Spectral Shift Function: Energy Levels Remember Interaction

“Spectral shift function can be viewed as the signature of Hamiltonian on energy levels.”

🎯 Core Proposition

Definition (Kreĭn Spectral Shift Function):

For a pair of self-adjoint operators $(H,H_0)$, satisfying trace-class or quasi-trace-class perturbation conditions, there exists unique real function $\xi (\lambda )$ such that:

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

Holds for all appropriate test functions $f$.

Birman-Kreĭn formula:

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

Where $S(\omega )$ is scattering matrix.

Relative density of states:

$$\boxed{{\rho }_{\text{rel}}(\omega )=-\frac{d\xi (\omega )}{d\omega }}$$

Physical meaning:

• $\xi (\omega )$: reflects change in energy level count caused by interaction
• ${\rho }_{\text{rel}}(\omega )$: relative density of states, describes density of energy level “shifts”
• Relationship: Birman-Kreĭn formula establishes a connection between scattering and spectral properties.

💡 Intuitive Image: String Tuning

Analogy: Violin with Damping

Imagine an ideal string ($H_0$) with natural frequencies:

$${\omega }_n^{(0)}=n\pi v/L,n=1,2,3,\dots$$

Now add damping and tension perturbation ($H=H_0+V$), frequencies become:

$${\omega }_n={\omega }_n^{(0)}+\Delta {\omega }_n$$

Energy level counting:

How many resonances below frequency $\omega$?

• No perturbation: $N_0(\omega )=\lfloor L\omega /(\pi v)\rfloor$
• With perturbation: $N(\omega )$

Spectral shift function:

$$\xi (\omega )=N(\omega )-N_0(\omega )$$

Physical meaning: $\xi$ records how many energy levels are “pushed past” $\omega$.

graph TB
    H0["Free String<br/>ω₁⁽⁰⁾, ω₂⁽⁰⁾, ω₃⁽⁰⁾, ..."] --> V["Add Perturbation<br/>V (Damping/Tension)"]
    V --> H["Perturbed String<br/>ω₁, ω₂, ω₃, ..."]
    H --> XI["Spectral Shift<br/>ξ(ω) = N(ω) - N₀(ω)"]
    XI --> RHO["Density of States<br/>ρ_rel = -dξ/dω"]

    style H0 fill:#e1f5ff
    style V fill:#ffe1e1
    style XI fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style RHO fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px

Energy Level Shifts

Example: Potential barrier scattering

Without potential ($V=0$):

• Energy levels continuous, $E\in [0,\infty )$
• No bound states

With potential ($V(x)\ne 0$):

• Bound states $E_b<0$ may appear
• Continuous spectrum energy levels undergo “shift”

Spectral shift:

• Each added bound state increases $\xi (\omega )$ by 1 as $\omega \to 0$
• In continuous spectrum, $\xi (\omega )$ measures “phase shift accumulation”

📐 Mathematical Definition

Kreĭn Trace Formula

Setup:

• $H_0$: free Hamiltonian
• $H=H_0+V$: perturbed Hamiltonian
• Assumption: $V$ such that $(H+i{)}^{-1}-(H_0+i{)}^{-1}\in {\mathfrak{S}}_1$ (trace class)

Definition: For test function $f$ (e.g., $f(x)=(x-z{)}^{-1}$), we have:

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

Example: $f(x)=(x-z{)}^{-1}$

$$\text{tr}\left[(H-z{)}^{-1}-(H_0-z{)}^{-1}\right]=-{\int }_{-\infty }^{\infty }\frac{\xi (\lambda )}{(\lambda -z{)}^2}d\lambda$$

Uniqueness: $\xi (\lambda )$ is uniquely determined by this integral equation.

Physical Interpretation

For energy level counting function:

$$N(\lambda )=\text{tr}{1}_{(-\infty ,\lambda ]}(H)$$

That is: number of eigenstates with energy $\le \lambda$.

Spectral shift function:

$$\xi (\lambda )=N(\lambda )-N_0(\lambda )$$

Integral form:

$$N(\lambda )={\int }_{-\infty }^{\lambda }\rho (E)dE$$

Where $\rho (E)={\sum }_n\delta (E-E_n)$ is density of states.

