Problem 7: Tensor Products of Operators and Their Spectra Background: Let H₁ = L²([0, 1]) and H₂ = L²([0, 1]), both Hilbert spaces. Consider the self- adjoint operators A: H₁ →H₁ and B: H2 → H₂ defined by d²ƒ (Af)(x) = da² (Bf)(x) = xf(x), with appropriate domains ensuring self-adjointness. Tasks: a) Individual Spectra: • Determine the spectra (A) and (B) of the operators A and B, respectively. ⚫ Plot Requirement: Plot the spectra σ(A) and (B) on the real line, indicating their key features. b) Tensor Product Operator: • Define the tensor product operator C = A&I+IB on H = H₁ & H₂. • Explain the significance of the tensor product in this context and its impact on the operator's properties. c) Spectrum of the Tensor Product Operator: • Using the spectra of A and B, determine the spectrum (C) of the operator C. • Plot Requirement: Visualize (C) in the complex plane or on the real line, as appropriate. Highlight how the tensor product affects the combined spectrum. d) Eigenfunctions and Basis: • Describe the eigenfunctions of C' in terms of the eigenfunctions of A and B. • Plot Requirement: If possible, plot a few eigenfunctions of C to illustrate their structure, especially focusing on how they combine the properties of A and B. e) Perturbation and Spectral Stability: • Introduce a perturbation D = C + EA & B, where € is a small real parameter. • Analyze how the spectrum (D) changes as e varies in [-0.1, 0.1]. • Plot Requirement: Create plots showing σ(D) for selected values of e. Discuss the stability of the spectrum under such tensor product perturbations.
Problem 7: Tensor Products of Operators and Their Spectra Background: Let H₁ = L²([0, 1]) and H₂ = L²([0, 1]), both Hilbert spaces. Consider the self- adjoint operators A: H₁ →H₁ and B: H2 → H₂ defined by d²ƒ (Af)(x) = da² (Bf)(x) = xf(x), with appropriate domains ensuring self-adjointness. Tasks: a) Individual Spectra: • Determine the spectra (A) and (B) of the operators A and B, respectively. ⚫ Plot Requirement: Plot the spectra σ(A) and (B) on the real line, indicating their key features. b) Tensor Product Operator: • Define the tensor product operator C = A&I+IB on H = H₁ & H₂. • Explain the significance of the tensor product in this context and its impact on the operator's properties. c) Spectrum of the Tensor Product Operator: • Using the spectra of A and B, determine the spectrum (C) of the operator C. • Plot Requirement: Visualize (C) in the complex plane or on the real line, as appropriate. Highlight how the tensor product affects the combined spectrum. d) Eigenfunctions and Basis: • Describe the eigenfunctions of C' in terms of the eigenfunctions of A and B. • Plot Requirement: If possible, plot a few eigenfunctions of C to illustrate their structure, especially focusing on how they combine the properties of A and B. e) Perturbation and Spectral Stability: • Introduce a perturbation D = C + EA & B, where € is a small real parameter. • Analyze how the spectrum (D) changes as e varies in [-0.1, 0.1]. • Plot Requirement: Create plots showing σ(D) for selected values of e. Discuss the stability of the spectrum under such tensor product perturbations.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 15EQ
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![Problem 7: Tensor Products of Operators and Their Spectra
Background: Let H₁ = L²([0, 1]) and H₂ = L²([0, 1]), both Hilbert spaces. Consider the self-
adjoint operators A: H₁ →H₁ and B: H2 → H₂ defined by
d²ƒ
(Af)(x)
=
da²
(Bf)(x) = xf(x),
with appropriate domains ensuring self-adjointness.
Tasks:
a) Individual Spectra:
•
Determine the spectra (A) and (B) of the operators A and B, respectively.
⚫ Plot Requirement: Plot the spectra σ(A) and (B) on the real line, indicating their key
features.
b) Tensor Product Operator:
• Define the tensor product operator C = A&I+IB on H = H₁ & H₂.
• Explain the significance of the tensor product in this context and its impact on the operator's
properties.
c) Spectrum of the Tensor Product Operator:
• Using the spectra of A and B, determine the spectrum (C) of the operator C.
• Plot Requirement: Visualize (C) in the complex plane or on the real line, as appropriate.
Highlight how the tensor product affects the combined spectrum.
d) Eigenfunctions and Basis:
• Describe the eigenfunctions of C' in terms of the eigenfunctions of A and B.
•
Plot Requirement: If possible, plot a few eigenfunctions of C to illustrate their structure,
especially focusing on how they combine the properties of A and B.
e) Perturbation and Spectral Stability:
•
Introduce a perturbation D = C + EA & B, where € is a small real parameter.
• Analyze how the spectrum (D) changes as e varies in [-0.1, 0.1].
•
Plot Requirement: Create plots showing σ(D) for selected values of e. Discuss the stability of
the spectrum under such tensor product perturbations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe3e95596-0b68-4149-932b-c1a9acbb7141%2F26e047f9-48ab-4e93-9cb9-a1d06f7c7fed%2F47dhqkr_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 7: Tensor Products of Operators and Their Spectra
Background: Let H₁ = L²([0, 1]) and H₂ = L²([0, 1]), both Hilbert spaces. Consider the self-
adjoint operators A: H₁ →H₁ and B: H2 → H₂ defined by
d²ƒ
(Af)(x)
=
da²
(Bf)(x) = xf(x),
with appropriate domains ensuring self-adjointness.
Tasks:
a) Individual Spectra:
•
Determine the spectra (A) and (B) of the operators A and B, respectively.
⚫ Plot Requirement: Plot the spectra σ(A) and (B) on the real line, indicating their key
features.
b) Tensor Product Operator:
• Define the tensor product operator C = A&I+IB on H = H₁ & H₂.
• Explain the significance of the tensor product in this context and its impact on the operator's
properties.
c) Spectrum of the Tensor Product Operator:
• Using the spectra of A and B, determine the spectrum (C) of the operator C.
• Plot Requirement: Visualize (C) in the complex plane or on the real line, as appropriate.
Highlight how the tensor product affects the combined spectrum.
d) Eigenfunctions and Basis:
• Describe the eigenfunctions of C' in terms of the eigenfunctions of A and B.
•
Plot Requirement: If possible, plot a few eigenfunctions of C to illustrate their structure,
especially focusing on how they combine the properties of A and B.
e) Perturbation and Spectral Stability:
•
Introduce a perturbation D = C + EA & B, where € is a small real parameter.
• Analyze how the spectrum (D) changes as e varies in [-0.1, 0.1].
•
Plot Requirement: Create plots showing σ(D) for selected values of e. Discuss the stability of
the spectrum under such tensor product perturbations.
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