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Renormalized oscillation theory for singular linear Hamiltonian pencils

Peter Howard, Alim Sukhtayev

TL;DR

Renormalized oscillation theory via the Maslov index is developed to analyze spectra of linear Hamiltonian pencils $J y' = \mathbb{B}(x;\lambda) y$ with singular endpoints and nonlinear dependence on $\lambda$. The core contributions are two theorems (regular-singular and singular endpoints) that bound and sometimes exactly count eigenvalues on $[\lambda_1,\lambda_2)$ in terms of Maslov indices between evolving frames $\ell_\alpha$, $\ell_a$, and $\ell_b$, under Assumptions (A)–(F). The method builds self-adjoint pencils $\mathcal{T}(\lambda)$ and $\mathcal{T}^{\alpha}(\lambda)$, boundary data via Niessen-type constructions, and a Green's-identity-based framework, then carries out Maslov-index calculations along a Maslov box. It then verifies the framework in three canonical cases (linear-in-$\lambda$, quadratic Schrödinger systems, degenerate Sturm–Liouville) and demonstrates application to a Quadratic Schrödinger Equation, illustrating how boundary conditions and spectral parameter dependence shape the spectrum. The results provide a geometric, spectrally robust tool for eigenvalue counting in broad singular settings with nonlinear spectral dependence.

Abstract

For many applications, critical information about system dynamics is encoded in associated eigenvalue problems that can be posed as linear Hamiltonian systems with suitable boundary conditions. Motivated by examples from hydrodynamics, quantum mechanics, and magnetohydrodynamics (MHD), we develop a general framework for analyzing a broad class of linear Hamiltonian systems with at least one singular boundary condition and possible nonlinear dependence on the spectral parameter. We show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated with appropriately chosen paths of Lagrangian subspaces of $\mathbb{C}^{2n}$. This extends previous work by the authors for regular linear Hamiltonian systems that depend nonlinearly on the spectral parameter and singular linear Hamiltonian systems that depend linearly on the spectral parameter. We conclude the study by using our framework to study the spectrum in the setting of each of our motivating examples.

Renormalized oscillation theory for singular linear Hamiltonian pencils

TL;DR

Renormalized oscillation theory via the Maslov index is developed to analyze spectra of linear Hamiltonian pencils with singular endpoints and nonlinear dependence on . The core contributions are two theorems (regular-singular and singular endpoints) that bound and sometimes exactly count eigenvalues on in terms of Maslov indices between evolving frames , , and , under Assumptions (A)–(F). The method builds self-adjoint pencils and , boundary data via Niessen-type constructions, and a Green's-identity-based framework, then carries out Maslov-index calculations along a Maslov box. It then verifies the framework in three canonical cases (linear-in-, quadratic Schrödinger systems, degenerate Sturm–Liouville) and demonstrates application to a Quadratic Schrödinger Equation, illustrating how boundary conditions and spectral parameter dependence shape the spectrum. The results provide a geometric, spectrally robust tool for eigenvalue counting in broad singular settings with nonlinear spectral dependence.

Abstract

For many applications, critical information about system dynamics is encoded in associated eigenvalue problems that can be posed as linear Hamiltonian systems with suitable boundary conditions. Motivated by examples from hydrodynamics, quantum mechanics, and magnetohydrodynamics (MHD), we develop a general framework for analyzing a broad class of linear Hamiltonian systems with at least one singular boundary condition and possible nonlinear dependence on the spectral parameter. We show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated with appropriately chosen paths of Lagrangian subspaces of . This extends previous work by the authors for regular linear Hamiltonian systems that depend nonlinearly on the spectral parameter and singular linear Hamiltonian systems that depend linearly on the spectral parameter. We conclude the study by using our framework to study the spectrum in the setting of each of our motivating examples.
Paper Structure (21 sections, 28 theorems, 700 equations, 7 figures)

This paper contains 21 sections, 28 theorems, 700 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Statements of the Main Results
  3. The Self-Adjoint Operator Pencils $\mathcal{T} (\cdot)$ and $\mathcal{T}^{\alpha} (\cdot)$
  4. Continuation to $\mathbb{R}$
  5. The Maslov Index
  6. Proofs of the Main Theorems
  7. Proof of Theorem \ref{['regular-singular-theorem']}
  8. Proof of Theorem \ref{['singular-theorem']}
  9. Applications
  10. Verification of Assumptions (A) through (F)
  11. The Linear Case
  12. Quadratic Schrödinger Systems
  13. Degenerate Sturm-Liouville Systems
  14. Specific Applications
  15. Quadratic Schrödinger Equation
  16. ...and 6 more sections

Key Result

Lemma 2.7

(i) Let Assumptions (A) through (D) hold, and let $(\mu_0, \lambda_0) \in (\mathbb{C} \backslash \mathbb{R}) \times I$ be fixed. Then there exists a selection of $n$ solutions $\{u^a_j (x; \mu_0, \lambda_0)\}_{j=1}^n$ to (linear-hammy) (with $(\mu, \lambda) = (\mu_0, \lambda_0)$) that lie left in is a self-adjoint operator. We will denote this operator $\mathcal{T} (\lambda)$. (ii) In addition

Figures (7)

  • Figure 6.1: The Maslov Box.
  • Figure 6.2: Triangular path in the $(\lambda, \mu)$-plane for Claim \ref{['triangle-claim-alpha']}.
  • Figure 7.1: The Full Maslov Box for $H$ on $[-3, -7/12] \times [0, 10]$ with $\delta = 0$.
  • Figure 7.2: The Full Maslov Box for $H$ on $[-3, -7/12] \times [0, 10]$ with $\delta = 1$.
  • Figure 7.3: The Full Maslov Box for $H$ on $[-3, -7/12] \times [0, 10]$ with $\delta = 5$.
  • ...and 2 more figures

Theorems & Definitions (66)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Lemma 2.7
  • Remark 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 56 more