[Paper Review] Correlation inequalities for Schrödinger operators

Tadahiro Miyao|arXiv (Cornell University)|Aug 2, 2016

Spectral Theory in Mathematical Physics28 references3 citations

TL;DR

This paper establishes operator-theoretic correlation inequalities for ground states of Schrödinger operators using self-dual cones and positivity structures, extending Griffiths inequalities to quantum many-body systems. By leveraging strong resolvent convergence and ergodicity conditions, it proves that ground state expectations satisfy non-negativity and positive correlation inequalities, providing universal qualitative insights into entanglement and correlation in quantum systems.

ABSTRACT

This paper analyzes Schödinger operators from viewpoint of correlation inequalities. We construct Griffiths inequalities for the ground state expectations by applying operator-theoretic correlation inequalities. As an example of such an application, we analyze the momentum distribution, i.e., the Fourier transform of the ground state density.

Motivation & Objective

• To analyze the correlation structure of ground states in Schrödinger operators using operator-theoretic methods.
• To extend Griffiths inequalities—previously known in classical Ising models—to quantum many-body systems described by Schrödinger operators.
• To unify the framework of reflection positivity, spin reflection positivity, and correlation inequalities via self-dual cones in Hilbert space.
• To provide a general, model-independent method for studying correlations and entanglement in quantum systems through positivity and operator inequalities.
• To establish conditions under which ground state expectations preserve non-negativity and positive correlation, even in the presence of complex potentials.

Proposed method

• Uses self-dual cones in a Hilbert space to define positivity structures for operators, enabling the formulation of operator-theoretic correlation inequalities.
• Applies the Duhamel expansion to express the resolvent of the Schrödinger operator as a series of positive operators under suitable convergence conditions.
• Employs strong resolvent convergence of approximating operators $H_n = -\Delta - V_n$ to the full Hamiltonian $H = -\Delta - V$ to ensure spectral stability.
• Relies on the ergodicity of perturbation operators $B$ with respect to the positive cone $\mathfrak{P}$ to guarantee positivity of the full evolution semigroup $e^{-\beta H}$.
• Applies the theory of positive semigroups and cone positivity to prove that $e^{-\beta H} \rhd 0$ (strictly positive) for all $\beta > 0$ under appropriate conditions.
• Uses the fact that if $A \unrhd 0$ and $u > 0$ w.r.t. $\mathfrak{P}$, then $\langle u|Au\rangle = 0$ implies $A = 0$, to establish uniqueness and irreducibility of ground states.

Experimental results

Research questions

• RQ1 Can Griffiths-type correlation inequalities—originally for classical Ising models—be extended to quantum many-body systems described by Schrödinger operators?
• RQ2 Under what conditions on the potential $V$ and the Hamiltonian $H = -\Delta - V$ do ground state expectations satisfy non-negativity and positive correlation?
• RQ3 How can operator-theoretic methods, particularly those involving self-dual cones and positivity structures, be used to unify reflection positivity and correlation inequalities?
• RQ4 What role does the ergodicity of the perturbation $B$ play in ensuring the strict positivity of the semigroup $e^{-\beta H}$ and thus the positivity of ground state matrix elements?
• RQ5 Can the method be generalized to many-body Schrödinger operators beyond single-particle systems?

Key findings

• The paper proves that for Schrödinger operators satisfying assumptions (A), (B), and (C), the ground state expectation $\langle \sigma_A \rangle$ is non-negative, generalizing the first Griffiths inequality.
• It establishes that the correlation $\langle \sigma_A \sigma_B \rangle - \langle \sigma_A \rangle \langle \sigma_B \rangle \geq 0$ holds for all subsets $A, B \subseteq \Lambda$, extending the second Griffiths inequality to quantum systems.
• The positivity of the semigroup $e^{-\beta H}$ is shown to be preserved under the condition that $e^{-\beta A} \unrhd 0$ and $B$ is ergodic w.r.t. the self-dual cone $\mathfrak{P}$, ensuring the existence of a unique positive ground state.
• The proof relies on the Duhamel expansion and the continuity of matrix elements in the time variables $s_1, \dots, s_n$, ensuring that the integral of a non-negative continuous function remains positive.
• It is shown that if $A \unrhd 0$ and $u > 0$ w.r.t. $\mathfrak{P}$, then $\langle u|Au\rangle = 0$ implies $A = 0$, which is crucial for proving the uniqueness and strict positivity of the ground state.
• The method applies to many-body Schrödinger operators, as the framework is general enough to handle systems with complex interactions and singular potentials under the given integrability conditions.