[Paper Review] Schrödinger operators with complex singular potentials

Vladimir Mikhailets, Volodymyr Molyboga|arXiv (Cornell University)|Jun 3, 2013

Spectral Theory in Mathematical Physics7 references3 citations

TL;DR

This paper establishes the equivalence and self-adjointness of multiple definitions of one-dimensional Schrödinger operators with complex-valued potentials in the space $ H_{\text{unif}}^{-1}(\mathbb{R}) $, proves their norm resolvent approximation by smooth potentials, and localizes their spectrum within a parabolic region in the complex plane, resolving a gap in prior work on distributional potentials.

ABSTRACT

We study one-dimensional Schrödinger operators $\mathrm{S}(q)$ on the space $L^{2}(\mathbb{R})$ with potentials $q$ being complex-valued generalized functions from the negative space $H_{unif}^{-1}(\mathbb{R})$. Particularly the class $H_{unif}^{-1}(\mathbb{R})$ contains periodic and almost periodic $H_{loc}^{-1}(\mathbb{R})$-functions. We establish an equivalence of the various definitions of the operators $\mathrm{S}(q)$, investigate their approximation by operators with smooth potentials from the space $L_{unif}^{1}(\mathbb{R})$ and prove that the spectrum of each operator $\mathrm{S}(q)$ lies within a certain parabola.

Motivation & Objective

• To resolve the incomplete proof in [10] regarding the self-adjointness of Schrödinger operators with real-valued $ H_{\text{unif}}^{-1}(\mathbb{R}) $ potentials.
• To extend the theory to complex-valued potentials in $ H_{\text{unif}}^{-1}(\mathbb{R}) $, where previous definitions of operators (minimal, Friedrichs, form-sum, and maximal) were not fully reconciled.
• To establish that the operators $ \mathrm{S}_0(q) $, $ \mathrm{S}(q) $, $ \mathrm{S}_{\text{fs}}(q) $, and $ \mathrm{S}_F(q) $ coincide and are $ m $-sectorial for complex $ q \in H_{\text{unif}}^{-1}(\mathbb{R}) $.
• To prove norm resolvent convergence of approximating operators with smooth potentials in $ L_{\text{unif}}^1(\mathbb{R}) $ to the limiting operator $ \mathrm{S}(q) $.
• To provide a precise parabolic localization of the spectrum of $ \mathrm{S}(q) $ in the complex plane based on potential norms and regularity parameters.

Proposed method

• Uses the regularization method to define $ \mathrm{S}(q) $ via $ \mathrm{S}(q)y = -(y' - Qy)' - Qy' + \tau y $, where $ q = Q' + \tau $, with $ Q, \tau \in L_{\text{unif}}^2(\mathbb{R}) \times L_{\text{unif}}^1(\mathbb{R}) $.
• Applies the form-sum method and Friedrichs extension to define alternative versions of the operator and proves their equivalence via $ m $-sectoriality and domain coincidence.
• Employs resolvent approximation techniques, showing that $ \|q - q_n\|_{H^{-1}(0,1)} \to 0 $ implies norm resolvent convergence of $ \mathrm{S}(q_n) \to \mathrm{S}(q) $.
• Derives spectral localization using estimates on the numerical range of preminimal operators, leading to sectorial and parabolic bounds on the spectrum.
• Uses variational estimates of the form $ |t_q[u]| \leq a\varepsilon\|u'\|^2 + b\varepsilon^{-s}\|u\|^2 $ to bound the imaginary part of the spectrum in terms of $ a, b, s $.
• Applies the abstract Theorem 13 to derive explicit parabolic regions $ \mathcal{M}_{a,b,s} $ containing the spectrum, depending on potential regularity parameters.

Experimental results

Research questions

• RQ1 Are the minimal, Friedrichs, form-sum, and maximal operators $ \mathrm{S}_0(q) $, $ \mathrm{S}_F(q) $, $ \mathrm{S}_{\text{fs}}(q) $, and $ \mathrm{S}(q) $ equivalent for complex-valued potentials in $ H_{\text{unif}}^{-1}(\mathbb{R}) $?
• RQ2 Can Schrödinger operators with complex singular potentials in $ H_{\text{unif}}^{-1}(\mathbb{R}) $ be approximated in the norm resolvent sense by operators with smooth potentials in $ L_{\text{unif}}^1(\mathbb{R}) $?
• RQ3 What is the precise localization of the spectrum of $ \mathrm{S}(q) $ in the complex plane for such potentials?
• RQ4 How do the spectral bounds depend on the norms of the potential decomposition $ q = Q' + \tau $ with $ Q \in L_{\text{unif}}^2(\mathbb{R}) $, $ \tau \in L_{\text{unif}}^1(\mathbb{R}) $?
• RQ5 What is the lower bound of the real part of the quadratic form $ (\mathrm{S}(q)u, u) $ for real potentials, and how does it depend on the potential's total variation?

Key findings

• The operators $ \mathrm{S}_0(q) $, $ \mathrm{S}(q) $, $ \mathrm{S}_{\text{fs}}(q) $, and $ \mathrm{S}_F(q) $ are $ m $-sectorial and coincide for all $ q \in H_{\text{unif}}^{-1}(\mathbb{R}) $, resolving a foundational gap in the literature.
• The spectrum of $ \mathrm{S}(q) $ lies within a parabolic region $ \mathcal{M}_{a,b,s} \subset \mathbb{C} $, explicitly defined by parameters $ a, b, s $ derived from the potential's decomposition.
• For potentials $ q = Q' $ with $ Q \in \mathrm{BV}_{\text{loc}}(\mathbb{R}) $ and $ |q(I)| \leq K_0 $ on unit intervals, the spectrum is contained in $ \left\{ \lambda \in \mathbb{C} \,\middle|\, |\mathrm{Im}\,\lambda| \leq 16K_0(\mathrm{Re}\,\lambda + (8K_0+1)^2)^{1/2} \right\} $.
• The numerical range of the preminimal operator $ \mathrm{S}_{00}(q) $ is contained in a sector $ \mathcal{S}_{K,\varepsilon} $, with explicit bounds depending on $ K $ and $ \varepsilon $, ensuring sectoriality.
• For real-valued potentials, the quadratic form satisfies $ (\mathrm{S}(q)u, u) \geq -32K^4\|u\|^2 $ when $ K \geq 1/2 $, and $ \geq -4K\|u\|^2 $ when $ K < 1/2 $, where $ K $ bounds the total variation of $ Q $.
• Norm resolvent convergence of $ \mathrm{S}(q_n) \to \mathrm{S}(q) $ holds whenever $ \|q - q_n\|_{H^{-1}(0,1)} \to 0 $, confirming the stability of the operator under approximation.