Fourier transform, null variety, and Laplacian's eigenvalues specgeom/ transform, null variety, and...

download Fourier transform, null variety, and Laplacian's eigenvalues specgeom/ transform, null variety, and Laplacian’s eigenvalues Michael Levitin Reading University Spectral Geometry Conference,

of 98

  • date post

    10-May-2018
  • Category

    Documents

  • view

    215
  • download

    1

Embed Size (px)

Transcript of Fourier transform, null variety, and Laplacian's eigenvalues specgeom/ transform, null variety, and...

  • Fourier transform, null variety, and Laplacianseigenvalues

    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

  • Objects of study in BenguriaL.Parnovski (2009)

    Rd simply connected bounded domain with connectedboundary ;

    (x) =

    {1, if x ,0 if x 6

    the characteristic function of ;

    () = F []() :=

    eix dx its Fourier transform;

    NC() := { Cd : () = 0} its complex null variety, or nullset;

    C() := dist(NC(), 0) = min{|| : NC()};Also, in particular for balanced (e.g. centrally symmetric domains) welook at N () := NC() Rd = { Rd : () = 0} = { Rd :

    cos( x) dx = 0} and () := dist(N (), 0);(0

  • Objects of study in BenguriaL.Parnovski (2009)

    Rd simply connected bounded domain with connectedboundary ;

    (x) =

    {1, if x ,0 if x 6

    the characteristic function of ;

    () = F []() :=

    eix dx its Fourier transform;

    NC() := { Cd : () = 0} its complex null variety, or nullset;

    C() := dist(NC(), 0) = min{|| : NC()};Also, in particular for balanced (e.g. centrally symmetric domains) welook at N () := NC() Rd = { Rd : () = 0} = { Rd :

    cos( x) dx = 0} and () := dist(N (), 0);(0

  • Objects of study in BenguriaL.Parnovski (2009)

    Rd simply connected bounded domain with connectedboundary ;

    (x) =

    {1, if x ,0 if x 6

    the characteristic function of ;

    () = F []() :=

    eix dx its Fourier transform;

    NC() := { Cd : () = 0} its complex null variety, or nullset;

    C() := dist(NC(), 0) = min{|| : NC()};Also, in particular for balanced (e.g. centrally symmetric domains) welook at N () := NC() Rd = { Rd : () = 0} = { Rd :

    cos( x) dx = 0} and () := dist(N (), 0);(0

  • Objects of study in BenguriaL.Parnovski (2009)

    Rd simply connected bounded domain with connectedboundary ;

    (x) =

    {1, if x ,0 if x 6

    the characteristic function of ;

    () = F []() :=

    eix dx its Fourier transform;

    NC() := { Cd : () = 0} its complex null variety, or nullset;

    C() := dist(NC(), 0) = min{|| : NC()};Also, in particular for balanced (e.g. centrally symmetric domains) welook at N () := NC() Rd = { Rd : () = 0} = { Rd :

    cos( x) dx = 0} and () := dist(N (), 0);(0

  • Objects of study in BenguriaL.Parnovski (2009)

    Rd simply connected bounded domain with connectedboundary ;

    (x) =

    {1, if x ,0 if x 6

    the characteristic function of ;

    () = F []() :=

    eix dx its Fourier transform;

    NC() := { Cd : () = 0} its complex null variety, or nullset;

    C() := dist(NC(), 0) = min{|| : NC()};

    Also, in particular for balanced (e.g. centrally symmetric domains) welook at N () := NC() Rd = { Rd : () = 0} = { Rd :

    cos( x) dx = 0} and () := dist(N (), 0);(0

  • Objects of study in BenguriaL.Parnovski (2009)

    Rd simply connected bounded domain with connectedboundary ;

    (x) =

    {1, if x ,0 if x 6

    the characteristic function of ;

    () = F []() :=

    eix dx its Fourier transform;

    NC() := { Cd : () = 0} its complex null variety, or nullset;

    C() := dist(NC(), 0) = min{|| : NC()};Also, in particular for balanced (e.g. centrally symmetric domains) welook at N () := NC() Rd = { Rd : () = 0} = { Rd :

    cos( x) dx = 0} and () := dist(N (), 0);

    (0

  • Objects of study in BenguriaL.Parnovski (2009)

    Rd simply connected bounded domain with connectedboundary ;

    (x) =

    {1, if x ,0 if x 6

    the characteristic function of ;

    () = F []() :=

    eix dx its Fourier transform;

    NC() := { Cd : () = 0} its complex null variety, or nullset;

    C() := dist(NC(), 0) = min{|| : NC()};Also, in particular for balanced (e.g. centrally symmetric domains) welook at N () := NC() Rd = { Rd : () = 0} = { Rd :

    cos( x) dx = 0} and () := dist(N (), 0);(0

  • What does N () look like?

