[Paper Review] Weyl function of a Hermitian operator and its connection with characteristic function

Vladimir Derkach, M. M. Malamud|arXiv (Cornell University)|Mar 31, 2015

Spectral Theory in Mathematical Physics5 references13 citations

TL;DR

This paper introduces a novel approach to the characteristic operator function of Hermitian operators using abstract boundary triplets and the second Green's formula, establishing a direct link between the Weyl function and the characteristic function. The key contribution is proving that the Weyl function serves as a $Q$-function and uniquely determines the spectral properties of almost solvable extensions, with explicit formulas derived for differential operators and Laplacians in domains with singular boundaries.

ABSTRACT

Let $A$ be a densely defined symmetric operator with equal deficiency indices in a Hilbert space. We introduce the notion of a Weyl function $M(z)$ of $A$ corresponding to an ordinary boundary triplet of the operator $A^*$ and then investigate its basic properties. In particular, a connection with Krein-Langer Q-functions and Krein's type formula for resolvents is discovered. Using this new connection, we show that the resolvent comparability of two proper extensions is equivalent to that of the corresponding boundary operators. Moreover, we show that the number of negative eigenvalues of a self-adjoint extension $A_B=A_B^*$ of a non-negative operator $A$ equals the number of negative eigenvalues of $B-M(0-)$, where $B$ is the boundary operator of $A_B$ and $M(0-)$ is the left limit of the Weyl function at zero. Also, we introduce the class of almost solvable extensions of $A$. A characteristic function (in the sense of A. V. Shtraus) of an almost solvable extension is expressed by means of the Weyl function and the corresponding boundary operator. Analytic properties of characteristic functions are completely characterized. The main results are applied to ordinary differential operators, Sturm-Liouville operators with unbounded operator potentials, Shrödinger operators and Laplacians on domains with a non-smooth boundary. These results were substantially elaborated and published later in the following papers: 1. V.A. Derkach and M.M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal. 95 (1991), 1-95. 2. --- Characteristic functions of almost solvable extensions of a Hermitian operators, Ukr. Mat. Zh. 44 (1992), 435-459. 3. --- The extension theory of Hermitian operators and the moment problem, J. Math. Sci. 73 (1995), 141-242.

Motivation & Objective

• To develop a new, more natural framework for the characteristic operator function of symmetric operators using abstract boundary triplets.
• To clarify the role of the Weyl function as a $Q$-function and its spectral determination of almost solvable extensions.
• To establish a unified connection between the Weyl function and the characteristic function across various differential operators.
• To provide explicit expressions for the Weyl function and characteristic function in singular domains, such as those with incoming angles or singular potentials.

Proposed method

• Introduces the concept of a boundary triplet $\{\mathcal{H}, \Gamma_0, \Gamma_1\}$ for the adjoint of a symmetric operator $A^*$, satisfying abstract Green's identity and surjectivity.
• Defines the Weyl function $M(z)$ via $M(z)\Gamma_0 f_z = \Gamma_1 f_z$ for $f_z \in \mathfrak{N}_z$, showing it is holomorphic and belongs to the $R_\mathcal{H}$ class.
• Establishes that $M(z)$ is a $Q$-function in the sense of Kreín and Langer, uniquely determining the pair $\{A, A_0\}$ up to unitary equivalence when $A$ is simple.
• Uses the abstract second Green's formula to define the Weyl function and derive its properties, including holomorphy and positivity of imaginary part.
• Applies the framework to differential operators with bounded or unbounded coefficients, and to Schrödinger and Laplace operators in domains with piecewise smooth or singular boundaries.
• Derives explicit expressions for the Weyl function and characteristic function in specific cases, such as $M(z) = C_\beta z^\beta$ for the Laplacian in a wedge domain.

Experimental results

Research questions

• RQ1 How can the characteristic operator function be redefined using boundary triplets and the abstract second Green's formula to yield a more natural and general framework?
• RQ2 What is the precise relationship between the Weyl function $M(z)$ and the characteristic function $\Theta(z)$ of an almost solvable extension?
• RQ3 Under what conditions does the Weyl function $M(z)$ belong to the Stieltjes class, and how does this relate to the finiteness of the negative spectrum?
• RQ4 How can the Weyl function be explicitly computed for differential operators with singular coefficients or in domains with singular geometries, such as corners or cusps?
• RQ5 What is the spectral characterization of self-adjoint extensions via the Weyl function, particularly in terms of the Friedrichs and Kreín extensions?

Key findings

• The Weyl function $M(z)$ is a $Q$-function in the sense of Kreín and Langer, belonging to the class $(R_\mathcal{H})$, and uniquely determines the pair $\{A, A_0\}$ up to unitary equivalence when $A$ is simple.
• For a Laplacian in a wedge domain with angle $\pi/\beta$, the Weyl function is explicitly given by $M(z) = C_\beta z^\beta$, where $C_\beta = \exp(-i\beta\pi)4^{-\beta}\Gamma(1-\beta)/\Gamma(1+\beta)$.
• The characteristic function of the extension $L_h$ with domain $\ker(\Gamma_1 - C_\beta h \Gamma_0)$ is $\Theta(z) = \frac{z^\beta + h}{z^\beta + \bar{h}}$, linking the spectral parameter to the boundary condition.
• The Friedrichs extension corresponds to $\ker\Gamma_0$ and the Kreín extension to $\ker\Gamma_1$, with $M(0) = 0$ ensuring the latter is well-defined.
• For nonnegative operators, a criterion for $M(z)$ to belong to the Stieltjes class $(S)$ is established, linking positivity and monotonicity to spectral properties.
• In the case of a minimal differential operator of order $2n$ on the half-line, the Weyl function $M(z)$ coincides with the characteristic matrix of a certain self-adjoint extension, generalizing classical results.