PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
1997 | 122 | 1 | 15-37
Tytuł artykułu

Initial value problem for the time dependent Schrödinger equation on the Heisenberg group

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let L be the full laplacian on the Heisenberg group $ℍ^{n}$ of arbitrary dimension n. Then for $f ∈ L^{2}(ℍ^{n})$ such that $(I-L)^{s/2}f ∈ L^{2}(ℍ^{n})$, s > 3/4, for a $ϕ ∈ C_{c}(ℍ^{n})$ we have $ʃ_{ℍ^{n}} |ϕ(x)| sup_{0 < t≤1} |e^{(√-1)tL}f(x)|^{2} dx ≤ C_{ϕ} ∥f∥_{W^{s}}^{2}$. On the other hand, the above maximal estimate fails for s < 1/4. If Δ is the sublaplacian on the Heisenberg group $ℍ^{n}$, then for every s < 1 there exists a sequence $f_{n} ∈ L^{2}(ℍ^{n})$ and $C_{n} > 0$ such that $(I-L)^{s/2} f_{n} ∈ L^{2}(ℍ^{n})$ and for a $ϕ ∈ C_{c}(ℍ^{n})$ we have $ʃ_{ℍ^{n}} |ϕ(x)| sup_{0 < t≤1} |e^{(√-1)tΔ} f_{n}(x)|^{2} dx ≥ C_{n} ∥f_{n}∥_{W^{s}}^{2}, lim_{n→∞}C_{n} = +∞$.
Słowa kluczowe
Czasopismo
Rocznik
Tom
122
Numer
1
Strony
15-37
Opis fizyczny
Daty
wydano
1997
otrzymano
1995-09-20
Twórcy
  • Institute of Mathematics, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland, zenek@math.uni.wroc.pl
Bibliografia
  • [B] J. Bourgain, A remark on Schrödinger operators, Israel J. Math. 77 (1992), 1-16.
  • [Car] A. Carbery, Radial Fourier multipliers and associated maximal functions, in: Recent Progress in Fourier Analysis, North-Holland Math. Stud. 111, North-Holland, 1985, 49-56.
  • [C] L. Carleson, Some analytic problems related to statistical mechanics, in: Euclidean Harmonic Analysis, Lecture Notes in Math. 779, Springer, 1980, 5-45.
  • [Cw] M. Cowling, Pointwise behaviour of solutions to Schrödinger equations, in: Harmonic Analysis, Lecture Notes in Math. 992, Springer, 1983, 83-90.
  • [DK] B. E. J. Dahlberg and C. E. Kenig, A note on the almost everywhere behavior of solutions to the Schrödinger equation, in: Harmonic Analysis, Lecture Notes in Math. 908, Springer, 1982, 205-209.
  • [E1] A. Erdélyi, Asymptotic forms for Laguerre polynomials, J. Indian Math. Soc. 24 (1960), 235-250.
  • [E2] A. Erdélyi, W. Magnus, F. Oberhettinger and G. F. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1953.
  • [HR] A. Hulanicki and F. Ricci, A Tauberian theorem and tangential convergence of bounded harmonic functions on balls in $ℂ^n$, Invent. Math. 62 (1980), 325-331.
  • [KPV1] C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), 33-69.
  • [KPV2] C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc. 4 (1991), 323-347.
  • [KR] C. E. Kenig and A. Ruiz, A strong type (2,2) estimate for a maximal operator associated to the Schrödinger equation, Trans. Amer. Math. Soc. 280 (1983), 239-246.
  • [M] D. Müller, A restriction theorem for the Heisenberg group, Ann. of Math. 131 (1990), 567-587.
  • [MR] D. Müller and F. Ricci, Analysis of second order differential operators on Heisenberg groups I, Invent. Math. 101 (1990), 545-582.
  • [NS] E. Nelson and W. F. Stinespring, Representations of elliptic operators in an enveloping algebra, Amer. J. Math. 81 (1959), 547-560.
  • [P] E. Prestini, Radial functions and regularity of solutions to the Schrödinger equation, Monatsh. Math. 109 (1990), 135-143.
  • [SS] P. Sjögren and P. Sjölin, Convergence properties for the time dependent Schrödinger equation, Ann. Acad. Sci. Fenn. Ser. AI Math. 14 (1989), 13-25.
  • [S1] P. Sjölin, Regularity of solutions to the Schrödinger equations, Duke Math. J. 55 (1987), 699-715.
  • [S2] P. Sjölin, Global maximal estimates for solutions to the Schrödinger equation, Studia Math. 110 (1994), 105-114.
  • [S3] P. Sjölin, Radial functions and maximal estimates for solutions to the Schrödinger equation, J. Austral. Math. Soc. 59 (1995), 134-142.
  • [Sz] G. Szegő, Orthogonal Polynomials, Colloq. Publ. 23, Amer. Math. Soc., 1939.
  • [V] L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), 874-878.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv122i1p15bwm
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.