scholarly journals Symmetric multiparameter problems and deficiency index theory

1988 ◽  
Vol 31 (3) ◽  
pp. 481-488 ◽  
Author(s):  
Patrick J. Browne ◽  
Hamlet Isaev
Keyword(s):  
Hilbert Space ◽  
Half Plane ◽  
Classical Result ◽  
Symmetric Operator ◽  
Null Space ◽  
Index Theory ◽  
Lower Half ◽  
Deficiency Index ◽  
Lower Half Plane ◽  
Adjoint Extension

In this article we study the multiparameter generalization of standard deficiency index theory. A classical result in this area states that if T is a symmetric operator in a Hilbert space then the dimension of the null space of T*−λI, λ∈ℂ, is constant for λ belonging to the upper (or lower) half-plane and further, when these two constants are equal, T admits a self-adjoint extension.

EXISTENCE OF ALMOST EXPONENTIALLY DECAYING STATES FOR BARRIER POTENTIALS

2001 ◽  
Vol 13 (03) ◽  
pp. 267-305 ◽  
Author(s):  
RICHARD LAVINE
Keyword(s):  
Error Estimate ◽  
Half Plane ◽  
Imaginary Part ◽  
Schrödinger Operator ◽  
Small Error ◽  
Lower Half ◽  
Half Line ◽  
Schrodinger Operator ◽  
Lower Half Plane ◽  
Decaying States

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‖.

The Stieltjes Moment Problem

1981 ◽  
Vol 24 (3) ◽  
pp. 279-282
Author(s):  
G. Klambauer
Keyword(s):  
Hilbert Space ◽  
Moment Problem ◽  
Symmetric Operator ◽  
Extension Theorem ◽  
Spectral Theorem ◽  
Friedrichs Extension ◽  
Stieltjes Moment Problem ◽  
Operator Version ◽  
Adjoint Extension ◽  
Adjoint Operators

We shall apply the spectral theorem for self adjoint operators in Hilbert space to study an operator version of the Stieltjes moment problem [1]. In the course of the work we shall make use of the Friedrichs extension theorem which states that any non-negative symmetric operator in Hilbert space has a non-negative self adjoint extension.

scholarly journals On the S-matrix of Schrödinger operator with nonlocal δ-interaction

2021 ◽  
Vol 41 (3) ◽  
pp. 413-435
Author(s):  
Anna Główczyk ◽  
Sergiusz Kużel
Keyword(s):  
Half Plane ◽  
Schrödinger Operator ◽  
Scattering Theory ◽  
Lower Half ◽  
Reflection And Transmission Coefficients ◽  
Schrodinger Operator ◽  
Lower Half Plane ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
S Matrix

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.

scholarly journals The Wedge-of-the-edge Theorem: Edge-of-the-wedge Type Phenomenon Within the Common Real Boundary

2019 ◽  
Vol 62 (02) ◽  
pp. 417-427
Author(s):  
J. E. Pascoe
Keyword(s):  
Half Plane ◽  
Real Space ◽  
Lower Half ◽  
Lower Half Plane ◽  
Fixed Size ◽  
Upper Half Plane ◽  
The Common ◽  
Set Of Points ◽  
Edge Theorem ◽  
Small Overlap

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.

Formally Normal Operators Having no Normal Extensions

1965 ◽  
Vol 17 ◽  
pp. 1030-1040 ◽  
Author(s):  
Earl A. Coddington
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Null Space ◽  
Normal Operator ◽  
Closed Linear Operator ◽  
Normal Operators ◽  
The Given ◽  
Densely Defined ◽  
Made In

The domain and null space of an operator A in a Hilbert space will be denoted by and , respectively. A formally normal operatorN in is a densely defined closed (linear) operator such that , and for all A normal operator in is a formally normal operator N satisfying 35 . A study of the possibility of extending a formally normal operator N to a normal operator in the given , or in a larger Hilbert space, was made in (1).

Finite-Dimensional Extensions of Certain Symmetric Operators(1)

1973 ◽  
Vol 16 (3) ◽  
pp. 455-456
Author(s):  
I. M. Michael
Keyword(s):  
Hilbert Space ◽  
Symmetric Operator ◽  
Inner Product ◽  
Selfadjoint Extension ◽  
Von Neumann ◽  
Deficiency Indices ◽  
Finite Dimensional ◽  
Symmetric Operators

Let H be a Hilbert space with inner product 〈,). A well-known theorem of von Neumann states that, if S is a symmetric operator in H, then S has a selfadjoint extension in H if and only if S has equal deficiency indices. This result was extended by Naimark, who proved that, even if the deficiency indices of S are unequal, there always exists a Hilbert space H1 such that H ⊆ H1 and S has a selfadjoint extension in H1.

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