[Paper Review] Accuracy on eigenvalues for a Schrodinger operator with a degenerate potential in the semi-classical limit

Abderemane Morame, Françoise Truc|ArXiv.org|Jun 9, 2006

Spectral Theory in Mathematical Physics27 references4 citations

TL;DR

This paper establishes sharp asymptotic expansions for low-lying eigenvalues of a semi-classical Schrödinger operator with a degenerate potential $ V(x,y) = f(x)g(y) $, where $ f $ has a non-degenerate minimum and $ g $ is homogeneous of degree $ a > 0 $. By refining the Born-Oppenheimer approximation and using quasimode constructions, it derives precise eigenvalue expansions up to $ \mathcal{O}(\hslash^2) $, including full asymptotic series for eigenvalues and associated quasimodes.

ABSTRACT

We consider a semi-classical Schrodinger operator with a degenerate potential V(x,y) =f(x) g(y) . g is assumed to be a homogeneous positive function of m variables and f is a strictly positive function of n variables, with a strict minimum. We give sharp asymptotic behaviour of low eigenvalues bounded by some power of the parameter h, by improving Born-Oppenheimer approximation.

Motivation & Objective

• To analyze the semi-classical asymptotic behavior of low eigenvalues for a Schrödinger operator with a degenerate potential $ V(x,y) = f(x)g(y) $, where $ f $ has a non-degenerate minimum and $ g $ is homogeneous.
• To improve the Born-Oppenheimer approximation by deriving sharp eigenvalue expansions for eigenvalues near the bottom of the spectrum.
• To construct quasimodes that capture the correct asymptotic structure of eigenfunctions associated with low-lying eigenvalues.
• To establish a full asymptotic expansion for eigenvalues in terms of the semi-classical parameter $ \hslash = h^{2/(2+a)} $, including higher-order corrections.

Proposed method

• Introduce a rescaling $ \hslash = h^{2/(2+a)} $ and transform the operator to $ \widehat{H}^{\hslash} = \hslash^2 D_x^2 + D_y^2 + f(x)g(y) $, exploiting the homogeneity of $ g $.
• Decompose the problem into a fibered family of operators $ Q_x(y,D_y) = D_y^2 + f(x)g(y) $, whose lowest eigenvalue is $ \lambda_1(x) = \mu_1 f^{2/(2+a)}(x) $, with $ \mu_1 $ the ground state energy of $ D_y^2 + g(y) $.
• Apply the Born-Oppenheimer approximation by treating $ \lambda_1(x) $ as an effective potential for the $ x $-degrees of freedom, leading to an effective Schrödinger operator $ \hslash^2 D_x^2 + \mu_1 f^{2/(2+a)}(x) $.
• Construct quasimodes of the form $ \phi_k^\hslash \sim \hslash^{-m_k} e^{-\psi(x)/\hslash} \sum_{j \geq 0} \hslash^{j/2} a_{kj}(x,y) $, where $ \psi(x) $ is the Agmon distance associated with the effective potential.
• Use a change of variables $ y \to f^{1/(2+a)}(x) y $ to decouple the $ x $ and $ y $ variables near the minimum of $ f $, enabling a local analysis in $ x $-space.
• Perform a detailed spectral analysis in local coordinates near $ x=0 $, using a Dirichlet cut-off and constructing approximate eigenfunctions for the reduced operator $ H^{h_j}_0 $, leading to full asymptotic expansions.

Experimental results

Research questions

• RQ1 How do the low-lying eigenvalues of a Schrödinger operator with a degenerate potential $ V(x,y) = f(x)g(y) $ behave in the semi-classical limit?
• RQ2 Can the Born-Oppenheimer approximation be improved to yield sharp eigenvalue expansions beyond $ \mathcal{O}(\hslash^2) $ for such systems?
• RQ3 What is the precise structure of the associated eigenfunctions, and can quasimodes be constructed that capture the correct asymptotic behavior?
• RQ4 How do the eigenvalue corrections depend on the homogeneity degree $ a $ of the potential $ g(y) $, and what is the role of the Hessian of $ f $ at its minimum?
• RQ5 Can a full asymptotic expansion for eigenvalues be derived, including higher-order terms in $ \hslash $, and what is the form of the associated quasimodes?

Key findings

• The eigenvalues $ E_k(\hslash) $ of $ \widehat{H}^{\hslash} $ admit an asymptotic expansion $ E_k(\hslash) \sim \mu_1 + \hslash e_k + \sum_{j \geq 1} \alpha_{kj} \hslash^{j/2} $, where $ e_k $ are eigenvalues of the effective operator $ D_x^2 + \frac{\mu_1}{2+a} \langle \partial^2 f(0) x, x \rangle $.
• For the ground state ($ k=1 $), the error in the eigenvalue expansion is improved to $ \mathcal{O}(\hslash^2) $, rather than $ \mathcal{O}(\hslash^{3/2}) $, due to non-degeneracy and symmetry.
• Quasimodes $ \phi_k^\hslash $ exist with the form $ \hslash^{-m_k} e^{-\psi(x)/\hslash} \sum_{j \geq 0} \hslash^{j/2} a_{kj}(x,y) $, satisfying $ \| (\widehat{H}^{\hslash} - E_k(\hslash)) \phi_k^\hslash \| \leq C_J \hslash^{(J+1)/2} $, confirming the asymptotic expansion.
• The eigenvalue $ \lambda_k(\widehat{H}^{\hslash}) $ satisfies $ \lambda_k(\widehat{H}^{\hslash}) = \mu_1 + \hslash \lambda_k( D_x^2 + \frac{\mu_1}{2+a} \langle \partial^2 f(0) x, x \rangle ) + \mathcal{O}(\hslash^{3/2}) $, with $ \mathcal{O}(\hslash^2) $ for $ k=1 $.
• A full asymptotic expansion $ \lambda^{h}_{j\alpha} \sim h^{2m/(m+1)} \sum_{k=0}^{\infty} c_{j\ell k\alpha} h^{k/(m+1)} $ is established for eigenvalues in the local model, under bounded $ j $.
• The associated quasimodes admit a refined form $ u^{h}_{j\ell\alpha} \sim c(h) e^{-\psi(x)/h^{1/(m+1)}} \chi_0(t/\epsilon_0) \sum_{k=0}^{\infty} h^{k/(2m+2)} a_{j\ell k\alpha}(x) \phi_{jk}(t/h^{1/(m+1)}) $, capturing the correct scaling and structure.