Spectral shift and density of states:

$$\xi (\lambda )={\int }_{-\infty }^{\lambda }[\rho (E)-{\rho }_0(E)]dE$$

Derivative:

$$\frac{d\xi }{d\lambda }=\rho (\lambda )-{\rho }_0(\lambda )=:-{\rho }_{\text{rel}}(\lambda )$$

(Negative sign is convention)

graph LR
    N["Energy Level Count<br/>N(E)"] --> XI["Spectral Shift Function<br/>ξ(E) = N - N₀"]
    XI --> D["Differentiate<br/>dξ/dE"]
    D --> RHO["Relative Density of States<br/>ρ_rel = -dξ/dE"]

    style N fill:#e1f5ff
    style XI fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style RHO fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px

🌀 Birman-Kreĭn Formula

Determinant of Scattering Matrix

In scattering theory, $S(\omega )$ is unitary matrix:

$$S(\omega )=I+2\pi iT(\omega )$$

Where $T(\omega )$ is transition operator.

Birman-Kreĭn theorem (1962):

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

Proof idea (heuristic):

Using Fredholm determinant theory:

$$\det (I+A)=e^{\text{tr}\ln (I+A)}$$

For $(H-z{)}^{-1}-(H_0-z{)}^{-1}$, through analytic continuation and boundary conditions, can derive relationship between scattering matrix determinant and spectral shift.

Rigorous proof: Requires Hilbert-Schmidt operator theory and Cauchy theorem (see Birman & Yafaev, 1993).

Total Scattering Phase

Recall $\Phi (\omega )=\arg \det S(\omega )$, from Birman-Kreĭn formula:

$$e^{i\Phi (\omega )}=e^{-2\pi i\xi (\omega )}$$

Taking phase (choosing continuous branch):

$$\boxed{\Phi (\omega )=-2\pi \xi (\omega )}$$

Differentiating:

$$\frac{\partial \Phi }{\partial \omega }=-2\pi \frac{d\xi }{d\omega }=2\pi {\rho }_{\text{rel}}(\omega )$$

Combining with previous article’s group delay formula:

$$\text{tr}Q(\omega )=\frac{\partial \Phi }{\partial \omega }$$

We get:

$$\boxed{\text{tr}Q(\omega )=2\pi {\rho }_{\text{rel}}(\omega )}$$

Or:

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

Conclusion: Scattering, spectral shift, and density of states show unity in this framework.

graph TB
    S["Scattering Matrix<br/>S(ω)"] --> D["Determinant<br/>det S"]
    D --> BK["Birman-Kreĭn<br/>det S = e^(-2πiξ)"]
    BK --> XI["Spectral Shift Function<br/>ξ(ω)"]
    XI --> RHO["Relative Density of States<br/>ρ_rel = -dξ/dω"]

    PH["Total Phase<br/>Φ = arg det S"] --> REL["Relationship<br/>Φ = -2πξ"]
    REL --> XI

    Q["Group Delay<br/>Q = -iS†∂_ωS"] --> TR["Trace<br/>tr Q = ∂_ωΦ"]
    TR --> ID["Identity<br/>tr Q = 2πρ_rel"]
    RHO --> ID

    style XI fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style ID fill:#e1ffe1,stroke:#ff6b6b,stroke-width:4px

🧮 Single-Channel Scattering Example

One-Dimensional Potential Barrier

Setup: Particle scattered by potential $V(x)$ ($V(x)=0$ as $|x|\to \infty$).

Scattering matrix (single channel):

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

Where $\delta (k)$ is phase shift, $k=\sqrt{2mE}/\hslash$ is wavenumber.

Total phase:

$$\Phi (k)=2\delta (k)$$

Birman-Kreĭn formula:

$$e^{2i\delta (k)}=e^{-2\pi i\xi (E)}$$

Choosing continuous phase:

$$2\delta (k)=-2\pi \xi (E)+2\pi n$$

Ignoring integer $n$ (phase winding):

$$\boxed{\xi (E)=-\frac{\delta (k)}{\pi }}$$

Levinson theorem:

If potential well supports $N_b$ bound states, then:

$$\delta (0)-\delta (\infty )=N_b\pi$$

From $\xi (E)=-\delta (k)/\pi$:

$$\xi (0)-\xi (\infty )=N_b$$

Physical meaning: Total change of spectral shift function equals number of bound states.

