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Problem Statement: Consider the Hilbert space L² (R) and the differential operator D = -id defined on the domain H¹(R), the Sobolev space of square-integrable functions with square-integrable first derivatives. 1. Self-Adjointness and Spectrum of D: Prove that D is self-adjoint and determine its spectrum. Show that the spectrum is absolutely continuous and covers the entire real line. 2. Spectral Measure of D: Construct the spectral measure E for D and demonstrate that it corresponds to the Fourier transform, i.e., E(B) projects onto the subspace of functions whose Fourier transforms are supported in BCR. 3. Measure-Theoretic Properties and Functional Calculus: Use the spectral measure E to define the functional calculus for D. Prove that for any bounded measurable function f : RC, the operator f(D) acts as multiplication by f($) in the Fourier transform domain, and analyze the measure-theoretic implications of this representation. Requirements: • Analyze differential operators within the spectral theory framework. Employ Fourier analysis and measure theory to construct and interpret spectral measures. • Apply functional calculus in the context of unbounded, self-adjoint operators.

Problem Statement: Consider the Hilbert space L² (R) and the differential operator D = -id defined on the domain H¹(R), the Sobolev space of square-integrable functions with square-integrable first derivatives. 1. Self-Adjointness and Spectrum of D: Prove that D is self-adjoint and determine its spectrum. Show that the spectrum is absolutely continuous and covers the entire real line. 2. Spectral Measure of D: Construct the spectral measure E for D and demonstrate that it corresponds to the Fourier transform, i.e., E(B) projects onto the subspace of functions whose Fourier transforms are supported in BCR. 3. Measure-Theoretic Properties and Functional Calculus: Use the spectral measure E to define the functional calculus for D. Prove that for any bounded measurable function f : RC, the operator f(D) acts as multiplication by f($) in the Fourier transform domain, and analyze the measure-theoretic implications of this representation. Requirements: • Analyze differential operators within the spectral theory framework. Employ Fourier analysis and measure theory to construct and interpret spectral measures. • Apply functional calculus in the context of unbounded, self-adjoint operators.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 78E
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Problem Statement: Consider the Hilbert space L² (R) and the differential operator D = -id defined on the domain H¹(R), the Sobolev space of square-integrable functions with square-integrable first derivatives. 1. Self-Adjointness and Spectrum of D: Prove that D is self-adjoint and determine its spectrum. Show that the spectrum is absolutely continuous and covers the entire real line. 2. Spectral Measure of D: Construct the spectral measure E for D and demonstrate that it corresponds to the Fourier transform, i.e., E(B) projects onto the subspace of functions whose Fourier transforms are supported in BCR. 3. Measure-Theoretic Properties and Functional Calculus: Use the spectral measure E to define the functional calculus for D. Prove that for any bounded measurable function f : RC, the operator f(D) acts as multiplication by f($) in the Fourier transform domain, and analyze the measure-theoretic implications of this representation. Requirements: • Analyze differential operators within the spectral theory framework. Employ Fourier analysis and measure theory to construct and interpret spectral measures. • Apply functional calculus in the context of unbounded, self-adjoint operators.
Transcribed Image Text:Problem Statement: Consider the Hilbert space L² (R) and the differential operator D = -id defined on the domain H¹(R), the Sobolev space of square-integrable functions with square-integrable first derivatives. 1. Self-Adjointness and Spectrum of D: Prove that D is self-adjoint and determine its spectrum. Show that the spectrum is absolutely continuous and covers the entire real line. 2. Spectral Measure of D: Construct the spectral measure E for D and demonstrate that it corresponds to the Fourier transform, i.e., E(B) projects onto the subspace of functions whose Fourier transforms are supported in BCR. 3. Measure-Theoretic Properties and Functional Calculus: Use the spectral measure E to define the functional calculus for D. Prove that for any bounded measurable function f : RC, the operator f(D) acts as multiplication by f($) in the Fourier transform domain, and analyze the measure-theoretic implications of this representation. Requirements: • Analyze differential operators within the spectral theory framework. Employ Fourier analysis and measure theory to construct and interpret spectral measures. • Apply functional calculus in the context of unbounded, self-adjoint operators.
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