    Far field zero intensity diffraction pattern from the aperture ! Physicalmotivation.Also, it is of importance for inverse problems and image recognition. It isknown that the structure of N () far from the origin determines theshape of a convex set .

    M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 3 / 22

  • What does N () look like?

    Far field zero intensity diffraction pattern from the aperture ! Physicalmotivation.Also, it is of importance for inverse problems and image recognition. It isknown that the structure of N () far from the origin determines theshape of a convex set .

    M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 3 / 22

  • What does N () look like?

    Farfield zero intensity diffraction pattern from the aperture ! Physicalmotivation.Also, it is of importance for inverse problems and image recognition. It isknown that the structure of N () far from the origin determines theshape of a convex set .

    M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 3 / 22

  • What does N () look like?

    Far field zero intensity diffraction pattern from the aperture ! Physicalmotivation.

    Also, it is of importance for inverse problems and image recognition. It isknown that the structure of N () far from the origin determines theshape of a convex set .

    M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 3 / 22

  • What does N () look like?

    Far field zero intensity diffraction pattern from the aperture ! Physicalmotivation.Also, it is of importance for inverse problems and image recognition. It isknown that the structure of N () far from the origin determines theshape of a convex set .

    M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 3 / 22

  • Questions and Motivation

    Main question: study () and its relations to the eigenvalues.

    Known links between N () and spectral theory: two unsolved problems

    Pompeius Problem

    Let M(d) be a group of rigidmotions of Rd , and be abounded simply connected domainwith piecewise smooth connectedboundary. Prove that the existenceof a non-zero continuous functionf : Rd Rd such thatm() f (x) dx = 0 for all

    m M(d) implies that is a ball.

    Schiffers conjecture

    The existence of an eigenfunction v(corresponding to a non-zeroeigenvalue ) of a NeumannLaplacian on a domain such thatv const along the boundary (or, in other words, the existence ofa non-constant solution v to theover-determined problemv = v , v/n| = 0,v | = 1) implies that is a ball.

    M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 4 / 22

  • Questions and Motivation

    Main question: study () and its relations to the eigenvalues.Known links between N () and spectral theory: two unsolved problems

    Pompeius Problem

    Let M(d) be a group of rigidmotions of Rd , and be abounded simply connected domainwith piecewise smooth connectedboundary. Prove that the existenceof a non-zero continuous functionf : Rd Rd such thatm() f (x) dx = 0 for all

    m M(d) implies that is a ball.

    Schiffers conjecture

    The existence of an eigenfunction v(corresponding to a non-zeroeigenvalue ) of a NeumannLaplacian on a domain such thatv const along the boundary (or, in other words, the existence ofa non-constant solution v to theover-determined problemv = v , v/n| = 0,v | = 1) implies that is a ball.

    M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 4 / 22

  • Questions and Motivation

    Main question: study () and its relations to the eigenvalues.Known links between N () and spectral theory: two unsolved problems

    Pompeius Problem

    Let M(d) be a group of rigidmotions of Rd , and be abounded simply connected domainwith piecewise smooth connectedboundary. Prove that the existenceof a non-zero continuous functionf : Rd Rd such thatm() f (x) dx = 0 for all

    m M(d) implies that is a ball.

    Schiffers conjecture

    The existence of an eigenfunction v(corresponding to a non-zeroeigenvalue ) of a NeumannLaplacian on a domain such thatv const along the boundary (or, in other words, the existence ofa non-constant solution v to theover-determined problemv = v , v/n| = 0,v | = 1) implies that is a ball.

    M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 4 / 22

  • Questions and Motivation

    Main question: study () and its relations to the eigenvalues.Known links between N () and spectral theory: two unsolved problems

    Pompeius Problem

    Let M(d) be a group of rigidmotions of Rd , and be abounded simply connected domainwith piecewise smooth connectedboundary. Prove that the existenceof a non-zero continuous functionf : Rd Rd such thatm() f (x) dx = 0 for all

    m M(d) implies that is a ball.

    Schiffers conjecture

    The existence of an eigenfunction v(corresponding to a non-zeroeigenvalue ) of a NeumannLaplacian on a domain such thatv const along the boundary (or, in other words, the existence ofa non-constant solution v to theover-determined problemv = v , v/n| = 0,v | = 1) implies that is a ball.

    M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 4 / 22

  • Motivation (contd.)

    It is known that

    the positive answer to the Pompeiu problem Schiffers conjecture 6 and r > 0 such that NC() { Cd :

    dj=1

    2j = r

    2} .

    Thus, the interest in NC(). Also, it is of importance for inverse problems determining the shape of . A lot of publications, e.g. Agranovsky,Aviles, Berens