Fourier transform, null variety, and Laplacian's eigenvalues

98 Pages·2010·0.77 MB·English

by  Michael Levitin

Fourier transform, null variety, and Laplacian’s
eigenvalues
Michael Levitin
Reading University
Spectral Geometry Conference, 19 July 2010
joint work with Rafael Benguria (PUC Santiago) and Leonid Parnovski (UCL)
M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 1 / 22
{
1, if x ∈ Ω,
χΩ(x) = — the characteristic function of Ω;
0 if x ∉ Ω
∫
iξ·x
χ̂Ω(ξ) = F[χΩ](ξ) :=
Ω e dx — its Fourier transform;
d
NC(Ω) := {ξ ∈ C : χ̂Ω(ξ) = 0} — its complex null variety, or null
set;
κC(Ω) := dist(NC(Ω), 0) = min{|ξ| : ξ ∈ NC(Ω)};
Also, in particular for balanced (e.g. centrally symmetric domains) we
d d d
look at N(Ω) := NC(Ω) ∩ R = {ξ ∈ R : χ̂Ω(ξ) = 0} = {ξ ∈ R :
∫
cos(ξ · x) dx = 0} and κ(Ω) := dist(N(Ω), 0);
Ω
(0 <)λ1(Ω) < λ2(Ω) ≤ . . . — Dirichlet Laplacian’s eigenvalues,
(0 =)µ1(Ω) < µ2(Ω) ≤ . . . — Neumann Laplacian’s eigenvalues.
Objects of study in Benguria–L.–Parnovski (2009)
d
Ω ⊂ R — simply connected bounded domain with connected
boundary ∂Ω;
M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 2 / 22
∫
iξ·x
χ̂Ω(ξ) = F[χΩ](ξ) :=
Ω e dx — its Fourier transform;
d
NC(Ω) := {ξ ∈ C : χ̂Ω(ξ) = 0} — its complex null variety, or null
set;
κC(Ω) := dist(NC(Ω), 0) = min{|ξ| : ξ ∈ NC(Ω)};
Also, in particular for balanced (e.g. centrally symmetric domains) we
d d d
look at N(Ω) := NC(Ω) ∩ R = {ξ ∈ R : χ̂Ω(ξ) = 0} = {ξ ∈ R :
∫
cos(ξ · x) dx = 0} and κ(Ω) := dist(N(Ω), 0);
Ω
(0 <)λ1(Ω) < λ2(Ω) ≤ . . . — Dirichlet Laplacian’s eigenvalues,
(0 =)µ1(Ω) < µ2(Ω) ≤ . . . — Neumann Laplacian’s eigenvalues.
Objects of study in Benguria–L.–Parnovski (2009)
d
Ω ⊂ R — simply connected bounded domain with connected
boundary ∂Ω;
{
1, if x ∈ Ω,
χΩ(x) = — the characteristic function of Ω;
0 if x ∉ Ω
M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 2 / 22
d
NC(Ω) := {ξ ∈ C : χ̂Ω(ξ) = 0} — its complex null variety, or null
set;
κC(Ω) := dist(NC(Ω), 0) = min{|ξ| : ξ ∈ NC(Ω)};
Also, in particular for balanced (e.g. centrally symmetric domains) we
d d d
look at N(Ω) := NC(Ω) ∩ R = {ξ ∈ R : χ̂Ω(ξ) = 0} = {ξ ∈ R :
∫
cos(ξ · x) dx = 0} and κ(Ω) := dist(N(Ω), 0);
Ω
(0 <)λ1(Ω) < λ2(Ω) ≤ . . . — Dirichlet Laplacian’s eigenvalues,
(0 =)µ1(Ω) < µ2(Ω) ≤ . . . — Neumann Laplacian’s eigenvalues.
Objects of study in Benguria–L.–Parnovski (2009)
d
Ω ⊂ R — simply connected bounded domain with connected
boundary ∂Ω;
{
1, if x ∈ Ω,
χΩ(x) = — the characteristic function of Ω;
0 if x ∉ Ω
∫
iξ·x
χ̂Ω(ξ) = F[χΩ](ξ) :=
Ω e dx — its Fourier transform;
M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 2 / 22
κC(Ω) := dist(NC(Ω), 0) = min{|ξ| : ξ ∈ NC(Ω)};
Also, in particular for balanced (e.g. centrally symmetric domains) we
d d d
look at N(Ω) := NC(Ω) ∩ R = {ξ ∈ R : χ̂Ω(ξ) = 0} = {ξ ∈ R :
∫
cos(ξ · x) dx = 0} and κ(Ω) := dist(N(Ω), 0);
Ω
(0 <)λ1(Ω) < λ2(Ω) ≤ . . . — Dirichlet Laplacian’s eigenvalues,
(0 =)µ1(Ω) < µ2(Ω) ≤ . . . — Neumann Laplacian’s eigenvalues.
Objects of study in Benguria–L.–Parnovski (2009)
d
Ω ⊂ R — simply connected bounded domain with connected
boundary ∂Ω;
{
1, if x ∈ Ω,
χΩ(x) = — the characteristic function of Ω;
