[Paper Review] Semiclassical analysis of low and zero energy scattering for one dimensional Schrödinger operators with inverse square potentials

Ovidiu Costin, Wilhelm Schlag+2 more|ArXiv.org|Apr 14, 2008

Spectral Theory in Mathematical Physics21 references3 citations

TL;DR

This paper provides a semiclassical analysis of low and zero-energy scattering for one-dimensional Schrödinger operators with inverse-square potentials at infinity. It establishes that the scattering matrix entries admit a WKB-type approximation with uniform $ ext{O}(ar{ ext{h}}) $ corrections, and derives precise asymptotic control of the error terms as energy $ E \to 0^+ $, under the assumption of no zero-energy resonance due to positivity of the potential.

ABSTRACT

This paper studies the scattering matrix $Σ(E;\hbar)$ of the problem \[ -\hbar^2 ψ''(x) + V(x) ψ(x) = Eψ(x) \] for positive potentials $V\in C^\infty(\R)$ with inverse square behavior as $x o\pm\infty$. It is shown that each entry takes the form $Σ_{ij}(E;\hbar)=Σ_{ij}^{(0)}(E;\hbar)(1+\hbar σ_{ij}(E;\hbar))$ where $Σ_{ij}^{(0)}(E;\hbar)$ is the WKB approximation relative to the {\em modified potential} $V(x)+\frac{\hbar^2}{4} \la x a^{-2}$ and the correction terms $σ_{ij}$ satisfy $|\partial_E^k σ_{ij}(E;\hbar)| \le C_k E^{-k}$ for all $k\ge0$ and uniformly in $(E,\hbar)\in (0,E_0) imes (0,\hbar_0)$ where $E_0,\hbar_0$ are small constants. This asymptotic behavior is not universal: if $-\hbar^2\partial_x^2 + V$ has a {\em zero energy resonance}, then $Σ(E;\hbar)$ exhibits different asymptotic behavior as $E o0$. The resonant case is excluded here due to $V>0$.

Motivation & Objective

• To analyze the structure of the scattering matrix $ \mathbb{S}(E;\hbar) $ for one-dimensional Schrödinger operators with smooth, positive potentials decaying as $ |x|^{-2} $ at infinity.
• To derive uniform asymptotic expansions of the scattering matrix entries in the limit $ E \to 0^+ $, with $ \hbar $ small, under the condition that there is no zero-energy resonance.
• To quantify the error in the WKB approximation of the scattering matrix, showing that corrections are uniformly bounded by $ \hbar \sigma_{ij}(E;\hbar) $ with controlled $ E $-dependence.
• To establish that the asymptotic behavior is not universal, and that resonant cases (excluded here) would yield different low-energy limits.
• To provide a rigorous foundation for the 'large angular momentum' problem in scattering theory, as part of a broader program in mathematical physics.

Proposed method

• Transform the original Schrödinger equation using a change of variables that maps the turning point to a fixed location and rescales energy, reducing the problem to a modified potential with $ \hbar^2 \langle x \rangle^{-2} $ correction.
• Introduce a new independent variable $ \xi = \xi(y,E) $ such that the potential becomes asymptotically close to a Bessel-type operator, enabling the use of special functions.
• Use the Volterra integral formulation to express solutions of the perturbed equation in terms of the homogeneous Bessel solutions (Hankel or Bessel functions of order $ n = \hbar^{-1} $).
• Apply asymptotic expansions of Bessel functions $ J_n(n\xi) $, $ Y_n(n\xi) $ for large $ n $, valid uniformly on intervals $ \xi > \xi_0 > 0 $, to control the behavior of the Green's function.
• Derive a perturbative equation for the scattering matrix entries via the Volterra integral equation involving the resolvent kernel and a remainder potential $ W_0 $, whose derivatives satisfy uniform decay estimates.
• Establish uniform bounds on the derivatives of the correction terms $ \sigma_{ij}(E;\hbar) $, showing $ |\partial_E^k \sigma_{ij}| \leq C_k E^{-k} $ for all $ k \geq 0 $, uniformly in $ (E,\hbar) \in (0,E_0) \times (0,\hbar_0) $.

Experimental results

Research questions

• RQ1 How does the scattering matrix $ \mathbb{S}(E;\hbar) $ behave as energy $ E \to 0^+ $ for one-dimensional Schrödinger operators with inverse-square decaying potentials?
• RQ2 Can the WKB approximation for the scattering matrix be systematically improved with an $ \hbar $-dependent correction term, and what is the uniformity of this correction in $ E $ and $ \hbar $?
• RQ3 What is the precise asymptotic structure of the scattering matrix entries in the low-energy, small-$ \hbar $ regime, and how does it differ from the universal case?
• RQ4 How do the derivatives of the correction terms $ \sigma_{ij}(E;\hbar) $ behave as $ E \to 0^+ $, and can they be uniformly bounded in terms of inverse powers of $ E $?
• RQ5 Why is the absence of a zero-energy resonance critical for the derived asymptotic form, and what changes if such a resonance is present?

Key findings

• The scattering matrix entries admit the decomposition $ \mathbb{S}_{ij}(E;\hbar) = \mathbb{S}_{ij}^{(0)}(E;\hbar)(1 + \hbar \sigma_{ij}(E;\hbar)) $, where $ \mathbb{S}_{ij}^{(0)} $ is the WKB approximation relative to the modified potential $ V(x) + \frac{\hbar^2}{4}\langle x \rangle^{-2} $.
• The correction terms $ \sigma_{ij}(E;\hbar) $ satisfy uniform bounds: $ |\partial_E^k \sigma_{ij}(E;\hbar)| \leq C_k E^{-k} $ for all $ k \geq 0 $, uniformly in $ (E,\hbar) \in (0,E_0) \times (0,\hbar_0) $.
• The asymptotic structure is not universal: if a zero-energy resonance exists, the behavior of $ \mathbb{S}(E;\hbar) $ as $ E \to 0 $ would differ, but this case is excluded here due to $ V > 0 $.
• The analysis relies on transforming the Schrödinger equation into a Bessel-type equation via a change of variables that fixes the turning point and rescales energy, enabling the use of Hankel functions as a fundamental solution basis.
• The remainder potential $ W_0 $ in the perturbed equation satisfies $ |\partial_E^k \partial_\xi^\ell W_0| \leq C_{k,\ell} E^{-k} \xi^{-3-\ell} $, ensuring uniform control over the perturbation.
• The method establishes uniform $ \text{O}(\hbar) $ error bounds in the scattering matrix approximation, with the error terms controlled independently of energy down to zero.