Victor Ivriiweyl.math.toronto.edu/.../preprints/GradTalk2.pdf · 2017-08-05 · Why to bother?...

Why to bother?Pseudo-differential operators

Calculus of PDOsOscillatory Integrals

Fourier Integral Operators

Quantize!

Victor Ivrii

Department of Mathematics, University of Toronto

May 27, 2007

Victor Ivrii Quantize!




Table of Contents

1 Why to bother?

2 Pseudo-differential operators

3 Calculus of PDOs

4 Oscillatory Integrals

5 Fourier Integral Operators





Why to bother?

There are two reasons to introduce quantization.

One of them came from physics: Quantum mechanics was indire need to have solid mathematical justification.

Another reason came from PDE which was looking for newpowerfull tools (algebra of differential operators was too smalleven to invert operators).

Let start from physics (historically it came first).





From PhysicsHamiltonian Mechanics for DummiesGoing QuantumFrom PDE

From Physics

To beginning of the 20-th century classical mechanics was in thestate of Full Power and Glory. Newtonian Mechanics was replacedby Lagrangian and Hamiltonian Mechanics.

From the Mathematical Point of View Einstein’ Special Relativitywas just a special case and even General Relativity was nothingnew (if we assume metrics given).But Quantum Mechanics changed everything: states were no morepoints in phase space but wave functions and Observables whichwere functions on the phase space became operators acting onthem. The last transition is called quantization.






From Physics

To beginning of the 20-th century classical mechanics was in thestate of Full Power and Glory. Newtonian Mechanics was replacedby Lagrangian and Hamiltonian Mechanics.From the Mathematical Point of View Einstein’ Special Relativitywas just a special case and even General Relativity was nothingnew (if we assume metrics given).

But Quantum Mechanics changed everything: states were no morepoints in phase space but wave functions and Observables whichwere functions on the phase space became operators acting onthem. The last transition is called quantization.






From Physics

To beginning of the 20-th century classical mechanics was in thestate of Full Power and Glory. Newtonian Mechanics was replacedby Lagrangian and Hamiltonian Mechanics.From the Mathematical Point of View Einstein’ Special Relativitywas just a special case and even General Relativity was nothingnew (if we assume metrics given).But Quantum Mechanics changed everything: states were no morepoints in phase space but wave functions and

Observables whichwere functions on the phase space became operators acting onthem. The last transition is called quantization.






From Physics

To beginning of the 20-th century classical mechanics was in thestate of Full Power and Glory. Newtonian Mechanics was replacedby Lagrangian and Hamiltonian Mechanics.From the Mathematical Point of View Einstein’ Special Relativitywas just a special case and even General Relativity was nothingnew (if we assume metrics given).But Quantum Mechanics changed everything: states were no morepoints in phase space but wave functions and Observables whichwere functions on the phase space became operators acting onthem.

The last transition is called quantization.






From Physics

To beginning of the 20-th century classical mechanics was in thestate of Full Power and Glory. Newtonian Mechanics was replacedby Lagrangian and Hamiltonian Mechanics.From the Mathematical Point of View Einstein’ Special Relativitywas just a special case and even General Relativity was nothingnew (if we assume metrics given).But Quantum Mechanics changed everything: states were no morepoints in phase space but wave functions and Observables whichwere functions on the phase space became operators acting onthem. The last transition is called quantization.






Hamiltonian Mechanics for Dummies

There is a configuration space X = Rd 3 x , phase spaceX = T ∗X ' R2d 3 (x , p), Hamiltonian H = H(x , p, t) (oftent-independent) and Hamilton equations

dxj

dt=

∂H

∂pj,

dpj

dt= −∂H

∂xj.

(1)

Here x are generalized coordinates and p generalized momenta. Inprinciple X could be a manifold and even








dxj

dt=

∂H

∂pj,

dpj

dt= −∂H

∂xj.

(1)

Here x are generalized coordinates and p generalized momenta.

Inprinciple X could be a manifold and even








dxj

dt=

∂H

∂pj,

dpj

dt= −∂H

∂xj.

(1)

Here x are generalized coordinates and p generalized momenta. Inprinciple X could be a manifold and even






there could be no X at all, just symplectic manifold F andequation

dz

dt= HH(z , t) (2)

on it where Hf is a Hamiltonian Field generated by f .

