Spectral inequality for Schrödinger's equation with multipoint potential

P. G. Grinevich   ; R. G. Novikov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 77 (2022) no. 6, pp. 1021-1028

Résumé

Schrödinger's equation with potential that is a sum of a regular function and a finite set of point scatterers of Bethe–Peierls type is under consideration. For this equation the spectral problem with homogeneous linear boundary conditions is considered, which covers the Dirichlet, Neumann, and Robin cases. It is shown that when the energy $E$ is an eigenvalue with multiplicity $m$, it remains an eigenvalue with multiplicity at least $m-n$ after adding $n$ point scatterers. As a consequence, because for the zero potential all values of the energy are transmission eigenvalues with infinite multiplicity, this property also holds for $n$-point potentials, as discovered originally in a recent paper by the authors. Bibliography: 33 titles.

Détail

Keywords: Schrödinger's equation, multipoint potentials, spectral problems
Mots-clés : transmisson eigenvalue.

Bibliographie

[1] L. D. Landau and E. M. Lifshitz, Course of theoretical physics, v. III, 3d ed., Nauka, Moscow, 1974, 752 pp. ; English transl. of 1st ed. v. 3, Addison-Wesley Series in Advanced Physics, Quantum mechanics: non-relativistic theory, Pergamon Press Ltd., London–Paris; Addison-Wesley Publishing Co., Inc., Reading, MA, 1958, xii+515 pp. | MR | MR | Zbl

[2] S. P. Novikov, S. V. Manakov, L. P. Pitaevskiĭ, and V. E. Zakharov, Theory of solitons. The inverse scattering method, Nauka, Moscow, 1980, 320 pp. ; English transl. Contemp. Soviet Math., Consultants Bureau [Plenum], New York, 1984, xi+276 pp. | MR | Zbl | MR | Zbl

[3] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable models in quantum mechanics, Texts Monogr. Phys., Springer-Verlag, New York, 1988, xiv+452 pp. | DOI | MR | Zbl

[4] L. Faddeev, “Instructive history of the quantum inverse scattering method”, KdV '95 (Amsterdam 1995), Acta Appl. Math., 39:1-3 (1995), 69–84 | DOI | MR | Zbl

[5] P. G. Grinevich and R. G. Novikov, “Transparent potentials at fixed energy in dimension two. Fixed-energy dispersion relations for the fast decaying potentials”, Comm. Math. Phys., 174:2 (1995), 409–446 | DOI | MR | Zbl

[6] P. G. Grinevich, “Scattering transformation at fixed non-zero energy for the two-dimensional Schrödinger operator with potential decaying at infinity”, Uspekhi Mat. Nauk, 55:6(336) (2000), 3–70 ; English transl. in Russian Math. Surveys, 55:6 (2000), 1015–1083 | DOI | MR | Zbl | DOI

[7] I. A. Taimanov and S. P. Tsarëv, “On the Moutard transformation and its applications to spectral theory and soliton equations”, 5th International Conference on Differential and Functional Differential Equations, Part 1 (Moscow 2008), Sovrem. Mat. Fund. Napravl., 35, RUDN University, Moscow, 2010, 101–117 ; English transl. in J. Math. Sci. (N.Y.), 170:3 (2010), 371–387 | MR | Zbl | DOI

[8] R. G. Novikov, I. A. Taimanov, and S. P. Tsarev, “Two-dimensional von Neumann–Wigner potentials with a multiple positive eigenvalue”, Funktsional. Anal. i Prilozhen., 48:4 (2014), 74–77 ; English transl. in Funct. Anal. Appl., 48:4 (2014), 295–297 | DOI | MR | Zbl | DOI

[9] H. Bethe and R. Peierls, “Quantum theory of the diplon”, Proc. Roy. Soc. London Ser. A, 148:863 (1935), 146–156 | DOI | Zbl

[10] L. H. Thomas, “The interaction between a neutron and a proton and the structure of $\mathbf H^3$”, Phys. Rev. (2), 47:12 (1935), 903–909 | DOI | Zbl

[11] E. Fermi, “Sul moto dei neutroni nelle sostanze idrogenate”, Ricerca Sci., 7(2) (1936), 13–52 ; Collected papers (Note e memorie), V. I: Italy, 1921–1938, Univ. of Chicago Press, Chicago, IL, 1971 | Zbl | Zbl

[12] Ya. B. Zel'dovich, “Scattering by a singular potential in perturbation theory and in the momentum representation”, Zh. Eksper. Teoret. Fiz., 38:3 (1960), 819–824 ; English transl. in Soviet Physics. JETP, 11:3 (1960), 594–597 | MR

[13] F. A. Berezin and L. D. Faddeev, “A remark on Schrödinger's equation with a singular potential”, Dokl. Akad. Nauk SSSR, 137:5 (1961), 1011–1014 ; English transl. in Soviet Math. Dokl., 2 (1961), 372–375 | MR | Zbl