Resonance Scattering

Near resonance energy $E_r$:

$$\delta (E)\approx {\delta }_{\text{bg}}+\arctan \frac{\Gamma /2}{E-E_r}$$

Spectral shift:

$$\xi (E)=-\frac{1}{\pi }\arctan \frac{\Gamma /2}{E-E_r}$$

Density of states:

$${\rho }_{\text{rel}}(E)=-\frac{d\xi }{dE}=\frac{1}{\pi }\frac{\Gamma /2}{(E-E_r{)}^2+(\Gamma /2{)}^2}$$

This is Lorentzian line shape.

Integral:

$${\int }_{-\infty }^{\infty }{\rho }_{\text{rel}}(E)dE=-[\xi (\infty )-\xi (-\infty )]=1$$

Result: One resonance contributes density of states integral of 1.

graph TB
    subgraph "Bound States"
        B["ξ(∞) - ξ(0) = N_b"]
    end

    subgraph "Resonance"
        R["ρ_rel(E) ~ Lorentz Peak<br/>∫ρ_rel dE = 1"]
    end

    subgraph "Levinson Theorem"
        L["δ(0) - δ(∞) = N_b π"]
    end

    B --> XI["Spectral Shift Function<br/>ξ(E) = -δ/π"]
    R --> XI
    L --> XI

    style XI fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px

🔬 Multi-Channel Scattering

N×N Scattering Matrix

For multi-channel scattering, $S(\omega )$ is $N\times N$ unitary matrix.

Birman-Kreĭn formula still holds:

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

Total phase:

$$\Phi (\omega )=\arg \det S(\omega )=\sum _{j=1}^N{\delta }_j(\omega )$$

Where ${\delta }_j$ are eigenvalues phases of $S$.

Spectral shift:

$$\xi (\omega )=-\frac{1}{2\pi }\sum _{j=1}^N{\delta }_j(\omega )$$

Relative density of states:

$${\rho }_{\text{rel}}(\omega )=-\frac{d\xi }{d\omega }=\frac{1}{2\pi }\sum _{j=1}^N\frac{d{\delta }_j}{d\omega }$$

Group delay:

$$\text{tr}Q(\omega )=\sum _{j=1}^N{\tau }_j(\omega )$$

Where ${\tau }_j$ are eigenvalues of $Q$.

Relationship (from Birman-Kreĭn):

$$\sum _j{\tau }_j=\frac{\partial \Phi }{\partial \omega }=2\pi {\rho }_{\text{rel}}$$

Theoretical framework is self-consistent.

💡 Physical Meaning

Three Understandings of Spectral Shift

1. Energy level counting:

$$\xi (E)=\#\{\text{energy levels}\le E{\}}_{\text{perturbed}}-\#\{\text{energy levels}\le E{\}}_{\text{free}}$$

2. Phase memory:

$$\xi (E)=-\frac{\Phi (E)}{2\pi }=-\frac{1}{2\pi }\arg \det S(E)$$

3. Density of states integral:

$$\xi (E)=-{\int }_{-\infty }^E{\rho }_{\text{rel}}(E^{\prime })d{E}^{\prime }$$

These three are mathematically equivalent.

Why Important?

1. Connects quantum and classical:

• Quantum: energy levels, phase, scattering
• Classical: time delay, orbit deflection

Bridge: $\xi$ connects both through Birman-Kreĭn formula

2. Observability:

• $\xi$ not directly measurable
• But ${\rho }_{\text{rel}}=-d\xi /dE$ can be extracted from scattering data
• $\text{tr}Q=2\pi {\rho }_{\text{rel}}$ is measurable

3. Topological information:

• $\xi (\infty )-\xi (-\infty )=N_b$ (Levinson theorem)
• Topological invariant: Even if perturbation changes, bound state number unchanged

graph TB
    XI["Spectral Shift Function<br/>ξ(E)"] --> C["Energy Level Count<br/>ΔN(E)"]
    XI --> P["Phase Memory<br/>-Φ/2π"]
    XI --> D["Density of States Integral<br/>-∫ρ_rel"]

    C --> O["Observable<br/>Scattering Experiment"]
    P --> O
    D --> O

    XI --> T["Topology<br/>N_b = ξ(∞)-ξ(-∞)"]

    style XI fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style O fill:#e1ffe1
    style T fill:#e1f5ff

🌊 Time Scale Identity Derivation

Now we can completely derive unified time scale formula.