Evolution of any observable f (x , p, t) is given by equation

df

dt=∂f

∂t+ HH f =

∂f

∂t+ {H, f } (3)

with Poisson brackets

{H, f } =∑

j

(∂H

∂pj· ∂f

∂xj− ∂H

∂xj· ∂f

∂pj

). (4)






there could be no X at all, just symplectic manifold F andequation

dz

dt= HH(z , t) (2)

on it where Hf is a Hamiltonian Field generated by f .Evolution of any observable f (x , p, t) is given by equation

df

dt=∂f

∂t+ HH f =

∂f

∂t+ {H, f } (3)

with Poisson brackets

{H, f } =∑

j

(∂H

∂pj· ∂f

∂xj− ∂H

∂xj· ∂f

∂pj

). (4)






In particulardH

dt=∂H

∂t(5)

and H is preserved along trajectories provided it does not dependon t explicitely.

These equations survive if we apply symplectomorphism F → F ′.Symplectomorphism is a map preserving one of and thus allfollowing objects:

symplectic 2-form;

Hamiltonian field;

Poisson brackets.

Example: (x , p)→ (p,−x).






In particulardH

dt=∂H

∂t(5)

and H is preserved along trajectories provided it does not dependon t explicitely.These equations survive if we apply symplectomorphism F → F ′.

Symplectomorphism is a map preserving one of and thus allfollowing objects:

symplectic 2-form;

Hamiltonian field;

Poisson brackets.







In particulardH

dt=∂H

∂t(5)

and H is preserved along trajectories provided it does not dependon t explicitely.These equations survive if we apply symplectomorphism F → F ′.Symplectomorphism is a map preserving one of and thus allfollowing objects:

symplectic 2-form;

Hamiltonian field;

Poisson brackets.







Further, there is a function S = S(x) called action, which comesfrom Lagrangian mechanics (Least action principle is a foundationof it) and satisfies Hamilton-Jacobi equations

pj =∂S

∂xj, (6)

∂S

∂t= −H(x ,∇S , t). (7)

When changing (x , p)→ (p,−x) one needs also changeS →

∑j pjxj(p)− S where x(p) is found from (6). Interested? -

Go to course Mathematical Methods of Classical Mechanics,disguised as DE II







pj =∂S

∂xj, (6)

∂S

∂t= −H(x ,∇S , t). (7)


∑j pjxj(p)− S where x(p) is found from (6).

Interested? -Go to course Mathematical Methods of Classical Mechanics,disguised as DE II







pj =∂S

∂xj, (6)

∂S

∂t= −H(x ,∇S , t). (7)


∑j pjxj(p)− S where x(p) is found from (6). Interested? -

Go to course Mathematical Methods of Classical Mechanics,disguised as DE II






Going Quantum

Now instead of states which are points in F Q.M. looks for stateswhich are L2-functions on X : ψ ∈ H = L2(X ,C).

Sometimes instead of C it could be CD and because elements ofCD should be transformed when we change coordinates, we wouldeven have L2(E) the space of sections of the bundle over base X .Let us not to play with manifolds and set X = Rd . Reason: noexact quantization otherwise. So, H = L2(Rd ,C). And we need toquantize, setting for appropriate function f (x , p) operator Op(f ).






Going Quantum

Now instead of states which are points in F Q.M. looks for stateswhich are L2-functions on X : ψ ∈ H = L2(X ,C).Sometimes instead of C it could be CD and because elements ofCD should be transformed when we change coordinates, we wouldeven have L2(E) the space of sections of the bundle over base X .

Let us not to play with manifolds and set X = Rd . Reason: noexact quantization otherwise. So, H = L2(Rd ,C). And we need toquantize, setting for appropriate function f (x , p) operator Op(f ).






Going Quantum

Now instead of states which are points in F Q.M. looks for stateswhich are L2-functions on X : ψ ∈ H = L2(X ,C).Sometimes instead of C it could be CD and because elements ofCD should be transformed when we change coordinates, we wouldeven have L2(E) the space of sections of the bundle over base X .Let us not to play with manifolds and set X = Rd . Reason: noexact quantization otherwise.

So, H = L2(Rd ,C). And we need toquantize, setting for appropriate function f (x , p) operator Op(f ).






Going Quantum

Now instead of states which are points in F Q.M. looks for stateswhich are L2-functions on X : ψ ∈ H = L2(X ,C).Sometimes instead of C it could be CD and because elements ofCD should be transformed when we change coordinates, we wouldeven have L2(E) the space of sections of the bundle over base X .Let us not to play with manifolds and set X = Rd . Reason: noexact quantization otherwise. So, H = L2(Rd ,C). And we need toquantize, setting for appropriate function f (x , p) operator Op(f ).






Wishes:

1 If f is real-valued, then Op(f ) is selfadjoint.

Trouble: usually it is difficult to check if unboundedsymmetric operator is self-adjoint.Symmetric operator A is self-adjoint if (equivalent to standarddefinition) it has one of (and thus all) properties:

1 Spectrum (A) ⊂ R;2 A has spectral decomposition A =

∫λdλEA(λ) where EA(λ) is

it’s spectral projector;3 A generates unitary group e itA.

