[Paper Review] Perturbation determinants and trace formulas for singular perturbations

M. M. Malamud, Hagen Neidhardt|arXiv (Cornell University)|Dec 31, 2012

Spectral Theory in Mathematical Physics25 references3 citations

TL;DR

This paper extends Krein's theory of perturbation determinants and trace formulas to singular perturbations—specifically, pairs of proper extensions of closed symmetric operators—using the boundary triplet framework. It introduces generalized perturbation determinants via abstract Weyl functions and characteristic functions of almost solvable extensions, deriving trace formulas for self-adjoint, dissipative, and other extension pairs, with new results for dissipative systems and applications to differential operators.

ABSTRACT

We use the boundary triplet approach to extend the classical concept of perturbation determinants to a more general setup. In particular, we examine the concept of perturbation determinants to pairs of proper extensions of closed symmetric operators. For an ordered pair of extensions we express the perturbation determinant in terms of the abstract Weyl function and the corresponding boundary operators. A crucial role in our approach plays so-called almost solvable extensions. We obtain trace formulas for pairs of self-adjoint, dissipative and other pairs of extensions and express the spectral shift function in terms of the abstract Weyl function and the characteristic function of almost solvable extensions. We emphasize that for pairs of dissipative extensions our results are new even for the case of additive perturbations. In this case we improve and complete some classical results of M.G. Krein for pairs of self-adjoint and dissipative operators. We apply the main results to ordinary differential operators and to elliptic operators as well.

Motivation & Objective

• To generalize Krein’s perturbation determinant theory from regular to singular pairs of operators, particularly pairs of proper extensions of a symmetric operator.
• To develop a framework for trace formulas and spectral shift functions in the context of singular perturbations using abstract boundary triplets.
• To establish new results for dissipative extensions, where existing classical theories are incomplete or inapplicable.
• To apply the abstract theory to concrete operators, including ordinary and elliptic differential operators.
• To characterize the spectral shift function in terms of the Weyl function and characteristic function of almost solvable extensions.

Proposed method

• Employ the boundary triplet approach to represent extensions of symmetric operators, using auxiliary Hilbert spaces and boundary maps Γ₀, Γ₁.
• Define perturbation determinants for pairs of extensions via the abstract Weyl function M(λ) and the characteristic function of almost solvable extensions.
• Use the Krein-type resolvent formula to relate the difference of resolvents to the Weyl function and boundary operators.
• Establish holomorphy and trace-class properties of perturbation determinants in the upper half-plane using functional calculus and H∞-functional calculus.
• Derive trace formulas by integrating the logarithmic derivative of the perturbation determinant, linking it to the spectral shift function.
• Apply the theory to Sturm-Liouville and second-order elliptic operators via explicit boundary conditions and characteristic functions.

Experimental results

Research questions

• RQ1 How can the classical concept of perturbation determinants be extended to singular pairs of operators, such as different proper extensions of a symmetric operator?
• RQ2 What is the role of the abstract Weyl function and characteristic function in expressing perturbation determinants for almost solvable extensions?
• RQ3 How can trace formulas for spectral shift functions be derived for pairs of self-adjoint, dissipative, and m-accumulative extensions?
• RQ4 In what way do the results improve or complete classical results of M.G. Krein for dissipative operators?
• RQ5 How can the abstract framework be applied to concrete differential operators like Sturm-Liouville and elliptic operators?

Key findings

• The paper introduces a new generalized perturbation determinant for singular pairs of extensions using boundary triplets, valid even when the resolvent difference is not trace-class.
• For pairs of self-adjoint extensions, the spectral shift function is expressed as the logarithmic derivative of the perturbation determinant in terms of the Weyl function.
• For dissipative extensions, the authors derive new trace formulas that extend and complete classical results of Krein, particularly in the case of additive perturbations.
• The characteristic function of almost solvable extensions plays a central role in expressing the perturbation determinant and spectral shift function.
• The theory is successfully applied to matrix Sturm-Liouville operators on ℝ₊, Sturm-Liouville operators on (0,b), and second-order elliptic operators with compact boundary.
• The framework allows the derivation of trace formulas even when the standard Krein theory fails due to lack of trace-class resolvent differences.