Step 1: Group Delay

From previous article, we know:

$$\text{tr}Q(\omega )=\frac{\partial \Phi (\omega )}{\partial \omega }$$

Step 2: Birman-Kreĭn

This article proved:

$$\Phi (\omega )=-2\pi \xi (\omega )$$

Step 3: Relative Density of States

Definition:

$${\rho }_{\text{rel}}(\omega )=-\frac{d\xi }{d\omega }$$

Step 4: Combine

$$\text{tr}Q=\frac{\partial \Phi }{\partial \omega }=\frac{\partial }{\partial \omega }(-2\pi \xi )=-2\pi \frac{d\xi }{d\omega }=2\pi {\rho }_{\text{rel}}$$

Step 5: Normalize

Define normalized time scale:

$$\kappa (\omega ):=\frac{1}{2\pi }\text{tr}Q(\omega )$$

Then:

$$\boxed{\kappa (\omega )={\rho }_{\text{rel}}(\omega )=-\frac{d\xi }{d\omega }}$$

This is the core part of time scale identity.

Step 6: With Phase Derivative

Define half-phase $\phi (\omega )=\Phi (\omega )/2=-\pi \xi (\omega )$:

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

Complete Identity

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

QED: The three are unified.

graph TB
    Q["Group Delay<br/>tr Q = ∂_ωΦ"] --> STEP1["Step 1"]
    BK["Birman-Kreĭn<br/>Φ = -2πξ"] --> STEP2["Step 2"]
    RHO["Density of States<br/>ρ_rel = -∂_ωξ"] --> STEP3["Step 3"]

    STEP1 --> MERGE["Combine<br/>tr Q = 2πρ_rel"]
    STEP2 --> MERGE
    STEP3 --> MERGE

    MERGE --> NORM["Normalize<br/>κ = tr Q / 2π"]
    NORM --> ID["Time Scale Identity<br/>φ'/π = ρ_rel = tr Q/2π"]

    style Q fill:#e1f5ff
    style BK fill:#ffe1e1
    style RHO fill:#e1ffe1
    style ID fill:#fff4e1,stroke:#ff6b6b,stroke-width:4px

📝 Derivation Chain Summary

🎓 Historical Notes

Kreĭn’s Contribution (1953)

M.G. Kreĭn first defined spectral shift function for trace formula:

$$\text{tr}[f(H)-f(H_0)]=\int f^{\prime }(\lambda )\xi (\lambda )d\lambda$$

Application: Perturbation theory, renormalization in quantum field theory

Birman’s Contribution (1962)

M.Sh. Birman proved relationship between scattering matrix and spectral shift:

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

Meaning: First connection between scattering (observable) and spectrum (mathematical).

Modern Developments

2000s: Generalizations to:

• Electromagnetic scattering (Strohmaier & Waters, 2021)
• Non-Hermitian systems
• Topological matter

GLS theory: Uses Birman-Kreĭn to unify time scale.

🤔 Exercises

1. Conceptual understanding:

   • What is the physical meaning of spectral shift function?
   • Why $\xi (\infty )-\xi (-\infty )=N_b$ (number of bound states)?
   • Why is Birman-Kreĭn formula important?

2. Calculation exercises:

   • For $S(k)=e^{2i\delta (k)}$, prove $\xi (E)=-\delta (k)/\pi$
   • Resonance $\delta =\arctan [\Gamma /(2(E-E_r))]$, calculate ${\rho }_{\text{rel}}(E)$
   • Verify ${\int }_{-\infty }^{\infty }{\rho }_{\text{rel}}(E)dE=1$ (single resonance)

3. Physical applications:

   • How to extract spectral shift function from scattering data?
   • How does Levinson theorem determine number of bound states?
   • In multi-channel scattering, how is $\xi$ defined?

4. Advanced thinking:

   • What is topological interpretation of Birman-Kreĭn formula?
   • How to generalize $\xi$ under non-trace-class perturbations?
   • What constraints does time-reversal symmetry impose on $\xi$?

Next step: We have understood phase-time (Article 1), group delay (Article 2), spectral shift (Article 3). Next article will completely prove time scale identity and reveal its profound meaning.

Navigation:

• Previous: 02-scattering-phase_en.md - Scattering Phase and Group Delay
• Next: 04-time-scale-identity_en.md - Time Scale Identity (⭐ Core)
• Overview: 00-time-overview_en.md - Unified Time Chapter Overview
• GLS theory: unified-time-scale-geometry.md