Sesquilinear forms corresponding to a non-semibounded Sturm–Liouville operator

Abstract

Let −DpD be a differential operator on the compact interval [−b, b] whose leading coefficient is positive on (0, b] and negative on [−b, 0), with fixed, separated, self-adjoint boundary conditions at b and −b and an additional interface condition at 0. The self-adjoint extensions of the corresponding minimal differential operator are non-semibounded and are related to non-semibounded sesquilinear forms by a generalization of Kato's representation theorems. The theory of non-semibounded sesquilinear forms is applied to this concrete situation. In particular, the generalized Friedrichs extension is obtained as the operator associated with the unique regular closure of the minimal sesquilinear form. Moreover, among all closed forms associated with the self-adjoint extensions, the regular closed forms are identified. As a consequence, eigenfunction expansion theorems are obtained for the differential operators as well as for certain indefinite Kreĭn–Feller operators with a single concentrated mass.

A Kato's second type representation theorem for solvable sesquilinear forms

“…This extends the representation theorems for sesquilinear forms considered by many authors, as for instance by Kato [11], McIntosh [12], Fleige et al [5,6,7], Grubisić et al [8] and Schmitz [15] (for a more complete list see the references of [2]). For a non-negative closed form Ω, with positive associated operator T , Kato also proved the so-called second representation theorem [11 In the case where Ω is a general sectorial closed form, Kato [10] generalized the representation as…”

“…[4,5,6,7] use the name "regular" instead of "hyper-solvable". We use a different terminology because we do not want to confuse hyper-solvable forms with Θ-regular forms (see Remark 3.10).…”

Non-semi-bounded closed symmetric forms associated with a generalized Friedrichs extension

“…The general theory of closed non-semi-bounded sesquilinear forms can be found in [5,7,9]. Here, some basic facts from this theory are recalled for the construction of the regular closed form associated with the generalized Friedrichs extension (if it exists).…”

“…The present theory has applications in indefinite Sturm-Liouville problems (see [1-7, 9, 18]). In particular, using the approach from [9], the above closed forms t F [·, ·] and t F [·, ·] may then be described more explicitly, and an example shows that t F [·, ·] need not be regular. The present paper shows that the situation described in [9] for the indefinite Sturm-Liouville setting also appears in general.…”

“…Whereas the first representation theorem holds, the second representation theorem does not hold in the new setting (see [5,7]). There may be closed non-semi-bounded forms associated with a non-semi-bounded selfadjoint operator T by (1.1) that do not satisfy (1.2) (see [7,9]). A closed form t [•, •] for which the identity (1.2) is satisfied is said to be regular.…”