Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T13:33:00.220Z Has data issue: false hasContentIssue false
  • English
  • Français

On Threshold Eigenvalues and Resonances for the Linearized NLSEquation

Published online by Cambridge University Press:  12 May 2010

V. Vougalter
V. Vougalter*
Affiliation:
University of Toronto, Department of Mathematics, Toronto, ON, M5S 2E4, Canada
*
* E-mail: vitali@math.toronto.edu
Get access

Abstract

We prove the instability of threshold resonances and eigenvalues of the linearized NLSoperator. We compute the asymptotic approximations of the eigenvalues appearing from theendpoint singularities in terms of the perturbations applied to the original NLS equation.Our method involves such techniques as the Birman-Schwinger principle and the Feshbachmap.

Keywords

NLS equationspectral stabilityBirman-Schwinger principleFeshbach map
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 4: Spectral problems. Issue dedicated to the memory of M. Birman , 2010 , pp. 448 - 469
Copyright
© EDP Sciences, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adachi, S.. A Positive solution of a nonhomogeneous elliptic equation in R N with G-invariant nonlinearity . Comm. PDE., 27 (2002), No. 1-2, 1–22. CrossRefGoogle Scholar
Bach, V., Fröhlich, J., Sigal, I.M.. Renormalization group analysis of spectral problems in quantum field theory . Adv. Math., 137 (1998), No. 2, 205–298. CrossRefGoogle Scholar
Berestycki, H., Lions, P.-L.. Nonlinear scalar field equations. I. Existence of a ground state , Arch. Rational Mech. Anal., 82 (1983), No. 4, 313–345. Google Scholar
Berestycki, H., Lions, P.-L.. Nonlinear scalar field equations. II. Existence of infinitely many solutions . Arch. Rational Mech. Anal., 82 (1983), No. 4, 347–375.Google Scholar
Berestycki, H., Lions, P.-L., Peletier, L.. An ODE approach to the existence of positive solutions for semilinear problems in ℝ N . Indiana Univ. Math. J., 30 (1981), No. 1, 141–157. CrossRefGoogle Scholar
Buslaev, V.S., Perelman, G.S.. Scattering for the nonlinear Schrödinger equation: states that are close to a soliton . St. Petersburg Math. J., 4 (1993), No. 6, 1111–1142. Google Scholar
Buslaev, V.S., Sulem, C.. On asymptotic stability of solitary waves for nonlinear Schrödinger equations . Ann. Inst. H. Poincare Anal. Non Lineaire, 20 (2003), No. 3, 419–475 CrossRefGoogle Scholar
Cuccagna, S.. On asymptotic stability of ground states of NLS , Rev. Math. Phys., 15 (2003), No. 8, 877–903. CrossRefGoogle Scholar
Chang, S.-M., Gustafson, S., Nakanishi, K., Tsai, T.-P.. Spectra of linearized operators for NLS solitary waves . SIAM J. Math. Anal., 39 (2007), No. 4, 1070–1111. CrossRefGoogle Scholar
S. Cuccagna, D. Pelinovsky. Bifurcations from the endpoints of the essential spectrum in the linearized nonlinear Schrödinger problem. J. Math. Phys., 46 (2005), No. 5, 053520, 15 pp.
Erdogan, B., Schlag, W.. Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II . J. Anal. Math., 99 (2006), 199–248. CrossRefGoogle Scholar
Floer, A., Weinstein, A.. Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential . J. Funct. Anal., 69 (1986), No. 3, 397–408. CrossRefGoogle Scholar
Cuccagna, S., Pelinovsky, D., Vougalter, V.. Spectra of positive and negative energies in the linearized NLS problem . Comm. Pure Appl. Math., 58 (2005), No. 1, 1–29. CrossRefGoogle Scholar
Grillakis, M.. Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system . Comm. Pure Appl. Math., 43 (1990), No. 3, 299–333. CrossRefGoogle Scholar
S. Gustafson, I.M. Sigal. Mathematical concepts of quantum mechanics. Springer–Verlag, Berlin, 2003.
Gang, Z., Sigal, I.M.. Asymptotic stability of nonlinear Schrödinger equations with potential , Rev. Math. Phys., 17 (2005), No. 10, 1143–1207. CrossRefGoogle Scholar
P.D. Hislop, I.M. Sigal. Introduction to spectral theory with applications to Schrödinger operators. Springer, 1996.
Kapitula, T., Sandstede, B.. Edge bifurcations for near integrable systems via Evans functions techniques . SIAM J. Math.Anal., 33 (2002), No. 5, 1117–1143. CrossRefGoogle Scholar
Kapitula, T., Sandstede, B.. Eigenvalues and resonances using the Evans functions . Discrete Contin. Dyn. Syst., 10 (2004), No. 4, 857–869. CrossRefGoogle Scholar
Klaus, M., Simon, B.. Coupling constant thresholds in nonrelativistic quantum mechanics. I. Short-range two-body case . Ann. Phys., 130 (1980), No. 2, 251–281. CrossRefGoogle Scholar
E. Lieb, M. Loss. Analysis. Graduate studies in Mathematics, 14. American Mathematical Society, Providence, 1997.
E.Lieb, B.Simon, A. Wightman. Book “Studies in mathematical physics: Essays in Honor of Valentine Bargmann.” Princeton University Press, 1976.
McLeod, K.. Uniqueness of positive radial solutions of Δu +f(u) = 0 in R n . II . Trans. Amer. Math. Soc., 339 (1993), No. 2, 495–505.Google Scholar
Pelinovsky, D., Kivshar, Y., Afanasjev, V.. Internal modes of envelope solitons , Phys. D, 116 (1998), No. 1–2, 121–142. CrossRefGoogle Scholar
Perelman, G.. Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations . Comm. Partial Differential Equations, 29 (2004), No. 7–8, 1051–1095. CrossRefGoogle Scholar
Strauss, W.. Existence of solitary waves in higher dimensions . Comm.Math.Phys., 55 (1977), No. 2, 149–162. CrossRefGoogle Scholar
B.Simon. Functional integration and quantum physics. Pure and Applied Mathematics, 86 (1979), Academic Press.
Schlag, W.. Stable manifolds for an orbitally unstable nonlinear Schrödinger equation . Ann. of Math. (2), 169 (2009), No. 1, 139–227. CrossRefGoogle Scholar
V. Vougalter. On the negative index theorem for the linearized NLS problem. To appear in Canad. Math. Bull.
V. Vougalter, D. Pelinovsky. Eigenvalues of zero energy in the linearized NLS problem. J. Math. Phys., 47 (2006), No. 6, 062701, 13 pp.
Weinstein, M.I.. Modulation stability of ground states of nonlinear Schrödinger equations . SIAM J. Math. Anal., 16 (1985), No. 3, 472–491. CrossRefGoogle Scholar