[Paper Review] Point Interactions: PT-Hermiticity and Reality of the Spectrum

Sergio Albeverio, S.M. Fei+1 more|ArXiv.org|Jun 18, 2002

Quantum Mechanics and Non-Hermitian Physics8 references4 citations

TL;DR

This paper investigates ${ackslash}mathcal{P}{ackslash}mathcal{T}$-self-adjoint point interactions for the one-dimensional Schrödinger operator, demonstrating that such non-Hermitian Hamiltonians can possess purely real spectra under specific boundary conditions. Using the method of boundary conditions and extending the notion of ${ackslash}mathcal{P}{ackslash}mathcal{T}$-symmetry to non-self-adjoint operators, the authors derive conditions under which the spectrum remains real, showing that ${ackslash}mathcal{P}{ackslash}mathcal{T}$-symmetry does not imply reality of the spectrum, and vice versa, but both properties can coexist in exactly solvable models.

ABSTRACT

General point interactions for the second derivative operator in one dimension are studied. In particular, ${\mathcal P \mathcal T}$-self-adjoint point interactions with the support at the origin and at points $\pm l$ are considered. The spectrum of such non-Hermitian operators is investigated and conditions when the spectrum is pure real are presented. The results are compared with those for standard self-adjoint point interactions.

Motivation & Objective

• To investigate the spectral properties of ${ackslash}mathcal{P}{ackslash}mathcal{T}$-self-adjoint point interactions in one-dimensional quantum mechanics.
• To determine under what conditions these non-Hermitian operators have purely real spectra.
• To clarify the relationship between ${ackslash}mathcal{P}{ackslash}mathcal{T}$-symmetry and spectral reality, showing that neither implies the other.
• To extend the framework of ${ackslash}mathcal{P}{ackslash}mathcal{T}$-symmetric quantum mechanics to exactly solvable point interaction models.
• To characterize the set of boundary conditions leading to real spectra in terms of parameterized families of coefficients.

Proposed method

• The method of boundary conditions is used to define non-self-adjoint extensions of the second derivative operator on $L_2({ackslash}mathbb{R})$, replacing classical von Neumann theory.
• The notion of ${ackslash}mathcal{P}{ackslash}mathcal{T}$-self-adjointness is introduced and adapted to the extension theory of linear operators.
• Eigenfunctions are constructed via an Ansatz involving exponential solutions on either side of the origin, with continuity and jump conditions encoded in a $2\times2$ matrix of boundary parameters.
• The dispersion equation $k^2\beta + ik(\alpha + \delta) - \gamma = 0$ is derived from the boundary conditions, determining the spectrum.
• Conditions for real spectrum are analyzed by requiring solutions $k$ to be either in the closed lower half-plane ($\Im k \leq 0$) or purely imaginary ($\Re k = 0$).
• Parameterization of coefficients $\alpha, \beta, \gamma, \delta$ is used to classify families of boundary conditions leading to real spectra, with phase alignment and inequality constraints derived.

Experimental results

Research questions

• RQ1 Does ${ackslash}mathcal{P}{ackslash}mathcal{T}$-symmetry guarantee a real spectrum for point interaction Hamiltonians?
• RQ2 Can non-${ackslash}mathcal{P}{ackslash}mathcal{T}$-symmetric operators still have real spectra?
• RQ3 What specific boundary conditions lead to purely real spectra in ${ackslash}mathcal{P}{ackslash}mathcal{T}$-self-adjoint point interaction models?
• RQ4 How does the family of ${ackslash}mathcal{P}{ackslash}mathcal{T}$-self-adjoint operators with real spectrum relate to self-adjoint and real-spectrum families?
• RQ5 What is the geometric and algebraic structure of the set of boundary condition parameters yielding real spectra?

Key findings

• The spectrum of a ${ackslash}mathcal{P}{ackslash}mathcal{T}$-self-adjoint point interaction Hamiltonian is purely real if and only if the solutions $k_{1,2}$ to the dispersion equation $k^2\beta + ik(\alpha + \delta) - \gamma = 0$ satisfy $\Im k_{1,2} \leq 0$ or $\Re k_{1,2} = 0$.
• Purely imaginary solutions $k$ correspond to discrete eigenvalues, and such solutions exist if and only if $\frac{\alpha + \delta}{\beta}, \frac{\gamma}{\beta} \in \mathbb{R}$ and $4\frac{c}{b} \leq \frac{t^2}{b^2}$, where $t = \alpha + \delta$, $b = \beta$, $c = \gamma$, all with the same phase.
• The family of ${ackslash}mathcal{P}{ackslash}mathcal{T}$-self-adjoint boundary conditions with real spectrum is contained within a 6-parameter family of boundary conditions leading to real spectra.
• The 4-parameter family of self-adjoint boundary conditions is embedded within the 6-parameter family of ${ackslash}mathcal{P}{ackslash}mathcal{T}$-self-adjoint operators with real spectrum.
• The set of parameters leading to real spectrum is not fully contained in the set of ${ackslash}mathcal{P}{ackslash}mathcal{T}$-self-adjoint operators, showing that ${ackslash}mathcal{P}{ackslash}mathcal{T}$-symmetry and real spectrum are independent properties.
• The results generalize to point interactions at multiple points, and the method is extendable to higher-dimensional spaces and higher-order differential operators.