On the spectrum of second-order differential operators with complex coefficients

Abstract

The main objective of this paper is to extend the pioneering work of Sims in [9] on secondorder linear differential equations with a complex coefficient, in which he obtains an analogue of the Titchmarsh-Weyl theory and classification. The generalisation considered exposes interesting features not visible in the special case in [9]. An m-function is constructed (which is either unique or a point on a "limit-circle") and the relationship between its properties and the spectrum of underlying maccretive differential operators analysed. The paper is a contribution to the study of non-self-adjoint operators; in general the spectral theory of such operators is rather fragmentary, and further study is being driven by important physical applications, to hydrodynamics, electro-magnetic theory and nuclear physics, for instance.

Classification of Sturm–Liouville differential equations with complex coefficients and operator realizations

“…With these definitions and using a nesting circle method based on the methods of both Weyl (1910) and Sims (1957), Brown et al (1999) divided equation (1.1) into three cases with respect to the corresponding half-planes L q,K as follows. The uniqueness referred to in the theorem and following sections is only up to constant multiples.…”

“…He considered the case where p(x) = w(x) ≡ 1, Im q(x) is semibounded and classified equation (1.1) into three cases. Recently, this work was extensively generalized by Brown et al (1999Brown et al ( , 2003.…”

“…In order to state clearly the classification given in Brown et al (1999), we introduce some notations. Define…”

Limit‐point / limit‐circle classification of second‐order differential operators arising in PT quantum mechanics

“…Via the parametrization z(x) := xe iφsgn(x) we obtain Sturm-Liouville differential equations on [0, ∞) and (−∞, 0]. We give a full classification in limit-point and limit-circle case for complex-valued potential as defined in [6], or more recent in [4] and [5]. With (1) we associate an operator in a L 2 (R) space.…”

“…[4]. Remark 3.3 One can show that the operator A + ⊕ A − with the coupling y (0+) = αy (0−) (α ∈ C) in zero is PTsymmetric if and only if |α| = 1.…”