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1. CJM 2003 (vol 55 pp. 449)
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)
Resonances for Slowly Varying Perturbations of a Periodic SchrÃ¶dinger 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)
Correction to: Upper Bounds for the Resonance Counting Function of SchrÃ¶dinger Operators in Odd Dimensions |
Correction to: Upper Bounds for the Resonance Counting Function of SchrÃ¶dinger 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)
Upper bounds for the resonance counting function of SchrÃ¶dinger 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 |