[Paper Review] Spectral shift function for perturbed periodic Schroedinger operators. The large-coupling constant limit case

Mouez Dimassi, Maher Zerzeri|arXiv (Cornell University)|Feb 11, 2011

Spectral Theory in Mathematical Physics14 references3 citations

TL;DR

This paper establishes a complete asymptotic expansion in powers of $\mu^{-1/\delta}$ for the derivative of the spectral shift function associated with perturbed periodic Schrödinger operators $P_\mu = P_0 + \mu W(x)$ in the large coupling constant limit ($\mu \to \infty$). Using semiclassical analysis and trace-class perturbation theory, it proves that $\xi'_{\mu}(\lambda)$ admits a pointwise asymptotic expansion when $W(x) \sim w_0(x/|x|)|x|^{-\delta}$ with $\delta > n$, generalizing prior results for the non-periodic case to the periodic setting.

ABSTRACT

In the large coupling constant limit, we obtain an asymptotic expansion in powers of $μ^{-\frac{1}δ}$ of the derivative of the spectral shift function corresponding to the pair $\big(P_μ=P_0+μW(x),P_0=-Δ+V(x)\big),$ where $W(x)$ is positive, $W(x)\sim w_0(\frac{x}{|x|})|x|^{-δ}$ near infinity for some $δ>n$ and $w_0\in {\mathcal C}^\infty(\mathbb S^{n-1};\,\mathbb R_+).$ Here $\mathbb S^{n-1}$ is the unite sphere of the space $\mathbb R^n$ and $μ$ is a large parameter. The potential $V$ is real-valued, smooth and periodic with respect to a lattice $Γ$ in ${\mathbb R}^n$.

Motivation & Objective

• To extend known asymptotic results for the spectral shift function (SSF) in the large coupling constant limit from non-periodic to periodic Schrödinger operators.
• To establish a complete asymptotic expansion in powers of $\mu^{-1/\delta}$ for $\xi'_{\mu}(\lambda)$, the derivative of the SSF, under decay and smoothness conditions on the perturbation $W(x)$.
• To analyze the spectral shift function for $P_\mu = P_0 + \mu W(x)$, where $P_0 = -\Delta + V(x)$ with $V$ smooth and periodic, and $W(x)$ positive and decaying like $|x|^{-\delta}$ at infinity with $\delta > n$.
• To develop a semiclassical reference operator $Q = H(\mu^{-1/\delta})$ to reduce the problem to a regime where trace-class and resolvent estimates can be applied effectively.

Proposed method

• Construction of a semiclassical reference operator $Q = H(\mu^{-1/\delta})$ to model the large-$\mu$ behavior of $P_\mu$.
• Use of the limiting absorption principle and resolvent identities to control the spectral shift function via trace-class perturbation theory.
• Application of pseudodifferential calculus and weighted estimates with $\langle hx \rangle$ weights to control operator norms and trace-class behavior.
• Decomposition of the trace of resolvent differences into two parts ($I_1, I_2$) and estimation of each using distance conditions between operators and support properties of $\widetilde{W}$.
• Employment of the cyclicity of the trace and dyadic decomposition techniques to bound trace norms uniformly in the spectral parameter $z$.
• Establishment of $\mathcal{O}(h^\infty)$ error bounds for trace terms, leading to a weak and then pointwise asymptotic expansion of $\xi'_{\mu}(\lambda)$.

Experimental results

Research questions

• RQ1 Can a complete asymptotic expansion in powers of $\mu^{-1/\delta}$ be established for the derivative of the spectral shift function $\xi'_{\mu}(\lambda)$ in the large coupling constant limit for perturbed periodic Schrödinger operators?
• RQ2 How does the periodicity of the background potential $V(x)$ affect the asymptotic structure of the spectral shift function compared to the non-periodic case?
• RQ3 What is the role of the decay rate $\delta > n$ of the perturbation $W(x)$ in determining the asymptotic scaling of the SSF derivative?
• RQ4 Can semiclassical methods and trace-class perturbation theory be adapted to derive pointwise asymptotics for $\xi'_{\mu}(\lambda)$ in the large-$\mu$ regime?

Key findings

• The derivative of the spectral shift function $\xi'_{\mu}(\lambda)$ admits a complete asymptotic expansion in powers of $\mu^{-1/\delta}$ as $\mu \to \infty$.
• The leading-order term in the expansion is determined by the angular behavior of $W(x)$ at infinity, encoded in $w_0(\frac{x}{|x|})$.
• The error in the asymptotic expansion is of order $\mathcal{O}(\mu^{-(k+1)/\delta})$ for any $k \geq 0$, uniformly in $\lambda$ on compact intervals.
• The proof relies on constructing a semiclassical reference operator $Q = H(\mu^{-1/\delta})$ and showing that $\xi'_{\mu}(\lambda) = \xi'_h(\lambda) + \mathcal{O}(h^\infty)$ uniformly in $\lambda \in [a,b]$.
• Trace estimates for resolvent differences are controlled via weighted norms and pseudodifferential calculus, ensuring uniform bounds in the spectral parameter.
• The method extends previous results for non-periodic Schrödinger operators to the periodic case, where $V(x)$ is smooth and periodic with respect to a lattice $\Gamma \subset \mathbb{R}^n$.