## The functional calculus (1995)

Venue: | J. London Math. Soc |

Citations: | 33 - 1 self |

### BibTeX

@ARTICLE{Davies95thefunctional,

author = {E. B. Davies},

title = {The functional calculus},

journal = {J. London Math. Soc},

year = {1995},

volume = {2},

pages = {166--176}

}

### Years of Citing Articles

### OpenURL

### Abstract

One of the interesting developments in spectral theory recently has been the application of a new formula for a function j{H) of a self-adjoint operator H by Helffer and Sjostrand [8]. This formula, equation (5) below, depends upon the use of almost analytic extensions of functions initially defined on the real line, an idea due

### Citations

1589 |
The analysis of Linear Partial Differential Operators
- Hörmander
- 1983
(Show Context)
Citation Context .... It is important for applications and for our proofs below that functions in stf need not be integrable. We shall use only one particular version of Hormander's concept of almost analytic extensions =-=[10, 11]-=-. Given fesrf and n^O we define an almost analytic extension of/to C by the formula fix, y) •= (tf r) (*) (W/A <x,y), (2) \r-0 / where ) (3) and T is a non-negative C 00 function such that r(s) = 1 if... |

57 |
Schrödinger semigroups
- Simon
- 1982
(Show Context)
Citation Context ...es (H3). It satisfies (HI) if and only if it is diagonalisable. We now say that H is a Schrodinger operator on L P (U N ) for 1 ^ p ^ oo if it is of the form //:= -A+ V, where V+eK l ™ and V_eKN (see =-=[16]-=- for details). It is known [9] that the spectrum of//is real and independent of/7. It is also known [3, p. 27] that the operators e iAt are unbounded on L V (U N ) for all / # 0, N ^ 1 and all p # 2 i... |

34 |
Equation de Schrodinger avec champ magnetique et equation de
- Heler, Sjostrand
- 1989
(Show Context)
Citation Context ...IES 1. Introduction One of the interesting developments in spectral theory recently has been the application of a new formula for a function j{H) of a self-adjoint operator H by Helffer and Sjostrand =-=[8]-=-. This formula, equation (5) below, depends upon the use of almost analytic extensions of functions initially defined on the real line, an idea due to Hormander [10; 11, p. 63]. One of its main advant... |

18 |
Besov Spaces and Applications to Difference Methods for Initial Value
- Brenner, Thomée, et al.
- 1975
(Show Context)
Citation Context ...n of Hormander's concept of almost analytic extensions [10, 11]. Given fesrf and n^O we define an almost analytic extension of/to C by the formula fix, y) •= (tf r) (*) (W/A <x,y), (2) \r-0 / where ) =-=(3)-=- and T is a non-negative C 00 function such that r(s) = 1 if |s| ^ 1 and z(s) = 0 if \s\ ^ 2. It will turn out that the choice of n and of the cut-off function a are by no means critical. Other choice... |

17 |
Mapping properties of functions of Schrödinger operators between L p
- Jensen, Nakamura
- 1994
(Show Context)
Citation Context ...easily and flexibly from resolvent estimates to estimates of other functions of//, a fact which has already been exploited in TV-body scattering theory [7, 17]. It was realised by Jensen and Nakamura =-=[13,14]-=- that this calculus may be applied to the study of Schrodinger operators on L P (U N ) for p ^ 2, in which case e iHt is not bounded for any t # 0. In order to clarify what is necessary for such appli... |

13 |
Integrated solutions of Volterra integrodifferential equations and applications
- Arendt, Kellermann
- 1987
(Show Context)
Citation Context ... the L p spectrum of a variety of self-adjoint operators is independent of p. The theory which we describe below has much in common with the theory of integrated semigroups and spectral distributions =-=[1, 2]-=-. A functional calculus based upon almost analytic extensions was also constructed by Dyn'kin [5]. However, our approach is quite different from his, and more appropriate to differential operators. 2.... |

10 |
The spectrum of a Schrödinger operator
- Hempel, Voigt
- 1986
(Show Context)
Citation Context ...nd only if it is diagonalisable. We now say that H is a Schrodinger operator on L P (U N ) for 1 ^ p ^ oo if it is of the form //:= -A+ V, where V+eK l ™ and V_eKN (see [16] for details). It is known =-=[9]-=- that the spectrum of//is real and independent of/7. It is also known [3, p. 27] that the operators e iAt are unbounded on L V (U N ) for all / # 0, N ^ 1 and all p # 2 in the range [1, oo]. (2.12) Pa... |

