[Paper Review] Isospectral operators

Mu-Fa Chen, Xu Zhang|arXiv (Cornell University)|Nov 22, 2014

Spectral Theory in Mathematical Physics4 citations

TL;DR

This paper introduces a systematic method to construct isospectral operators for a broad class of second-order differential and integral operators using h-transform duality. By leveraging harmonic functions and measure transformations, it establishes explicit isospectrality relations that extend spectral knowledge from simple operators (e.g., Ornstein-Uhlenbeck) to complex ones, solving an open problem on the positivity of the principal eigenvalue for birth–death processes.

ABSTRACT

For a large class of integral operators or second order differential operators, their isospectral (or cospectral) operators are constructed explicitly in terms of $h$-transform (duality). This provides us a simple way to extend the known knowledge on the spectrum (or the estimation of the principal eigenvalue) from a smaller class of operators to a much larger one. In particular, an open problem about the positivity of the principal eigenvalue for birth--death processes is solved in the paper.

Motivation & Objective

• To develop a general framework for constructing isospectral operators beyond known classes.
• To solve the open problem concerning the positivity of the principal eigenvalue for birth–death processes.
• To extend spectral information from a simple operator (e.g., Ornstein–Uhlenbeck) to a large class of more complex operators via isospectrality.
• To formalize the duality between operators through h-transforms and measure changes.
• To provide explicit formulas for isospectral operators in terms of the original operator and a harmonic function.

Proposed method

• Use of h-transform duality: define a new operator $ \widetilde{L} $ on a weighted measure $ \tilde{\mu} = h^2 \mu $, where $ h $ is $ L $-harmonic.
• Establish isospectrality via the isometry $ \tilde{f} = f/h $, ensuring $ (Lf, f)_\mu = (\widetilde{L}\tilde{f}, \tilde{f})_{\tilde{\mu}} $.
• Derive explicit formulas for the coefficients of $ L $ in terms of $ \widetilde{L} $, $ h $, and their derivatives: $ L = \widetilde{L} - \frac{2}{h}\langle \tilde{a}\nabla h, \nabla \rangle + \left[ \frac{2}{h^2}\langle \tilde{a}\nabla h, \nabla h \rangle - \frac{1}{h}\widetilde{L}h \right] $.
• Apply the method to construct a large class of operators isospectral to the Ornstein–Uhlenbeck operator, using $ h = \exp(\psi) $ with $ \psi' = b - x $.
• Use the Riccati equation approach in one dimension: solve $ \bar{a}\phi' + \bar{a}\phi^2 + \bar{b}\phi + \bar{c} = 0 $ to recover the dual operator.
• Extend results to higher dimensions, e.g., dual of $ L = \frac{1}{2}\sum_i(\partial_{ii}^2 + 1 - x_i^2) $ is $ \widetilde{L} = \frac{1}{2}\sum_i(\partial_{ii}^2 - 2x_i\partial_i) $ with $ h(x) = \exp(-|x|^2/2) $.

Experimental results

Research questions

• RQ1 Can isospectral operators be systematically constructed for a broad class of second-order differential operators using a unified transformation?
• RQ2 Does the h-transform duality method preserve spectral properties, particularly the principal eigenvalue, across different operator classes?
• RQ3 Can this method resolve the open problem on the positivity of the principal eigenvalue for birth–death processes?
• RQ4 How can spectral information from a simple operator (e.g., Ornstein–Uhlenbeck) be extended to more complex operators via isospectrality?
• RQ5 What is the explicit form of the dual operator $ L $ given $ \widetilde{L} $ and a harmonic function $ h $?

Key findings

• The Ornstein–Uhlenbeck operator $ \widetilde{L} = \frac{1}{2}\frac{d^2}{dx^2} - x\frac{d}{dx} $ has $ L^2 $-eigenvalues $ \lambda_n = -n $ with eigenfunctions $ g_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n}(e^{-x^2}) $.
• A large class of operators $ L^b = \frac{1}{2}\frac{d^2}{dx^2} - b(x)\frac{d}{dx} + \frac{1}{2}[b(x)^2 - b'(x) - x^2 + 1] $ are isospectral to the Ornstein–Uhlenbeck operator, sharing the same eigenvalues $ \lambda_n = -n $.
• For any $ h \in C^2(\mathbb{R}) $, $ h \neq 0 $ a.e., the operator $ L^h $ defined with $ \frac{h'}{h} $ terms has $ L^2 $-eigenvalues $ \lambda_n(L^h) = -n $ and eigenfunctions $ g_n(x) = (-1)^n h(x) e^{x^2} \frac{d^n}{dx^n}(e^{-x^2}) $.
• The dual operator $ L $ of $ \widetilde{L} $ is given explicitly by $ L = \widetilde{L} - 2\langle \tilde{a}\nabla\log h, \nabla \rangle + \left[ 2\langle \tilde{a}\nabla\log h, \nabla\log h \rangle - h^{-1}\widetilde{L}h \right] $, valid when $ h $ is $ L $-harmonic.
• In higher dimensions, the dual of $ L = \frac{1}{2}\sum_i(\partial_{ii}^2 + 1 - x_i^2) $ is $ \widetilde{L} = \frac{1}{2}\sum_i(\partial_{ii}^2 - 2x_i\partial_i) $, with $ h(x) = \exp(-|x|^2/2) $, and $ L $ has eigenvalue $ n $ with multiplicity $ \#\{(k_1,\dots,k_d) : \sum k_i = n\} $.
• The method resolves the open problem: the principal eigenvalue of birth–death processes is positive, as shown via isospectrality to known positive-eigenvalue operators.