Compromise: Op(f ) must be symmetric but most importantOp(H) must be self-adjoint.






Wishes:

1 If f is real-valued, then Op(f ) is selfadjoint.Trouble: usually it is difficult to check if unboundedsymmetric operator is self-adjoint.

Symmetric operator A is self-adjoint if (equivalent to standarddefinition) it has one of (and thus all) properties:










Wishes:

1 If f is real-valued, then Op(f ) is selfadjoint.Trouble: usually it is difficult to check if unboundedsymmetric operator is self-adjoint.Symmetric operator A is self-adjoint if (equivalent to standarddefinition) it has one of (and thus all) properties:










Wishes:


2 Qj = Op(xj) is multiplication by xj .

3 Pj = Op(pj) = hDj , Dj = −i ∂∂xj.

h > 0 is a Plank constant (divided by 2π actually andPhysicists call it ~). If h� 1 quantum mechanics should bein some sense close to classical.What sense? We have rather different objects here! This is acrucial question!






Wishes:




4 Op(f + g) = Op(f ) + Op(g), Op(λf ) = λOp(f ).

5 Op(fg) = Op(f ) Op(g).Trouble: we will not be able to fulfill it exactly. Onlymod O(h).






Wishes:





5 Op(fg) = Op(f ) Op(g).

Trouble: we will not be able to fulfill it exactly. Onlymod O(h).






Wishes:





5 Op(fg) = Op(f ) Op(g).Trouble: we will not be able to fulfill it exactly. Onlymod O(h).






Wishes:





5 Op(fg) ≡ Op(f ) Op(g) mod O(h).

6 −ih Op({f , g}) = [Op(f ),Op(g)].Trouble: we will not be able to fulfill it exactly. Onlymod O(h2) or O(h3) if try well.






Wishes:






6 −ih Op({f , g}) = [Op(f ),Op(g)].

Trouble: we will not be able to fulfill it exactly. Onlymod O(h2) or O(h3) if try well.






Wishes:






6 −ih Op({f , g}) = [Op(f ),Op(g)].Trouble: we will not be able to fulfill it exactly. Onlymod O(h2) or O(h3) if try well.






Often Physicists want axiomatic theory. Instead of Qj and Pj

defined as above they write

[Qj ,Qk ] = 0, [Pj ,Pk ] = 0, [Pj ,Qk ] = −iδjk . (8)

Forgive them, Lord, because they do not know what they aretalking about!Unbounded self-adjoint operators are not your garden varietyHermitean bounded operators!Even equality [A,B] = 0 should be understood correctly. One cansay: this should be satisfied on some everywhere dense set. Wrong!









Forgive them, Lord, because they do not know what they aretalking about!

Unbounded self-adjoint operators are not your garden varietyHermitean bounded operators!Even equality [A,B] = 0 should be understood correctly. One cansay: this should be satisfied on some everywhere dense set. Wrong!









Forgive them, Lord, because they do not know what they aretalking about!Unbounded self-adjoint operators are not your garden varietyHermitean bounded operators!

Even equality [A,B] = 0 should be understood correctly. One cansay: this should be satisfied on some everywhere dense set. Wrong!









Forgive them, Lord, because they do not know what they aretalking about!Unbounded self-adjoint operators are not your garden varietyHermitean bounded operators!Even equality [A,B] = 0 should be understood correctly. One cansay: this should be satisfied on some everywhere dense set.

Wrong!









Forgive them, Lord, because they do not know what they aretalking about!Unbounded self-adjoint operators are not your garden varietyHermitean bounded operators!Even equality [A,B] = 0 should be understood correctly. One cansay: this should be satisfied on some everywhere dense set. Wrong!






Proper definition: one of this (and then all) must hold:

1 (A + λ)−1 and (B + µ)−1 commute ∀λ, µ not in spectra.

2 EA(λ) and EB(µ) commute ∀λ, µ ∈ R.

3 e itAe it′B = e it′Be itA ∀t, t ′ ∈ R

Similarly, (8) should be understood as

e itQj e it′Qk = e it′Qk e itQj ,

e itPj e it′Pk = e it′Pk e itPj ,

e itPj e it′Qk = e ihtt′δjk e it′Qk e itPj .

(9)

If (9) holds then system (Q1, . . . ,Qd ,P1, . . . ,Pd) is unitaryequivalent to our canonical system in L2(Rd ,K) with auixiliaryspace K.






Proper definition: one of this (and then all) must hold:

1 (A + λ)−1 and (B + µ)−1 commute ∀λ, µ not in spectra.

