2011 Bulletin of the Australian Mathematical Society  
By modifying the inner product in the direct sum of the Hilbert spaces associated with each of two underlying intervals on which an even-order equation is defined, we generate self-adjoint realisations for boundary conditions with any real coupling matrix which are much more general than the coupling matrices from the 'unmodified' theory. 2010 Mathematics subject classification: primary 34B20, 34B24; secondary 47B25. As noted in [2], a simple way of getting self-adjoint operators in a
more » ... Hilbert space is to take the direct sum of self-adjoint operators from each of the separate Hilbert spaces. However, there are many self-adjoint operators which are not merely the sum of self-adjoint operators from each of the separate intervals. These 'new' self-adjoint operators involve interactions between the two intervals. Therefore in [2] the authors develop a 'two-interval' theory. Mukhtarov and Yakubov [6] observed that the set of two-interval self-adjoint realisations can be further enlarged by using different multiples of the usual inner products associated with each of the intervals. In [8] Sun et al. use the Mukhtarov-Yakubov modification of the Everitt-Zettl theory to characterise all self-adjoint realisations of regular two-interval problems. This characterisation is explicit and involves only the values of solutions and their quasiderivatives at the endpoints of the intervals and the multiple inner product parameters. In particular, for the second-order case with coupled boundary conditions and a real coupling matrix K, the method of [2] requires that det(K) = 1 whereas with the Mukhtarov-Yakubov modification in [8] it is only required that det(K) is positive. In this paper we develop a complete analogue of [7] when one endpoint of each interval (a 1 , b 1 ), (a 2 , b 2 ) is regular using Hilbert spaces but with the usual inner products replaced by appropriate multiples. The interplay of these multiples with the boundary conditions generates self-adjoint problems of even order with real coupling matrices K which are much more general than the coupling matrices from the 'unmodified' theory. We give a number of examples to illustrate this additional generality, among other things. From another perspective, instead of using multiples of the usual inner products, our approach can be described as using multiples of weight functions. Notation and basic facts for one interval Although we only consider even-order equations with real coefficients in this paper, we summarise some basic facts about general quasidifferential equations of even and odd order and real or complex coefficients for the convenience of the reader. Let J = (a, b) be an interval with −∞ ≤ a < b ≤ ∞ and let n be a positive integer (even or odd). For a given set S , M n (S ) denotes the set of n × n complex matrices with entries from S . (2.1) Let Q ∈ Z n (J). We define V 0 := {y : J → C, y is measurable} https://doi.Note that, from (2.1) it follows that if the above hold for some c ∈ J then they hold for any c ∈ J. We say that M is regular on J, or just M is regular, if M is regular at both endpoints. Notation and Basic Assumptions for Two Intervals Let J r = (a r , b r ), −∞ < a r < b r ≤ ∞, r = 1, 2. Define two differential expressions with real-valued coefficients by M r y = M Q r y := i n y [n] on J r , r = 1, 2, n = 2k, k > 1. https://doi.
doi:10.1017/s0004972711002991 fatcat:4ibzxspisndydb74hh2mgaewri