[14] Yu. N. Demkov and V. N. Ostrovskii, Zero-range potentials and their applications in atomic physics, Leninigrad University Publishing House, Leningrad, 1975, 240 pp.; English transl. Physics of Atoms and Molecules, Plenum Press, New York, 1988, vii+288 pp. | DOI

[15] V. A. Burov and S. A. Morozov, “Relationship between the amplitude and phase of a signal scattered by a point-like acoustic inhomogeneity”, Akustichaskii Zh., 47:6 (2001), 751–756; English transl. in Acoust. Phys., 47:6 (2001), 659–664 | DOI

[16] N. P. Badalyan, V. A. Burov, S. A. Morozov, and S. D. Rumyantseva, “Scattering by acoustic boundary scatterers with small wave sizes and their reconstruction”, Akushicheskii Zh., 55:1 (2009), 3–10; English transl. in Acoust. Phys., 55:1 (2009), 1–7 | DOI

[17] K. V. Dmitriev and O. D. Rumyantseva, “Features of solving the direct and inverse scattering problems for two sets of monopole scatterers”, J. Inverse Ill-Posed Probl., 29:5 (2021), 775–789 | DOI | MR | Zbl

[18] P. G. Grinevich and R. G. Novikov, “Faddeev eigenfunctions for point potentials in two dimensions”, Phys. Lett. A, 376:12-13 (2012), 1102–1106 | DOI | MR | Zbl

[19] P. G. Grinevich and R. G. Novikov, “Faddeev eigenfunctions for multipoint potentials”, Eurasian J. Math. Comput. Appl., 1:2 (2013), 76–91

[20] P. G. Grinevich and R. G. Novikov, “Multipoint scatterers with bound states at zero energy”, Teoret. Mat. Fiz., 193:2 (2017), 309–314 ; English transl. in Theoret. and Math. Phys., 193:2 (2017), 1675–1679 | DOI | MR | Zbl | DOI

[21] P. G. Grinevich and R. G. Novikov, “Creation and annihilation of point-potentials using Moutard-type transform in spectral variable”, J. Math. Phys., 61:9 (2020), 093501, 9 pp. | DOI | MR | Zbl

[22] P. G. Grinevich and R. G. Novikov, “Transmission eigenvalues for multipoint scatterers”, Eurasian J. Math. Comput. Appl., 9:4 (2021), 17–25 | DOI

[23] A. D. Agaltsov and R. G. Novikov, “Examples of solution of the inverse scattering problem and the equations of the Novikov–Veselov hierarchy from the scattering data of point potentials”, Uspekhi Mat. Nauk, 74:3(447) (2019), 3–16 ; English transl. in Russian Math. Surveys, 74:3 (2019), 373–386 | DOI | MR | Zbl | DOI

[24] R. G. Novikov, “Inverse scattering for the Bethe–Peierls model”, Eurasian J. Math. Comput. Appl., 6:1 (2018), 52–55 | DOI

[25] R. G. Novikov and I. A. Taimanov, “The Moutard transformation and two-dimensional multipoint delta-type potentials”, Uspekhi Mat. Nauk, 68:5(413) (2013), 181–182 ; English transl. in Russian Math. Surveys, 68:5 (2013), 957–959 | DOI | MR | Zbl | DOI

[26] D. S. Chashchin, “Example of point potential with inner structure”, Eurasian J. Math. Comput. Appl., 6:1 (2018), 4–10 | DOI

[27] E. Amaldi, O. D'Agostino, E. Fermi, B. Pontecorvo, F. Rasetti, and E. Segrè, “Artificial radioactivity produced by neutron bombardment–II”, Proc. Roy. Soc. London Ser. A, 149:868 (1935), 522–558 | DOI

[28] A. Kirsch, “The denseness of the far field patterns for the transmission problem”, IMA J. Appl. Math., 37:3 (1986), 213–225 | DOI | MR | Zbl

[29] D. Colton and P. Monk, “The inverse scattering problem for time-harmonic acoustic waves in an inhomogeneous medium”, Quart. J. Mech. Appl. Math., 41:1 (1988), 97–125 | DOI | MR | Zbl

[30] F. Cakoni and H. Haddar, “Transmission eigenvalues”, Inverse Problems, 29:10 (2013), 100201, 3 pp. | DOI | MR | Zbl

[31] B. P. Rynne and B. D. Sleeman, “The interior transmission problem and inverse scattering from inhomogeneous media”, SIAM J. Math. Anal., 22:6 (1991), 1755–1762 | DOI | MR | Zbl

[32] E. Lakshtanov and B. Vainberg, “Weyl type bound on positive interior transmission eigenvalues”, Comm. Partial Differential Equations, 39:9 (2014), 1729–1740 | DOI | MR | Zbl

[33] F. Cakoni amd H.-M. Nguyen, “On the discreteness of transmission eigenvalues for the Maxwell equations”, SIAM J. Math. Anal., 53:1 (2021), 888–913 | DOI | MR | Zbl