By Jan Janas, Pavel Kurasov, A. Laptev, Sergei Naboko, Günter Stolz
This quantity includes lectures introduced by way of the contributors of the foreign convention Operator concept and its functions in Mathematical Physics (OTAMP 2004), held on the Mathematical examine and convention middle in Bedlewo close to Poznan, Poland. the assumption in the back of those lectures used to be to give attention-grabbing ramifications of operator equipment in present examine of mathematical physics. the most subject matters are practical versions of non-selfadjoint operators, spectral homes of Dirac and Jacobi matrices, Dirichlet-to-Neumann suggestions, Lyapunov exponents equipment, and inverse spectral difficulties for quantum graphs.
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U. Schmidt, Mathematics of the quantum Zeno eﬀect. In “Mathematical Physics Research on Leading Edge” (Ch. , 2004; pp. 113-143. jp Sylwia Kondej Institute of Physics University of Zielona G´ ora ul. pl Operator Theory: Advances and Applications, Vol. 174, 35–50 c 2007 Birkh¨ auser Verlag Basel/Switzerland On the Spectrum of Partially Periodic Operators Rupert L. Frank and Roman G. Shterenberg Abstract. We consider Schr¨ odinger operators H = −Δ + V in L2 (Ω) where the domain Ω ⊂ Rd+1 and the potential V = V (x, y) are periodic with respect + to the variable x ∈ Rd .
We discuss the deﬁnition of a rank one singular perturbation of a non-self-adjoint operator L in Hilbert space H. Provided that the operator L is a non-self-adjoint perturbation of a self-adjoint operator A and that the spectrum of the operator L is absolutely continuous we are able to establish a concise resolvent formula for the singular perturbations of the class considered and to establish a model representation of it in the dilation space associated with the operator L. Mathematics Subject Classiﬁcation (2000).
To prove the theorem we need some preliminaries. For simplicity, we denote Ut ( ) := e−i t for a ﬁxed t > 0. It was shown in  that the operator Hα,β has at least one and at most n isolated eigenvalues. We denote them by αβ,k , k = 1, . . , l with l ≤ n, and use ψαβ,k as symbols for the corresponding (normalized) eigenfunctions. 1) where E(·) ≡ Eα,β (·) is the spectral measure of Hα,β . By assumption there are no embedded eigenvalues (cf. 1) and by  also the singularly continuous component is void, hence the second term is associated solely with σac (Hα,β ).