Darboux Moutard Transformations and Poincaré—Steklov Operators

Abstract

Formulas relating Poincare-Steklov operators for Schrödinger equations related by Darboux-Moutard transformations are derived. They can be used for testing algorithms of reconstruction of the potential from measurements at the boundary. * The work was supported by the French-Russian grant (RFBR 17-51-150001 NCNI a/PRC 1545 CNRS/RFBR) and done during the visit of the second author (I.A.T.) to

Moutard transforms for the conductivity equation

“…In the present work we show that the conductivity equation ( 1) admits Moutard-type transforms, going back to [12]. Such transforms were successfully used in studies of integrable systems of mathematical physics and differential geometry, in spectral theory and in complex analysis; see, for example, [10], [13], [22], [19], [14], [17], [18], [4], [5], [6], [11], [15], [16]. In particular, the present article can be considered as a direct continuation of our recent works [4]- [6] on Moutard-type transforms for the generalized analytic functions.…”

“…In addition, in dimension d = 2 a very large class of Schrödinger equations integrable at zero energy can be constructed via classical Moutard transform (see, e.g. [19], [14], [16]). This approach for constructing integrable conductivity equations will be developed elsewhere.…”

“…It remains to verify that (23) holds. This verification uses (16), (22) and consists of the following:…”

“…Using formulas (16), definition f + R in (22) and formulas (25) one can see that (53) is equivalent to (26).…”

Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory

“…A matrix formalism for Darboux transformations for Schrödinger equation was developed in the beginning of the twenty first century by Pecheritsin, Pupasov and Samsonov, see [43]. Darboux transformations (and their matrix formalisms) also appear in works of Novikov, Taimanov and Tsarev, see [39,40] and references therein.…”