0 if x ∉ Ω
∫
iξ·x
χ̂Ω(ξ) = F[χΩ](ξ) :=
Ω e dx — its Fourier transform;
d
NC(Ω) := {ξ ∈ C : χ̂Ω(ξ) = 0} — its complex null variety, or null
set;
M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 2 / 22
Also, in particular for balanced (e.g. centrally symmetric domains) we
d d d
look at N(Ω) := NC(Ω) ∩ R = {ξ ∈ R : χ̂Ω(ξ) = 0} = {ξ ∈ R :
∫
cos(ξ · x) dx = 0} and κ(Ω) := dist(N(Ω), 0);
Ω
(0 <)λ1(Ω) < λ2(Ω) ≤ . . . — Dirichlet Laplacian’s eigenvalues,
(0 =)µ1(Ω) < µ2(Ω) ≤ . . . — Neumann Laplacian’s eigenvalues.
Objects of study in Benguria–L.–Parnovski (2009)
d
Ω ⊂ R — simply connected bounded domain with connected
boundary ∂Ω;
{
1, if x ∈ Ω,
χΩ(x) = — the characteristic function of Ω;
0 if x ∉ Ω
∫
iξ·x
χ̂Ω(ξ) = F[χΩ](ξ) :=
Ω e dx — its Fourier transform;
d
NC(Ω) := {ξ ∈ C : χ̂Ω(ξ) = 0} — its complex null variety, or null
set;
κC(Ω) := dist(NC(Ω), 0) = min{|ξ| : ξ ∈ NC(Ω)};
M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 2 / 22
(0 <)λ1(Ω) < λ2(Ω) ≤ . . . — Dirichlet Laplacian’s eigenvalues,
(0 =)µ1(Ω) < µ2(Ω) ≤ . . . — Neumann Laplacian’s eigenvalues.
Objects of study in Benguria–L.–Parnovski (2009)
d
Ω ⊂ R — simply connected bounded domain with connected
boundary ∂Ω;
{
1, if x ∈ Ω,
χΩ(x) = — the characteristic function of Ω;
0 if x ∉ Ω
∫
iξ·x
χ̂Ω(ξ) = F[χΩ](ξ) :=
Ω e dx — its Fourier transform;
d
NC(Ω) := {ξ ∈ C : χ̂Ω(ξ) = 0} — its complex null variety, or null
set;
κC(Ω) := dist(NC(Ω), 0) = min{|ξ| : ξ ∈ NC(Ω)};
Also, in particular for balanced (e.g. centrally symmetric domains) we
d d d
look at N(Ω) := NC(Ω) ∩ R = {ξ ∈ R : χ̂Ω(ξ) = 0} = {ξ ∈ R :
∫
cos(ξ · x) dx = 0} and κ(Ω) := dist(N(Ω), 0);
Ω
M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 2 / 22
Objects of study in Benguria–L.–Parnovski (2009)
d
Ω ⊂ R — simply connected bounded domain with connected
boundary ∂Ω;
{
1, if x ∈ Ω,
χΩ(x) = — the characteristic function of Ω;
0 if x ∉ Ω
∫
iξ·x
χ̂Ω(ξ) = F[χΩ](ξ) :=
Ω e dx — its Fourier transform;
d
NC(Ω) := {ξ ∈ C : χ̂Ω(ξ) = 0} — its complex null variety, or null
set;
κC(Ω) := dist(NC(Ω), 0) = min{|ξ| : ξ ∈ NC(Ω)};
Also, in particular for balanced (e.g. centrally symmetric domains) we
d d d
look at N(Ω) := NC(Ω) ∩ R = {ξ ∈ R : χ̂Ω(ξ) = 0} = {ξ ∈ R :
∫
cos(ξ · x) dx = 0} and κ(Ω) := dist(N(Ω), 0);
Ω
(0 <)λ1(Ω) < λ2(Ω) ≤ . . . — Dirichlet Laplacian’s eigenvalues,
(0 =)µ1(Ω) < µ2(Ω) ≤ . . . — Neumann Laplacian’s eigenvalues.
M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 2 / 22
Far field zero intensity diffraction pattern from the aperture Ω! Physical
motivation.
Also, it is of importance for inverse problems and image recognition. It is
known that the structure of N(Ω) far from the origin determines the
shape of a convex set Ω.
What does N(Ω) look like?
M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 3 / 22
Far field zero intensity diffraction pattern from the aperture Ω! Physical
motivation.
Also, it is of importance for inverse problems and image recognition. It is
known that the structure of N(Ω) far from the origin determines the
shape of a convex set Ω.
What does N(Ω) look like?
M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 3 / 22