Spectral Theory: “Spectral Analysis” of Operators

“Just as light can be decomposed into a rainbow, operators can be decomposed into ‘eigenvalue spectra.’”

🎯 What is Spectral Theory?

Imagine you’re listening to music:

• Time domain: What you hear is sound waves changing over time
• Frequency domain: Actually this is superposition of many sine waves of different frequencies

Fourier transform decomposes time-domain signals into frequency-domain “spectrum”:

graph LR
    T["Time Domain Signal<br/>f(t)"] --> |"Fourier Transform"| F["Spectrum<br/>F(ω)"]
    F --> |"Inverse Transform"| T

    style T fill:#e1f5ff
    style F fill:#ffe1e1

Spectral theory does something similar, but for operators rather than functions:

Decompose complex operator $H$ into simple “eigenvalues” and “eigenvectors”.

🌈 Analogy: Prism Decomposing Light

graph LR
    W["White Light<br/>(Complex Operator H)"] --> P["Prism<br/>(Spectral Decomposition)"]
    P --> R["Red Light λ₁"]
    P --> O["Orange Light λ₂"]
    P --> Y["Yellow Light λ₃"]
    P --> G["Green Light λ₄"]
    P --> B["Blue Light λ₅"]
    P --> V["Violet Light λ₆"]

    style W fill:#fff4e1
    style P fill:#e1f5ff
    style R fill:#ffcccc
    style O fill:#ffd4b3
    style Y fill:#fff4b3
    style G fill:#ccffcc
    style B fill:#b3d9ff
    style V fill:#e6b3ff

• White light = Complex operator $H$
• Prism = Spectral decomposition
• Colored lights = Eigenvalues ${\lambda }_1,{\lambda }_2,\dots$

Each eigenvalue corresponds to a “pure color”—a simple mode of the system.

📐 Self-Adjoint Operators and Spectral Decomposition

What is a Self-Adjoint Operator?

In quantum mechanics, all observables are represented by self-adjoint operators:

$$H^{†}=H$$

(Here $†$ denotes conjugate transpose)

Why require self-adjoint?

Because eigenvalues of self-adjoint operators must be real—only then can they correspond to physically measurable quantities!

Spectral Theorem

Theorem: Any self-adjoint operator $H$ can be “diagonalized”—written as sum of eigenvalues and projections:

$$H={\int }_{\sigma (H)}\lambda dE(\lambda )$$

where:

• $\sigma (H)$: Spectrum of operator $H$ (set of all eigenvalues)
• $\lambda$: Eigenvalue
• $E(\lambda )$: Spectral measure (projection-valued measure)

Physical meaning:

Any measurement can be decomposed into projection measurements on various eigenstates!

Discrete Spectrum vs Continuous Spectrum

Discrete spectrum: Eigenvalues are countable (e.g., hydrogen atom energy levels)

$$H=\sum _{n=1}^{\infty }{E}_n|n⟩⟨n|$$

graph TB
    subgraph "Discrete Spectrum (Hydrogen Atom)"
        E1["E₁ = -13.6 eV"]
        E2["E₂ = -3.4 eV"]
        E3["E₃ = -1.5 eV"]
        E4["E₄ = ..."]
    end

    style E1 fill:#ffe1e1
    style E2 fill:#ffe1e1
    style E3 fill:#ffe1e1
    style E4 fill:#ffe1e1

Continuous spectrum: Eigenvalues are continuous (e.g., free particle momentum)

$$H={\int }_{-\infty }^{\infty }\frac{p^2}{2m}|p⟩⟨p|dp$$

graph TB
    subgraph "Continuous Spectrum (Free Particle)"
        C["Energy Can Take Any Positive Value<br/>E ∈ [0, ∞)"]
    end

    style C fill:#e1f5ff

🔬 Spectral Shift Function: “Fingerprint” of Scattering

From Free System to Scattering System

Consider two Hamiltonians:

• $H_0$: Free system (no interaction)
• $H=H_0+V$: Scattering system (with potential $V$)

Question: What is the relationship between spectrum of $H$ and spectrum of $H_0$?

Answer: Described by spectral shift function $\xi (\omega )$!

