Constructive Mathematics and Quantum Mechanics: Unbounded Operators and the Spectral Theorem

2021 ◽  
pp. 212-236
Keyword(s):  
Quantum Mechanics ◽  
Spectral Theorem ◽  
Constructive Mathematics ◽  
Unbounded Operators

scholarly journals ALGEBRAS OF UNBOUNDED OPERATORS AND PHYSICAL APPLICATIONS: A SURVEY

2007 ◽  
Vol 19 (03) ◽  
pp. 231-271 ◽  
Author(s):  
F. BAGARELLO
Keyword(s):  
Quantum Mechanics ◽  
Algebraic Approach ◽  
Unbounded Operators ◽  
Large Systems

After a historical introduction on the standard algebraic approach to quantum mechanics of large systems we review the basic mathematical aspects of the algebras of unbounded operators. After that we discuss in some details their relevance in physical applications.

Green's Functions for Singular Ordinary Differential Operators

1967 ◽  
Vol 19 ◽  
pp. 571-582 ◽  
Author(s):  
Fred Brauer
Keyword(s):  
Hilbert Space ◽  
Green's Function ◽  
Eigenfunction Expansion ◽  
Differential Operators ◽  
Good Deal ◽  
Linear Operators ◽  
Spectral Theorem ◽  
Unbounded Operators ◽  
Expansion Theorem ◽  
Ordinary Differential Operators

There are several ways to approach the eigenfunction expansion problem for ordinary differential operators via the spectral theorem for self-ad joint linear operators in Hilbert space. One can examine the resolvent, which requires a detailed study of the Green's function (4, 5, 7), or one can use the spectral theorem for unbounded operators (2, 3, 9). Since the eigenf unction expansion theorem also requires some multiplicity theory, unless one is prepared to use a rather powerful form of the spectral theorem for unbounded operators, as in (2, 9), the proof requires a good deal of work in addition to the spectral theorem.

Spherical Harmonics, the Weyl Transform and the Fourier Transform on the Heisenberg Group

1984 ◽  
Vol 36 (4) ◽  
pp. 615-684 ◽  
Author(s):  
Daryl Geller
Keyword(s):  
Hilbert Space ◽  
Fourier Transform ◽  
Quantum Mechanics ◽  
Spherical Harmonics ◽  
Separable Hilbert Space ◽  
Dense Subspace ◽  
Unbounded Operators ◽  
Weyl Transform ◽  
The Fourier Transform ◽  
Image Position

In the early days of quantum mechanics, Weyl asked the following question. Let λ be a non-zero real number, ℋa separable Hilbert space. Given certain (unbounded) operators W1,…,Wn,W1+, …, Wn+ on ℋ satisfying(on a dense subspace D of ℋ) with all other commutators vanishing. Given also a function where ζ ∈ Cn. Let W = (W1 …, Wn) W+ = (W1+ …, Wn+). How does one associate to f an operator f(W, W+)? (Actually, Weyl phrased the question in terms of p = Re ζ, q = Im ζ, P = Re W, Q = Im W+ which represent momentum and position. In this paper, however, we wish to exploit the unitary group on Cn and so prefer complex notation.)

Sign in / Sign up

Export Citation Format

Share Document