. CJM 2001 (vol 53 pp. 756)

Froese, Richard
Correction to: Upper Bounds for the Resonance Counting Function of Schrödinger Operators in Odd Dimensions
The proof of Lemma~3.4 in [F] relies on the incorrect equality $\mu_j (AB) = \mu_j (BA)$ for singular values (for a counterexample, see [S, p.~4]). Thus, Theorem~3.1 as stated has not been proven. However, with minor changes, we can obtain a bound for the counting function in terms of the growth of the Fourier transform of $|V|$.

Categories:47A10, 47A40, 81U05

4. CJM 1998 (vol 50 pp. 538)

Froese, Richard
Upper bounds for the resonance counting function of Schrödinger operators in odd dimensions
The purpose of this note is to provide a simple proof of the sharp polynomial upper bound for the resonance counting function of a Schr\"odinger operator in odd dimensions. At the same time we generalize the result to the class of super-exponentially decreasing potentials.

Categories:47A10, 47A40, 81U05

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