Definition of Spectral Shift Function

Intuitive idea:

How much more “spectral weight” does $H$ have compared to $H_0$ near energy $\omega$?

Mathematical definition (Krein formula):

For any smooth test function $f$:

$$\text{tr}(f(H)-f(H_0))={\int }_{-\infty }^{\infty }{f}^{\prime }(\omega )\xi (\omega )d\omega$$

Physical meaning:

$\xi (\omega )$ measures how much spectral weight the scattering potential $V$ “shifts” at energy $\omega$.

graph TB
    subgraph "Free System Spectrum"
        F["ρ₀(ω)<br/>Density of States"]
    end

    subgraph "Scattering System Spectrum"
        S["ρ(ω)<br/>Density of States"]
    end

    F --> |"Add Potential V"| S
    S -.-> |"Spectral Shift ξ(ω)"| D["ρ(ω) - ρ₀(ω)"]

    style F fill:#e1f5ff
    style S fill:#ffe1e1
    style D fill:#fff4e1

Spectral Shift and Density of States

Key relation:

$${\xi }^{\prime }(\omega )=\rho (\omega )-{\rho }_0(\omega )=:{\rho }_{\text{rel}}(\omega )$$

where:

• $\rho (\omega )=\text{tr}\delta (\omega -H)$: Density of states of $H$
• ${\rho }_0(\omega )=\text{tr}\delta (\omega -H_0)$: Density of states of $H_0$
• ${\rho }_{\text{rel}}(\omega )$: Relative density of states

Physical meaning:

Derivative of spectral shift function = Extra density of states that scattering system has compared to free system!

⚡ Birman-Kreĭn Formula: Core Tool

This is one of the most important mathematical formulas in GLS theory!

Relation Between S-Matrix and Spectral Shift

Birman-Kreĭn formula:

$$\boxed{\det S(\omega )=e^{-2\pi i\xi (\omega )}}$$

where:

• $S(\omega )$: Scattering matrix (depends on energy $\omega$)
• $\xi (\omega )$: Spectral shift function
• $\det$: Determinant

Corollary: Taking logarithm and differentiating w.r.t. $\omega$:

$$\frac{d}{d\omega }[-i\ln \det S(\omega )]=2\pi {\xi }^{\prime }(\omega )=-2\pi {\rho }_{\text{rel}}(\omega )$$

Define total scattering phase:

$$\Phi (\omega )=-i\ln \det S(\omega )$$

Then $\Phi (\omega )=-2\pi \xi (\omega )$, so

$$\boxed{\frac{{\Phi }^{\prime }(\omega )}{\pi }=-2{\xi }^{\prime }(\omega )=2{\rho }_{\text{rel}}(\omega )}$$

Or written as:

$$\boxed{\frac{{\phi }^{\prime }(\omega )}{\pi }={\rho }_{\text{rel}}(\omega )}$$

(Here $\phi =\Phi /2$ is half-phase)

Connection to Wigner-Smith Delay

Recall Wigner-Smith time delay matrix:

$$Q(\omega )=-iS(\omega {)}^{†}\frac{\partial S(\omega )}{\partial \omega }$$

Its trace is:

$$\text{tr}Q(\omega )=-i\text{tr}\left(S^{†}{\partial }_{\omega }S\right)=-i\frac{\partial }{\partial \omega }\text{tr}\ln S=\frac{\partial \Phi }{\partial \omega }$$

Combining Birman-Kreĭn formula with ${\Phi }^{\prime }(\omega )=-2\pi {\xi }^{\prime }(\omega )=2\pi {\rho }_{\text{rel}}(\omega )$, we get

$$\boxed{\frac{1}{2\pi }\text{tr}Q(\omega )=\frac{{\Phi }^{\prime }(\omega )}{2\pi }=-{\xi }^{\prime }(\omega )={\rho }_{\text{rel}}(\omega )}$$

This is the mathematical source of the Unified Time Scale Identity!

graph TB
    S["Scattering Matrix<br/>S(ω)"] --> |"Determinant"| D["det S(ω)"]
    D --> |"Birman-Kreĭn"| Xi["Spectral Shift Function<br/>ξ(ω)"]
    Xi --> |"Derivative"| Rho["Relative Density of States<br/>ρ_rel(ω)"]

