[Paper Review] Kato's Inequality for Magnetic Relativistic Schrödinger Operators

Fumio Hiroshima, Takashi Ichinose+1 more|arXiv (Cornell University)|Apr 13, 2016

Spectral Theory in Mathematical Physics15 references3 citations

TL;DR

This paper establishes Kato’s inequality for the magnetic relativistic Schrödinger operator $ H_{A,m} = ig( (-i abla - A(x))^2 + m^2 ig)^{1/2} $ with $ L^2_{ ext{loc}} $ vector potential $ A(x) $. Using operator-theoretic methods and modifications of Kato’s original proof, it proves the distributional inequality $ \operatorname{Re}[(\operatorname{sgn} u) H_{A,m}u] \geq H_{0,m}|u| $, which implies a diamagnetic inequality and essential selfadjointness of $ H_{A,m} + V $ for nonnegative $ V \in L^2_{\text{loc}} $.

ABSTRACT

Kato's inequality is shown for the magnetic relativistic Schrödinger operator $H_{A,m}$ defined as the operator theoretical {\it square root} of the selfadjoint, magnetic nonrelativistic Schrödinger operator $(-i abla-A(x))^2+m^2$ with an $L^{2}_{ ext{ m loc}}$ vector potential $A(x)$.

Motivation & Objective

• To establish Kato’s inequality for the magnetic relativistic Schrödinger operator $ H_{A,m} $, defined as the operator-theoretic square root of a selfadjoint nonrelativistic magnetic Hamiltonian.
• To extend the classical Kato inequality to the nonlocal, relativistic setting where $ H_{A,m} $ is not a differential or pseudo-differential operator.
• To prove a diamagnetic inequality for the semigroup $ e^{-t(H_{A,m}-m)} $, showing that the absolute value of matrix elements is dominated by the free case.
• To establish essential selfadjointness of $ H_{A,m} + V $ on $ C_0^rown({\mathbb{R}}^d) $ for nonnegative $ V \in L^2_{\text{loc}}({\mathbb{R}}^d) $, under minimal regularity assumptions on $ A $.
• To overcome the lack of regularity in weak solutions by developing a novel operator-theoretic approach that avoids reliance on pseudo-differential calculus.

Proposed method

• Adapting Kato’s original proof strategy for the nonrelativistic magnetic Schrödinger operator, modified to handle the nonlocal nature of $ H_{A,m} $.
• Using the operator-theoretic definition of $ H_{A,m} $ as the square root of $ (-i\nabla - A(x))^2 + m^2 $, which is selfadjoint and essentially selfadjoint on $ C_0^rown({\mathbb{R}}^d) $.
• Applying a limiting argument involving the semigroup $ e^{-t(H_{A,m}-m)} $, whose kernel is expressed via Bessel functions and integrals involving $ K_{\nu}(z) $, to derive the pointwise inequality.
• Employing the integral representation of the kernel $ k_0^{m,\alpha}(t,x) $ of $ e^{-t[(H_{0,m})^\alpha - m^\alpha]} $ and analyzing its derivative at $ t=0 $ to obtain the singular kernel $ n^{m,\alpha}(x) $.
• Using the K-transform formula from integral tables to evaluate the resulting integral and derive the explicit form of $ n^{m,\alpha}(x) $, which reduces to the known Bessel function kernel when $ \alpha=1 $.
• Establishing the inequality $ \operatorname{Re}[(\operatorname{sgn} u) H_{A,m}u] \geq H_{0,m}|u| $ in the distributional sense for $ u \in L^2({\mathbb{R}}^d) $ with $ H_{A,m}u \in L^1_{\text{loc}}({\mathbb{R}}^d) $, relying on weak solution properties and duality.

Experimental results

Research questions

• RQ1 Can Kato’s inequality be extended to the magnetic relativistic Schrödinger operator $ H_{A,m} $, which is a nonlocal operator defined as a square root?
• RQ2 How can one prove a distributional inequality of the form $ \operatorname{Re}[(\operatorname{sgn} u) H_{A,m}u] \geq H_{0,m}|u| $ without assuming regularity of the solution $ u $?
• RQ3 Does the diamagnetic inequality $ |(f, e^{-t(H_{A,m}-m)}g)| \leq (|f|, e^{-t(H_{0,m}-m)}|g|) $ hold for $ H_{A,m} $ with $ L^2_{\text{loc}} $ vector potentials?
• RQ4 Is the operator $ H_{A,m} + V $ essentially selfadjoint on $ C_0^\infty({\mathbb{R}}^d) $ when $ V \in L^2_{\text{loc}}({\mathbb{R}}^d) $ and $ V \geq 0 $ a.e.?
• RQ5 What is the precise form of the kernel of the semigroup $ e^{-t(H_{A,m}-m)} $, and how does it relate to Bessel functions and integral transforms?

Key findings

• The paper proves the distributional Kato inequality $ \operatorname{Re}[(\operatorname{sgn} u) H_{A,m}u] \geq H_{0,m}|u| $ for all $ u \in L^2({\mathbb{R}}^d) $ with $ H_{A,m}u \in L^1_{\text{loc}}({\mathbb{R}}^d) $, under the assumption $ A \in [L^2_{\text{loc}}({\mathbb{R}}^d)]^d $.
• The diamagnetic inequality $ |(f, e^{-t(H_{A,m}-m)}g)| \leq (|f|, e^{-t(H_{0,m}-m)}|g|) $ holds for all $ f,g \in L^2({\mathbb{R}}^d) $, which follows directly from the Kato inequality.
• The semigroup kernel $ k_0^{m,\alpha}(t,x) $ of $ e^{-t[(H_{0,m})^\alpha - m^\alpha]} $ is explicitly represented via an integral involving Bessel functions $ K_{\nu}(z) $.
• The singular kernel $ n^{m,\alpha}(x) = \lim_{t \downarrow 0} \frac{1}{t} k_0^{m,\alpha}(t,x) $ is derived as $ n^{m,\alpha}(x) = \frac{2^{1+\frac{\alpha}{2}} \sin(\frac{\alpha}{2}\pi) (2\pi)^{\frac{\alpha}{2}} \Gamma(\frac{\alpha}{2}+1)}{\pi} \left(\frac{m}{2\pi}\right)^{\frac{d+\alpha}{2}} \frac{K_{\frac{d+\alpha}{2}}(m|x|)}{|x|^{\frac{d+\alpha}{2}}} $, which reduces to the known form for $ \alpha=1 $.
• The essential selfadjointness of $ H_{A,m} + V $ on $ C_0^\infty({\mathbb{R}}^d) $ is established for $ A \in [L^2_{\text{loc}}({\mathbb{R}}^d)]^d $ and $ V \in L^2_{\text{loc}}({\mathbb{R}}^d) $ with $ V \geq 0 $ a.e., with the selfadjoint extension bounded below by $ m $.
• The proof avoids reliance on pseudo-differential calculus and instead uses operator-theoretic techniques and integral representations, making it robust even when weak solutions lack regularity.