lower half plane
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2021 ◽  
Vol 41 (3) ◽  
pp. 413-435
Author(s):  
Anna Główczyk ◽  
Sergiusz Kużel

Schrödinger operators with nonlocal \(\delta\)-interaction are studied with the use of the Lax-Phillips scattering theory methods. The condition of applicability of the Lax-Phillips approach in terms of non-cyclic functions is established. Two formulas for the \(S\)-matrix are obtained. The first one deals with the Krein-Naimark resolvent formula and the Weyl-Titchmarsh function, whereas the second one is based on modified reflection and transmission coefficients. The \(S\)-matrix \(S(z)\) is analytical in the lower half-plane \(\mathbb{C}_{−}\) when the Schrödinger operator with nonlocal \(\delta\)-interaction is positive self-adjoint. Otherwise, \(S(z)\) is a meromorphic matrix-valued function in \(\mathbb{C}_{−}\) and its properties are closely related to the properties of the corresponding Schrödinger operator. Examples of \(S\)-matrices are given.


2019 ◽  
Vol 16 (01) ◽  
pp. 29-64
Author(s):  
Joshua Males

We introduce and investigate an infinite family of functions which are shown to have generalized quantum modular properties. We realize their “companions” in the lower half plane both as double Eichler integrals and as non-holomorphic theta functions with coefficients given by double error functions. Further, we view these Eichler integrals in a modular setting as parts of certain weight two indefinite theta series.


2019 ◽  
Vol 62 (02) ◽  
pp. 417-427
Author(s):  
J. E. Pascoe

AbstractThe edge-of-the-wedge theorem in several complex variables gives the analytic continuation of functions defined on the poly upper half plane and the poly lower half plane, the set of points in $\mathbb{C}^{n}$ with all coordinates in the upper and lower half planes respectively, through a set in real space, $\mathbb{R}^{n}$ . The geometry of the set in the real space can force the function to analytically continue within the boundary itself, which is qualified in our wedge-of-the-edge theorem. For example, if a function extends to the union of two cubes in $\mathbb{R}^{n}$ that are positively oriented with some small overlap, the functions must analytically continue to a neighborhood of that overlap of a fixed size not depending of the size of the overlap.


2018 ◽  
Vol 6 (1) ◽  
pp. 4-11
Author(s):  
Le Xuan Quang

 In this paper the projection iterative method is used for solving the generalized Carleman problem A(t) F+(t) + B(t) F+(-t) + C(t) F-(t) = h(t), tÎR Where R is the real axis; A(t), B(t), C(t) are Holder continuous functions;  h(t)ÎL2, and  F+(t), F-(t) is a boundary value of analytic function upper (lower) half-plane.


2013 ◽  
Vol 34 (4) ◽  
pp. 1168-1183 ◽  
Author(s):  
KIRIL DATCHEV ◽  
SEMYON DYATLOV ◽  
MACIEJ ZWORSKI

AbstractWe give a sharp polynomial bound on the number of Pollicott–Ruelle resonances. These resonances, which are complex numbers in the lower half-plane, appear in expansions of correlations for Anosov contact flows. The bounds follow the tradition of upper bounds on the number of scattering resonances and improve a recent bound of Faure and Sjöstrand. The complex scaling method used in scattering theory is replaced by an approach using exponentially weighted spaces introduced by Helffer and Sjöstrand in scattering theory and by Faure and Sjöstrand in the theory of Anosov flows.


2006 ◽  
Vol 18 (01) ◽  
pp. 61-78 ◽  
Author(s):  
HELLMUT BAUMGÄRTEL

A Gelfand triplet for the Hamiltonian H of the Friedrichs model on ℝ with multiplicity space [Formula: see text], [Formula: see text], is constructed such that exactly the resonances (poles of the inverse of the Livšic-matrix) are (generalized) eigenvalues of H. The corresponding eigen(anti)linear forms are calculated explicitly. Using the wave matrices for the wave (Möller) operators the corresponding eigen(anti)linear forms on the Schwartz space [Formula: see text] for the unperturbed Hamiltonian H0 are also calculated. It turns out that they are of pure Dirac type and can be characterized by their corresponding Gamov vector λ → k/(ζ0 - λ)-1, ζ0 resonance, [Formula: see text], which is uniquely determined by restriction of [Formula: see text] to [Formula: see text], where [Formula: see text] denotes the Hardy space of the upper half-plane. Simultaneously this restriction yields a truncation of the generalized evolution to the well-known decay semigroup for t ≥ 0 of the Toeplitz type on [Formula: see text]. That is: Exactly those pre-Gamov vectors λ → k/(ζ - λ)-1, ζ from the lower half-plane, [Formula: see text], have an extension to a generalized eigenvector of H if ζ is a resonance and if k is from that subspace of [Formula: see text] which is uniquely determined by its corresponding Dirac type antilinear form.


2001 ◽  
Vol 13 (03) ◽  
pp. 267-305 ◽  
Author(s):  
RICHARD LAVINE

For a Schrödinger operator H on the half line whose potential has a trapping barrier, and is convex outside the barrier, there exists a φ, supported mostly inside the barrier, such that for t>0, <φ, e-iHtφ>~e-izt up to a small error, where φ is obtained by cutting off a nonnormalizable solution ψ of Hψ=zψ, and z is in the lower half-plane. The imaginary part of z is estimated explicitly, and the error estimate is explicitly proportional to | Im z log | Im z‖.


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