Boundary data maps and Krein’s resolvent formula for Sturm-Liouville operators on a finite interval.(English)Zbl 1311.34037

In this elaborate paper the authors cast a fresh look at operators about which everything seems to be known, namely, the selfadjoint realisations of a regular Sturm-Liouville operator. They start by expressing a general boundary trace map in terms of the Dirichlet and Neumann maps $$\gamma_D$$ and $$\gamma_N$$ which give an absolutely continuous function and its quasi-derivative their values at the interval ends. $$\gamma_D$$ and $$\gamma_N$$ are then used to show that there is a Cayley-type one-to-one correspondence between all selfadjoint realisations and all unitary $$2\times 2$$ matrices (Theorem 2.9).
Next they employ a Kreĭn resolvent formula (an appropriate abstract version of which is presented in an appendix) to relate the resolvents of different selfadjoint realisations with separated (Theorem 3.1) or non-separated boundary conditions (Theorem 3.2). The latter result is applied to characterise the von Neumann-Kreĭn extension of the minimal Sturm-Liouville operator (naturally assumed to be strictly positive) by means of non-separated boundary conditions imposed on a canonical fundamental system.
In §4 the concept of a general boundary data map is introduced. This is a $$2\times 2$$ matrix which arises from the solution of an auxiliary boundary value problem; it is analytic on the common resolvent set of two selfadjoint realisations. The Kreĭn resolvent formula can then be used to express the difference of two resolvents in terms of the inverse of such a general boundary data map and an associated boundary trace map (Theorem 4.9). The determinant of a general boundary data map can be related to a certain Fredholm determinant (Theorem 5.1), and the derivative of its logarithm equals the negative of Kreĭn’s trace formula for the resolvent difference of two selfadjoint realisations (Theorem 5.4). As a consequence there is also a connection between a general boundary data map and Kreĭn’s spectral shift function (Theorem 5.5). Finally, the well-known parametrisation of all selfadjoint realisations by means of John von Neumann’s formula is connected with a general boundary data map (Theorems 6.2 and 6.3).

MSC:

 34B05 Linear boundary value problems for ordinary differential equations 34B27 Green’s functions for ordinary differential equations 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 34B20 Weyl theory and its generalizations for ordinary differential equations 34L05 General spectral theory of ordinary differential operators 47A10 Spectrum, resolvent 47E05 General theory of ordinary differential operators
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