Prirodoslovno-matematiˇ cki fakultet Matematiˇ cki odsjek Sveuˇ ciliˇ ste u Zagrebu Contour integration methods for self adjoint operators OTKR December 19–22 2019 – TU Vienna Luka Grubiˇ si´ c Department of Mathematics, University of Zagreb luka.grubisic@math.hr Joint work with Jay Goplakrishnan and Jeffrey Ovall Luka Grubiˇ si´ c 1 / 36
Outline 1 About the problem 2 About the method 3 Discrete resolvent estimates 4 Numerical experiments Luka Grubiˇ si´ c 2 / 36
About this talk FEAST idea (Polizzi et al.) • Substitute numerical integration formula for Cauchy resolvent integral • Then subspace iterate the so obtained operator • Use an appropriate subspace extraction (eg. RR) to determine eigenvalue/vector approximation • Other contour integration approaches: Beyn, Sakurai & Sugiura, RIM of Sun et al. In this talk: Luka Grubiˇ si´ c 3 / 36
About this talk FEAST idea (Polizzi et al.) • Substitute numerical integration formula for Cauchy resolvent integral • Then subspace iterate the so obtained operator • Use an appropriate subspace extraction (eg. RR) to determine eigenvalue/vector approximation • Other contour integration approaches: Beyn, Sakurai & Sugiura, RIM of Sun et al. In this talk: • Looping contours is naturally dictated by perturbation theory Luka Grubiˇ si´ c 3 / 36
About this talk FEAST idea (Polizzi et al.) • Substitute numerical integration formula for Cauchy resolvent integral • Then subspace iterate the so obtained operator • Use an appropriate subspace extraction (eg. RR) to determine eigenvalue/vector approximation • Other contour integration approaches: Beyn, Sakurai & Sugiura, RIM of Sun et al. In this talk: • Looping contours is naturally dictated by perturbation theory • It will work – for self adjoint operators – with coarse integration formulae Luka Grubiˇ si´ c 3 / 36
About this talk FEAST idea (Polizzi et al.) • Substitute numerical integration formula for Cauchy resolvent integral • Then subspace iterate the so obtained operator • Use an appropriate subspace extraction (eg. RR) to determine eigenvalue/vector approximation • Other contour integration approaches: Beyn, Sakurai & Sugiura, RIM of Sun et al. In this talk: • Looping contours is naturally dictated by perturbation theory • It will work – for self adjoint operators – with coarse integration formulae • With very few iterations per refinement level (asymptotically speaking) Luka Grubiˇ si´ c 3 / 36
About this talk FEAST idea (Polizzi et al.) • Substitute numerical integration formula for Cauchy resolvent integral • Then subspace iterate the so obtained operator • Use an appropriate subspace extraction (eg. RR) to determine eigenvalue/vector approximation • Other contour integration approaches: Beyn, Sakurai & Sugiura, RIM of Sun et al. In this talk: • Looping contours is naturally dictated by perturbation theory • It will work – for self adjoint operators – with coarse integration formulae • With very few iterations per refinement level (asymptotically speaking) • Strong resolvent convergence with“contraction property”is enough Luka Grubiˇ si´ c 3 / 36
Model operator Model operator u ∈ H 1 A u = −△ u − Vu , ∗ (Ω) . • Ω ⊂ R d open and boundˇ zed • A self-adjoint and unbounded • ( ξ − A ) − 1 operator valued function • Spec( A ) is countable without finite accumulation points. • Notation: A ψ i = λ i ψ i , and −� V � ≤ λ 1 ≤ λ 2 ≤ · · · • We are counting eigenvalues with multiplicity. • Variational (energy) scalar product ( u , v ) V = a [ u , v ], u , v ∈ H 1 ∗ (Ω). About the problem Luka Grubiˇ si´ c 4 / 36
Model operator Model operator u ∈ H 1 A u = −△ u − Vu , ∗ (Ω) . • Ω= R d • A self-adjoint and unbounded • ( ξ − A ) − 1 operator valued function • It holds for the discrete spectrum Spec disc ( A ) ⊂ [inf V , 0 � • Notation: A ψ i = λ i ψ i , and −� V � ≤ λ 1 ≤ λ 2 ≤ · · · < 0 are only isolated eigenvalues call them Λ • We are counting eigenvalues with multiplicity. • Variational (energy) scalar product ( u , v ) V = a [ u , v ], u , v ∈ H 1 ∗ (Ω). • CT estimates: � χ B x ( ξ − A ) − 1 χ B y � ≤ C exp( − τ � x − y � ) About the problem Luka Grubiˇ si´ c 5 / 36
Bounded and extended states � � exp( − t 2 )+0 . 7 exp( − ( t − 3) 2 ) Quantum states from chebfun V ( t )= − 50 About the problem Luka Grubiˇ si´ c 6 / 36
