Methods of spectral analysis in mathematical physics: by Jan Janas, Pavel Kurasov, A. Laptev, Sergei Naboko, Gunter

By Jan Janas, Pavel Kurasov, A. Laptev, Sergei Naboko, Gunter Stolz

The quantity comprises the complaints of the OTAMP 2006 (Operator idea, research and Mathematical Physics) convention held at Lund college in June 2006. The convention was once dedicated to the tools of study and operator thought in glossy mathematical physics. the subsequent unique classes have been prepared through the convention: Spectral research of Schr?dinger operators; Jacobi and CMV matrices and orthogonal polynomials; Quasi-periodic and random Schr?dinger operators; Quantum graphs.

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Estimates on the difference between succeeding eigenvalues and Lifshitz tails for random Schr¨ odinger operators. In Stochastic processes – mathematics and 26 T. Antunovi´c and I. , pages 138–151. Springer, Berlin, 1987. [52] W. Kirsch and F. Martinelli. Large deviations and Lifshitz singularity of the integrated density of states of random Hamitonians. Commun. Math. , 89:27–40, 1983. [53] W. Kirsch and B. Metzger. The integrated density of states for random Schr¨odinger operators. In Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday, volume 76 of Proceedings of Symposia in Pure Mathematics, pages 649–698.

3. The differential equations If q is an integrable function on T define Q = diag(q1 , . . , qr ). We will consider the differential expression y → −y ′′ +Qy and the differential equation −y ′′ +Qy = λy. Define D = {y ∈ C : −yj′′ + qj yj ∈ L2 ([0, 1])}. We now define the operator L : D → L2 (T ) by (Ly)(ǫj (t)) = −yj′′ (t) + qj (t)yj (t). Again, this definition is independent of the orientation or the labeling of the edges. If y ∈ D satisfies Ly = λy we will call it a solution of Ly = λy signifying that both the differential equations and the interface conditions are satisfied.

Sr (λ, t)). A function y satisfying the differential equations may now be expressed as y = (C(λ, ·), S(λ, ·))ξ for an appropriate ξ ∈ C2r . In particular, the function y = (C(λ, ·), S(λ, ·))ξ satisfies the interface conditions (and hence is a solution of Ly = λy) precisely if ξ is in the kernel of the (2r − n0 ) × 2r-matrix J(λ) = I(C(λ, ·), S(λ, ·)). 4. Initial value problems Initial value problems do, in general, not have unique solutions on trees. This causes the main differences in the treatment of inverse problems on trees when compared to intervals.

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