    S --> |"Wigner-Smith"| Q["Delay Matrix<br/>Q(ω)"]
    Q --> |"Trace/2π"| Rho

    style S fill:#e1f5ff
    style Xi fill:#fff4e1
    style Rho fill:#ffe1e1,stroke:#ff6b6b,stroke-width:3px
    style Q fill:#e1ffe1

🧮 Simple Example: Single-Channel Scattering

Problem Setup

Consider one-dimensional scattering, single channel, scattering matrix is $1\times 1$ matrix (just a complex number):

$$S(\omega )=e^{2i\delta (\omega )}$$

where $\delta (\omega )$ is scattering phase shift.

Calculate Spectral Shift Function

By Birman-Kreĭn formula:

$$\det S(\omega )=S(\omega )=e^{2i\delta (\omega )}=e^{-2\pi i\xi (\omega )}$$

Comparing exponents:

$$2i\delta (\omega )=-2\pi i\xi (\omega )$$

We get:

$$\xi (\omega )=-\frac{\delta (\omega )}{\pi }$$

Calculate Density of States

Relative density of states:

$${\rho }_{\text{rel}}(\omega )={\xi }^{\prime }(\omega )=-\frac{1}{\pi }\frac{d\delta (\omega )}{d\omega }$$

This is the famous Friedel sum rule!

Calculate Time Delay

Wigner-Smith matrix (1×1 case):

$$Q(\omega )=-i{e}^{-2i\delta }\frac{d}{d\omega }{e}^{2i\delta }=-i\cdot 2i\frac{d\delta }{d\omega }=2\frac{d\delta }{d\omega }$$

Time delay:

$${\tau }_W=\frac{Q(\omega )}{2\pi }=\frac{1}{\pi }\frac{d\delta }{d\omega }=-{\rho }_{\text{rel}}(\omega )$$

Perfectly verifies the formula!

🔗 Applications in GLS Theory

1. Unified Time Scale

Birman-Kreĭn formula gives:

$$\kappa (\omega )={\rho }_{\text{rel}}(\omega )=\frac{1}{2\pi }\text{tr}Q(\omega )=\frac{{\phi }^{\prime }(\omega )}{\pi }$$

This is the mathematical foundation of the Unified Time Scale Identity!

2. Density of States and Entropy

Relative density of states ${\rho }_{\text{rel}}$ directly relates to entropy:

$$S=k_B\ln \Omega \approx k_B\int \rho (\omega )\ln \rho (\omega )d\omega$$

3. Causality and Spectrum

Non-negativity of spectrum ${\rho }_{\text{rel}}(\omega )\ge 0$ ensures monotonicity of time scale, which in turn is considered to guarantee causality.

📝 Key Formulas Summary

🎓 Further Reading

• Theory document: unified-time-scale-geometry.md Appendix A
• Original paper: Birman & Kreĭn, “On the theory of wave operators and scattering operators” (1962)
• Strohmaier & Waters, “The Birman-Krein formula for differential forms” (arXiv:2104.13589)
• Next: 02-noncommutative-geometry_en.md - Noncommutative Geometry

🤔 Exercises

1. Conceptual Understanding:

   • Why must eigenvalues of self-adjoint operators be real?
   • Why is spectral shift function $\xi (\omega )$ called “shift”?
   • Why is relative density of states important?

2. Calculation Exercises:

   • Verify: $\det (AB)=\det A\cdot \det B$
   • For $S(\omega )=e^{2i\delta (\omega )}$, calculate $Q(\omega )$
   • Prove: $\text{tr}\ln A=\ln \det A$ (finite-dimensional)

3. Physical Applications:

   • Are hydrogen atom energy levels discrete or continuous spectrum?
   • What is the spectrum of a free particle?
   • What is the physical meaning of scattering phase shift $\delta (\omega )$?

4. Advanced Thinking:

   • If $V$ is an attractive potential, what is the sign of ${\rho }_{\text{rel}}$?
   • How does Birman-Kreĭn formula generalize to multi-channel scattering?
   • What is the relationship between spectral shift function and Levinson’s theorem?

Next Step: After understanding spectral theory, we will learn Noncommutative Geometry—how to define geometry using algebra, the mathematical language of boundary theory!