Bounded and extended states � � exp( − t 2 )+0 . 7 exp( − ( t − 3) 2 ) Quantum states from chebfun V ( t )= − 50 • We consider A R on finite Ω R . About the problem Luka Grubiˇ si´ c 6 / 36
Bounded and extended states � � exp( − t 2 )+0 . 7 exp( − ( t − 3) 2 ) Quantum states from chebfun V ( t )= − 50 • We consider A R on finite Ω R . • Perturbation argument: put a contour on [ −� V � , 0 � , unwanted clustered eigenvalues in [0 , ∞� . About the problem Luka Grubiˇ si´ c 6 / 36
Also in 2D (thanks to NG Solve) About the problem Luka Grubiˇ si´ c 7 / 36
However! • We would like to spice things up • essential spectrum About the problem Luka Grubiˇ si´ c 8 / 36
However! • We would like to spice things up • essential spectrum extreme clustering • Enter About the problem Luka Grubiˇ si´ c 8 / 36
However! • We would like to spice things up • essential spectrum extreme clustering • Enter 1 Periodic potentials About the problem Luka Grubiˇ si´ c 8 / 36
However! • We would like to spice things up • essential spectrum extreme clustering • Enter 1 Periodic potentials 2 First order block operators, e.g. Dirac operators About the problem Luka Grubiˇ si´ c 8 / 36
However! • We would like to spice things up • essential spectrum extreme clustering • Enter 1 Periodic potentials 2 First order block operators, e.g. Dirac operators 3 Strong resolvent convergence. About the problem Luka Grubiˇ si´ c 8 / 36
Model operator – periodic potential e.g. V = cos( · ) Model operator u ∈ H 1 A u = −△ u − Vu , ∗ (Ω) . • Ω= R d • A self-adjoint and unbounded • ( ξ − A ) − 1 operator valued function • Only essential spectrum with band gaps. • Notation: A ψ i = λ i ψ i , for isolated eigenvalues call them Λ • We are counting eigenvalues with multiplicity. • Variational (energy) scalar product ( u , v ) V = a [ u , v ], u , v ∈ H 1 ∗ (Ω). About the problem Luka Grubiˇ si´ c 9 / 36
Quantum pendulum About the problem Luka Grubiˇ si´ c 10 / 36
Quantum pendulum – has band gaps About the problem Luka Grubiˇ si´ c 11 / 36
Model operator – V = cos( x ) − exp( − x 2 ) Model operator u ∈ H 1 A u = −△ u − Vu , ∗ (Ω) . • Ω= R d • A self-adjoint and unbounded • ( ξ − A ) − 1 operator valued function • The same essential spectrum with band gaps and isolated eigenvalues in the middle. • Notation: A ψ i = λ i ψ i , for isolated eigenvalues call them Λ • We are counting eigenvalues with multiplicity. • Variational (energy) scalar product ( u , v ) V = a [ u , v ], u , v ∈ H 1 ∗ (Ω). About the problem Luka Grubiˇ si´ c 12 / 36
Quantum pendulum – with a bump About the problem Luka Grubiˇ si´ c 13 / 36
Quantum pendulum – with a bump About the problem Luka Grubiˇ si´ c 13 / 36
Cascading spectrum The bump is a relatively compact perturbation About the problem Luka Grubiˇ si´ c 14 / 36
Cascading spectrum The bump is a relatively compact perturbation About the problem Luka Grubiˇ si´ c 14 / 36
Cascading spectrum The bump is a relatively compact perturbation About the problem Luka Grubiˇ si´ c 14 / 36
Cascading spectrum The bump is a relatively compact perturbation About the problem Luka Grubiˇ si´ c 14 / 36
Cascading spectrum The bump is a relatively compact perturbation About the problem Luka Grubiˇ si´ c 14 / 36
Cascading spectrum The bump is a relatively compact perturbation About the problem Luka Grubiˇ si´ c 14 / 36
Cascading spectrum The bump is a relatively compact perturbation About the problem Luka Grubiˇ si´ c 14 / 36
Cascading spectrum The bump is a relatively compact perturbation About the problem Luka Grubiˇ si´ c 14 / 36
First order operators are good candidates Dirac Op with radial symmetry and the Culomb potential • Defined by the block operator matrix � mc 2 − Z − ∂ r + κ � r r A = − mc 2 − Z ∂ r + κ r r with Dom( A ) = H 1 ( R ) ⊕ H 1 ( R ). • Z = 1 , 2 , ..., 137 is the electric charge number • κ is the spin orbit coupling parameter • See L. Boulton’s web page About the problem Luka Grubiˇ si´ c 15 / 36
Associated norms – not used in the standard analysis so far • L 2 norm � · � About the problem Luka Grubiˇ si´ c 16 / 36
Associated norms – not used in the standard analysis so far • L 2 norm � · � • Energy norm � A 1 / 2 · � About the problem Luka Grubiˇ si´ c 16 / 36
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