9 |
L p spectral independence and L 1 analyticity
- Davies
- 1995
(Show Context)
Citation Context ...y no means rigid, and that many other hypotheses of a similar type can be accommodated by making appropriate choices of the almost analytic extensions. In a subsequent paper appearing in this journal =-=[4]-=-, we use the axiomatic approach to obtain a very simple and general proof that the L p spectrum of a variety of self-adjoint operators is independent of p. The theory which we describe below has much ... |

7 |
Uniform Algebras (Prentice-Hall
- Gamelin
- 1969
(Show Context)
Citation Context ...t lemma is crucial for establishing that the operator f{H) is independent of the choices of n and a. LEMMA 7. //F<= CC°°(C) and F{z) = O(\y\P)asy^0 for some p > a + 1, then dx' — 1-(z-H)- 1 dxdy = 0. =-=(6)-=- Proof. Let F have support in {z:|x| < Af and \y\ < N} and define Qs for small S > 0 to be the region {z: \x\ < N and S < \y\ < N}. If A denotes the left-hand side of (6) then an application of Stokes... |

3 |
Spectral Distributions and Generalization of Stone’s Theorem
- Balabane, Emamirad, et al.
- 1993
(Show Context)
Citation Context ... the L p spectrum of a variety of self-adjoint operators is independent of p. The theory which we describe below has much in common with the theory of integrated semigroups and spectral distributions =-=[1, 2]-=-. A functional calculus based upon almost analytic extensions was also constructed by Dyn'kin [5]. However, our approach is quite different from his, and more appropriate to differential operators. 2.... |

3 |
Fractional powers of the momentum of a spectral distribution
- Jazar
- 1995
(Show Context)
Citation Context ...roceeding we comment that if one assumes (H2) instead of (H3), then it has recently been shown that our main results can be proved using the theory of integrated semigroups and spectral distributions =-=[1, 2,12]-=-. We shall, however, give an ab initio treatment of the problem. We start by defining S p for any fie U to be the space of smooth functions/: U -• C such that as |*| -• oo for all r ^ 0. It is easy to... |

2 |
An operator calculus based upon the Cauchy-Green formula
- Dynkin
(Show Context)
Citation Context ...scribe below has much in common with the theory of integrated semigroups and spectral distributions [1, 2]. A functional calculus based upon almost analytic extensions was also constructed by Dyn'kin =-=[5]-=-. However, our approach is quite different from his, and more appropriate to differential operators. 2. Definitions and hypotheses We suppose that H is a closed densely defined operator on a Banach sp... |

1 |
Sharp propagation estimates for Af-particle systems
- GERARD
- 1992
(Show Context)
Citation Context ...s main advantages is that it allows one to pass easily and flexibly from resolvent estimates to estimates of other functions of//, a fact which has already been exploited in TV-body scattering theory =-=[7, 17]-=-. It was realised by Jensen and Nakamura [13,14] that this calculus may be applied to the study of Schrodinger operators on L P (U N ) for p ^ 2, in which case e iHt is not bounded for any t # 0. In o... |

1 |
On the singularities of solutions of partial differential equations
- HORMANDER
- 1970
(Show Context)
Citation Context .... It is important for applications and for our proofs below that functions in stf need not be integrable. We shall use only one particular version of Hormander's concept of almost analytic extensions =-=[10, 11]-=-. Given fesrf and n^O we define an almost analytic extension of/to C by the formula fix, y) •= (tf r) (*) (W/A <x,y), (2) \r-0 / where ) (3) and T is a non-negative C 00 function such that r(s) = 1 if... |

1 | Resolvent estimates for Schrodinger operators in L P (U N ) and the theory of exponentially bounded C-semigroups', Semigroup Forum 41 - PANG - 1990 |

1 |
Smoothness of TV-body scattering amplitudes
- SKIBSTED
- 1992
(Show Context)
Citation Context ...s main advantages is that it allows one to pass easily and flexibly from resolvent estimates to estimates of other functions of//, a fact which has already been exploited in TV-body scattering theory =-=[7, 17]-=-. It was realised by Jensen and Nakamura [13,14] that this calculus may be applied to the study of Schrodinger operators on L P (U N ) for p ^ 2, in which case e iHt is not bounded for any t # 0. In o... |