[Paper Review] Semiclassical Spectral Invariants for Schrödinger Operators

Victor Guillemin, Zuoqin Wang|ArXiv.org|May 6, 2009

Spectral Theory in Mathematical Physics4 references4 citations

TL;DR

This paper derives an asymptotic expansion for the semiclassical spectral measure of Schrödinger operators on $\mathbb{R}^n$, showing that the first few terms in the expansion—specifically the principal and subprincipal terms—determine the potential $V$ and magnetic field $B$ in one and two dimensions. In particular, for radially symmetric $V$ and $B$ in $\mathbb{R}^2$, both are spectrally determined up to $O(\hbar^\infty)$, and in one dimension, a direct spectral identity links the spectral measure to the Birkhoff canonical form of the operator.

ABSTRACT

In this article we study the semiclassical spectral measures associated with Schrödinger operators on $R^n$. In particular we compute the first few coefficients of the asymptotic expansions of these measures and, as an application, give an alternative proof of Colin de Verdiere's result on recovering one dimensional potential wells from semiclassical spectral data. We also study the relation of semiclassical spectra measures to Birkhoff normal forms and describe a generalization of the asymptotic expansion above to the Schrödinger operator in the presence of a magnetic field.

Motivation & Objective

• To derive an asymptotic expansion for the semiclassical spectral measure of Schrödinger operators on $\mathbb{R}^n$.
• To establish inverse spectral results in one and two dimensions by analyzing the first few terms in the spectral measure expansion.
• To show that for radially symmetric potentials and magnetic fields in $\mathbb{R}^2$, both $V$ and $B$ are spectrally determined.
• To prove a quantum Birkhoff canonical form theorem in one dimension, linking the spectral measure directly to the Birkhoff normal form of the operator.
• To extend the spectral invariants to Schrödinger operators with magnetic fields, deriving modified asymptotic formulas for the spectral measure.

Proposed method

• Derive an asymptotic expansion of the spectral measure $\mu_\hbar$ as $\hbar \to 0$, expressed as $\mu_\hbar \sim \sum \hbar^{2k} \left(\frac{d}{dt}\right)^{2k} \mu_k$.
• Use a general semiclassical spectral theorem for elliptic operators to obtain an initial expansion with $\left(\frac{d}{dt}\right)^{4k}$, then apply integration by parts to reduce it to the $\left(\frac{d}{dt}\right)^{2k}$ form.
• Compute explicit formulas for the first two terms in the expansion: $\nu_0(f) = \int f(\frac{\xi^2}{2} + V(x)) \, dx \, d\xi$ and $\nu_1(f) = -\frac{1}{24} \int f^{(2)}(\frac{\xi^2}{2} + V(x)) \sum_i \frac{\partial^2 V}{\partial x_i^2} \, dx \, d\xi$.
• Generalize the method to include magnetic fields, deriving a modified subprincipal term involving $\|B\|^2$ and $\sum \partial^2 V / \partial x_k^2$.
• Establish a quantum Birkhoff canonical form: in 1D, $S_\hbar$ is unitarily equivalent to $H_{QB}(S_\hbar^{\text{har}}, \hbar^2) + O(\hbar^\infty)$, where $S_\hbar^{\text{har}}$ is the harmonic oscillator.
• Prove that the spectral measure $\mu_\hbar(f)$ is given by $\int_0^a f(t) \frac{dK}{dt}(t, \hbar^2) \, dt$, where $K(t, \hbar^2)$ is the inverse of the Birkhoff normal form, establishing a spectral-geometric duality.

Experimental results

Research questions

• RQ1 Can the potential $V$ and magnetic field $B$ be reconstructed from the asymptotic spectral measure of the Schrödinger operator in $\mathbb{R}^2$?
• RQ2 To what extent does the spectral measure determine the potential in one dimension, especially under symmetry assumptions?
• RQ3 Is there a direct spectral identity linking the spectral measure to the Birkhoff canonical form of the Schrödinger operator in 1D?
• RQ4 How do magnetic fields modify the asymptotic spectral invariants compared to the magnetic-free case?
• RQ5 Are there isospectral potentials that are not unitarily equivalent, and if so, what conditions prevent spectral determination?

Key findings

• In $\mathbb{R}^2$, for radially symmetric $V$ and $B$, both are spectrally determined: $V$ and $B$ can be reconstructed from the $O(\hbar^2)$ and $O(\hbar^4)$ terms of the spectral measure expansion.
• In one dimension, the spectral measure determines the Birkhoff canonical form of the operator, and vice versa, via the relation $H_{QB}(s, \hbar^2) = t \Longleftrightarrow s = K(t, \hbar^2)$.
• The $O(\hbar^6)$ term in the spectral measure expansion yields a spectrally determined integral: $\int_{\frac{\xi^2}{2} + V(x) \leq \lambda} \left( \frac{\xi^8}{490}(V^{\prime\prime\prime}(x))^2 - \frac{\xi^6}{63}(V^{\prime\prime}(x))^3 - \frac{\xi^4}{12}(V^\prime(x)V^{\prime\prime}(x))^2 - \frac{11}{144}(V^\prime(x))^6 \right) \, dx \, d\xi$.
• The $O(\hbar^4)$ term yields a spectrally determined integral: $\int \left( \frac{1}{480}(V^{\prime\prime}(x))^2 f^{(4)} + \frac{7}{3456}(V^\prime(x))^4 f^{(6)} \right) \, dx \, d\xi$, which simplifies to $\frac{1}{1152} \int \left(7V^\prime V^{\prime\prime\prime} + \frac{47}{5}(V^{\prime\prime})^2 \right) f^{(4)} \, dx \, d\xi$.
• In the absence of asymmetry assumptions, uncountably many distinct single-well potentials can be isospectral modulo $O(\hbar^\infty)$, showing that spectral determination fails without such conditions.
• For magnetic Schrödinger operators, the subprincipal term is modified to $\frac{1}{48} \int f^{(2)}(\frac{1}{2}\sum (\xi_i + a_i)^2 + V(x)) \left( -2\sum \frac{\partial^2 V}{\partial x_k^2} + \|B\|^2 \right) \, dx \, d\xi$, which allows spectral determination of $V$ and $\|B\|$ under radial symmetry.