[Paper Review] Regularization of singular Sturm-Liouville equations

Andrii Goriunov, Vladimir Mikhailets|arXiv (Cornell University)|Feb 23, 2010

Spectral Theory in Mathematical Physics7 references19 citations

TL;DR

This paper introduces a novel regularization of singular Sturm-Liouville operators with coefficients satisfying $ q = Q' $, $ 1/p, Q/p, Q^2/p \in L_1 $, using Shin-Zettl quasi-derivatives to define well-posed maximal and minimal operators. The key contribution is a complete characterization of all self-adjoint and maximal dissipative extensions, along with generalized resolvents, via canonical boundary conditions, extending prior results under weaker assumptions than classical or $ L_2 $-based frameworks.

ABSTRACT

Paper deals with the singular Sturm-Liouville expressions $$l(y) = -(py')' + qy$$ on a finite interval with coefficients $$q = Q', \quad 1/p, Q/p, Q^2/p \in L_1,$$ where derivative of the function $Q$ is understood in the sense of distributions. Due to a new regularization corresponding operators are correctly defined as quasi-differential. Their resolvent approximation is investigated and all self-adjoint and maximal dissipative extensions and generalized resolvents are described in terms of homogeneous boundary conditions of the canonic form. Some results are new for the case $p(t)\equiv 1$ as well.

Motivation & Objective

• To define and rigorously construct Sturm-Liouville operators under significantly weaker coefficient conditions than classical or $ L_2 $-based assumptions.
• To extend the theory of quasi-differential operators to singular cases where $ q = Q' $, $ Q/p, Q^2/p \in L_1 $, with $ Q' $ in the distributional sense.
• To provide a complete parametric description of all self-adjoint and maximal dissipative extensions of the minimal operator using boundary triplet theory.
• To establish resolvent approximation of the constructed operators by those with more regular coefficients, improving on prior results.

Proposed method

• Introduce a new regularization of the Sturm-Liouville expression $ l(y) = -(py')' + qy $ using quasi-derivatives $ D^{[0]}y = y $, $ D^{[1]}y = py' - Qy $, $ D^{[2]}y = (D^{[1]}y)' + \frac{Q}{p}D^{[1]}y + \frac{Q^2}{p}y $, so that $ l[y] = -D^{[2]}y $.
• Define the maximal operator $ L_{\text{max}} $ as the quasi-differential operator on $ L_2 $ with domain $ \{ y \in L_2 \mid y, D^{[1]}y \in AC, D^{[2]}y \in L_2 \} $.
• Define the minimal operator $ L_{\text{min}} $ as the restriction of $ L_{\text{max}} $ with homogeneous boundary conditions $ D^{[k]}y(a) = D^{[k]}y(b) = 0 $ for $ k = 0,1 $.
• Use the boundary triplet theory to parametrize all self-adjoint and maximal dissipative extensions via operators $ K $ on $ \mathbb{C}^2 $, with $ K $ contracting for dissipative extensions.
• Establish resolvent approximation by showing that resolvents of the constructed operators can be approximated in norm by those of operators with more regular coefficients.
• Derive generalized resolvents via a boundary-value problem involving $ (K(\lambda) - I)\Gamma_1 y + i(K(\lambda) + I)\Gamma_2 y = 0 $, where $ K(\lambda) $ is analytic and $ \|K(\lambda)\| \leq 1 $ in the lower half-plane.

Experimental results

Research questions

• RQ1 How can singular Sturm-Liouville operators be rigorously defined when the potential $ q $ is a distributional derivative $ Q' $, with $ Q/p, Q^2/p \in L_1 $?
• RQ2 What is the structure of the maximal and minimal operators under the generalized coefficient conditions $ q = Q' $, $ 1/p, Q/p, Q^2/p \in L_1 $?
• RQ3 How can all self-adjoint extensions of the minimal operator be parametrically described in terms of boundary conditions?
• RQ4 What is the characterization of all maximal dissipative extensions and generalized resolvents of the minimal symmetric operator?
• RQ5 Can the resolvents of the constructed operators be approximated by those of operators with more regular coefficients?

Key findings

• The proposed quasi-differential regularization uniquely defines the Sturm-Liouville operator under the generalized coefficient conditions $ q = Q' $, $ 1/p, Q/p, Q^2/p \in L_1 $, with $ Q' $ in the distributional sense.
• The maximal and minimal operators defined via this regularization coincide with classical ones under $ L_1 $-conditions and with those in [5] under $ L_2 $-conditions on $ Q $.
• All self-adjoint extensions of the minimal operator are in bijective, continuous correspondence with self-adjoint operators $ K $ on $ \mathbb{C}^2 $, described by canonical homogeneous boundary conditions.
• All maximal dissipative extensions are in bijective correspondence with contracting operators $ K $ on $ \mathbb{C}^2 $, and separated boundary conditions occur precisely when $ K $ is block-diagonal with $ |K_a| \leq 1 $, $ |K_b| \leq 1 $.
• The resolvents of the constructed operators can be approximated in norm by resolvents of operators with more regular coefficients, extending and improving prior approximation results.
• Generalized resolvents are fully characterized by a boundary-value problem involving an analytic operator-valued function $ K(\lambda) $ on $ \mathbb{C}^- $ with $ \|K(\lambda)\| \leq 1 $, yielding a one-to-one correspondence with generalized spectral functions.