Darboux--Moutard transformations and Poincar\'e--Steklov operators

R. G. Novikov ; I. A. Taimanov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Topology and physics, Tome 302 (2018), pp. 334-342.

Résumé

Formulas relating Poincaré–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.

Bibliographie

[1] Adilkhanov A. N., Taimanov I. A., “On numerical study of the discrete spectrum of a two-dimensional Schrödinger operator with soliton potential”, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 83–92 | DOI | MR

[2] Crum M. M., “Associated Sturm–Liouville systems”, Q. J. Math., 6 (1955), 121–127 | DOI | MR

[3] Darboux G., “Sur une proposition relative aux équations linéaires”, C. r. Acad. sci. Paris., 94 (1882), 1456–1459

[4] B. A. Dubrovin, I. M. Kričever, S. P. Novikov, “The Schrödinger equation in a periodic field and Riemann surfaces”, Sov. Math., Dokl., 17 (1976), 947–951 | MR

[5] P. G. Grinevich, R. G. Novikov, “Generalized analytic functions, Moutard-type transforms, and holomorphic maps”, Funct. Anal. Appl., 50:2 (2016), 150–152 | DOI | DOI | MR

[6] Grinevich P. G., Novikov R. G., “Moutard transform approach to generalized analytic functions with contour poles”, Bull. sci. math., 140:6 (2016), 638–656 | DOI | MR

[7] Grinevich P. G., Novikov R. G., “Moutard transform for generalized analytic functions”, J. Geom. Anal., 26:4 (2016), 2984–2995 | DOI | MR

[8] Grinevich P. G., Novikov R. G., Moutard transform for the conductivity equation, 2018, arXiv: 1801.00295 [math-ph] | MR

[9] P. G. Grinevich, S. P. Novikov, “Two-dimensional ‘inverse scattering problem’ for negative energies and generalized-analytic functions. I: Energies below the ground state”, Funct. Anal. Appl., 22:1 (1988), 19–27 | DOI | MR

[10] Hu H.-C., Lou S.-Y., Liu Q.-P., “Darboux transformation and variable separation approach: the Nizhnik–Novikov–Veselov equation”, Chin. Phys. Lett., 20:9 (2003), 1413–1415 | DOI

[11] R. M. Matuev, I. A. Taimanov, “The Moutard transformation of two-dimensional Dirac operators and the conformal geometry of surfaces in four-dimensional space”, Math. Notes, 100:6 (2016), 835–846 | DOI | DOI | MR

[12] Matveev V. B., “Darboux transformations, covariance theorems and integrable systems”, L. D. Faddeev's seminar on mathematical physics, AMS Transl. Ser. 2, 201, Amer. Math. Soc., Providence, RI, 2000, 179–209 | MR

[13] Matveev V. B., Salle M. A., Darboux transformations and solitons, Springer, Berlin, 1991 | MR

[14] Moutard Th., “Sur la construction des équations de la forme $\frac {1}{z}\frac {d^2 z}{dx dy} = \lambda (x,y)$, qui admettent une intégrale générale explicite”, J. Éc. Polytech., 28 (1878), 1–11

[15] R. G. Novikov, I. A. Taimanov, “The Moutard transformation and two-dimensional multipoint delta-type potentials”, Russ. Math. Surv., 68:5 (2013), 957–959 | DOI | DOI | MR

[16] R. G. Novikov, I. A. Taimanov, “Moutard type transformation for matrix generalized analytic functions and gauge transformations”, Russ. Math. Surv., 71:5 (2016), 970–972 | DOI | DOI | MR

[17] R. G. Novikov, I. A. Taimanov, S. P. Tsarev, “Two-dimensional von Neumann–Wigner potentials with a multiple positive eigenvalue”, Funct. Anal. Appl., 48:4 (2014), 295–297 | DOI | DOI | MR

[18] I. A. Taimanov, “The Moutard transformation of two-dimensional Dirac operators and Möbius geometry”, Math. Notes, 97:1 (2015), 124–135 | DOI | DOI | MR

[19] I. A. Taimanov, “A fast decaying solution to the modified Novikov–Veselov equation with a one-point singularity”, Dokl. Math., 91:1 (2015), 35–36 | DOI | MR

[20] I. A. Taimanov, “Blowing up solutions of the modified Novikov–Veselov equation and minimal surfaces”, Theor. Math. Phys., 182:2 (2015), 173–181 | DOI | DOI | MR

[21] I. A. Taimanov, S. P. Tsarev, “Two-dimensional Schrödinger operators with fast decaying potential and multidimensional $L_2$-kernel”, Russ. Math. Surv., 62:3 (2007), 631–633 | DOI | DOI | MR

[22] I. A. Taimanov, S. P. Tsarev, “Blowing up solutions of the Novikov–Veselov equation”, Dokl. Math., 77:3 (2008), 467–468 | DOI | MR

[23] I. A. Taimanov, S. P. Tsarev, “Two-dimensional rational solitons and their blowup via the Moutard transformation”, Theor. Math. Phys., 157:2 (2008), 1525–1541 | DOI | DOI | MR

[24] I. A. Taimanov, S. P. Tsarev, “On the Moutard transformation and its applications to spectral theory and soliton equations”, J. Math. Sci., 170:3 (2010), 371–387 | DOI | MR

[25] I. A. Taimanov, S. P. Tsarev, “Faddeev eigenfunctions for two-dimensional Schrödinger operators via the Moutard transformation”, Theor. Math. Phys., 176:3 (2013), 1176–1183 | DOI | DOI | MR

[26] A. P. Veselov, S. P. Novikov, “Finite-zone, two-dimensional, potential Schrödinger operators. Explicit formulas and evolution equations”, Sov. Math., Dokl., 30 (1984), 588–591 | MR | MR

[27] A. P. Veselov, S. P. Novikov, “Finite-zone, two-dimensional Schrödinger operators. Potential operators”, Sov. Math., Dokl., 30 (1984), 705–708 | MR

[28] Yu D., Liu Q. P., Wang S., “Darboux transformation for the modified Veselov–Novikov equation”, J. Phys. A: Math. Gen., 35:16 (2002), 3779–3785 | DOI | MR