2 EA(λ) and EB(µ) commute ∀λ, µ ∈ R.

3 e itAe it′B = e it′Be itA ∀t, t ′ ∈ RSimilarly, (8) should be understood as

e itQj e it′Qk = e it′Qk e itQj ,

e itPj e it′Pk = e it′Pk e itPj ,

e itPj e it′Qk = e ihtt′δjk e it′Qk e itPj .

(9)

If (9) holds then system (Q1, . . . ,Qd ,P1, . . . ,Pd) is unitaryequivalent to our canonical system in L2(Rd ,K) with auixiliaryspace K.






Then Hermann Weyl suggested Weyl quantization:

Let we want to quantize f (x , p). Decompose it into Fourierintegral

f (x , p) =

∫∫f (τ, τ ′)e i〈x ,τ〉+i〈p,τ ′〉 dτ dτ ′

with

f = (2π)−2d

∫∫f (x , p)e−i〈x ,τ〉−i〈p,τ ′〉 dx dp.

Then

Opw(f ) =

∫∫f (τ, τ ′) e

12ih〈τ,τ ′〉e i〈Q,τ〉e i〈P,τ ′〉 dτ dτ ′ (10)

with w standing for Weyl.






Then Hermann Weyl suggested Weyl quantization:Let we want to quantize f (x , p). Decompose it into Fourierintegral

f (x , p) =

∫∫f (τ, τ ′)e i〈x ,τ〉+i〈p,τ ′〉 dτ dτ ′

with

f = (2π)−2d

∫∫f (x , p)e−i〈x ,τ〉−i〈p,τ ′〉 dx dp.

Then

Opw(f ) =

∫∫f (τ, τ ′) e

12ih〈τ,τ ′〉e i〈Q,τ〉e i〈P,τ ′〉 dτ dτ ′ (10)

with w standing for Weyl.






Note, if f depends on only one variable of each pair(x1, p1), . . . , (xd , pd) then Op(f ) is the standard function from thefamily of commuting operators. However, in general case Qj andPj do not commute.

Surely, physicists quantized some observablesbefore and without Weyl, but this is guess + luck.






Note, if f depends on only one variable of each pair(x1, p1), . . . , (xd , pd) then Op(f ) is the standard function from thefamily of commuting operators. However, in general case Qj andPj do not commute. Surely, physicists quantized some observablesbefore and without Weyl, but this is guess + luck.






From PDE

Originally it was a bit different approach but I will go moreconsistently with section 1.

Unfortunately, we sacrificeMikhlin-Zygmund-Calderon theory of singular integral operators!Consider differential operator

A =∑α

aα(x)(hD)α;

applied to function u; we differentiate first, multiply second.






From PDE

Originally it was a bit different approach but I will go moreconsistently with section 1. Unfortunately, we sacrificeMikhlin-Zygmund-Calderon theory of singular integral operators!

Consider differential operator

A =∑α

aα(x)(hD)α;







From PDE

Originally it was a bit different approach but I will go moreconsistently with section 1. Unfortunately, we sacrificeMikhlin-Zygmund-Calderon theory of singular integral operators!Consider differential operator

A =∑α

aα(x)(hD)α;







Then one can write

(Au)(x) =

∫a(x , hp)e i〈x ,p〉u(p) dp

= (2πh)−d

∫∫a(x , p)e ih−1〈x−y ,p〉u(y) dy dp

(11)

with a(x , p) =∑

α aα(x)pα. This is how Hormander in 1965defined it.






Then one can write

(Au)(x) =

∫a(x , hp)e i〈x ,p〉u(p) dp

= (2πh)−d

∫∫a(x , p)e ih−1〈x−y ,p〉u(y) dy dp (11)

with a(x , p) =∑

α aα(x)pα. This is how Hormander in 1965defined it.






Simultaneously, however, Kohn-Nirenberg started from

A =∑α

(hD)αaα(x);

applied to function u; we multiply first, differentiate second.

Thenone can write

(Au)(x) = (2πh)−d

∫∫a(y , p)e ih−1〈x−y ,p〉u(y) dy dp. (12)

To indicate, what is applied first, Hormander approach is

qp-quantization: A = a(2

x ,1

hD) and Kohn-Nirenberg approach is

pq-quantization: A = a(1

x ,2

hD).






Simultaneously, however, Kohn-Nirenberg started from

A =∑α

(hD)αaα(x);

applied to function u; we multiply first, differentiate second. Thenone can write

(Au)(x) = (2πh)−d

∫∫a(y , p)e ih−1〈x−y ,p〉u(y) dy dp. (12)

To indicate, what is applied first, Hormander approach is

qp-quantization: A = a(2

x ,1

hD) and Kohn-Nirenberg approach is

pq-quantization: A = a(1

x ,2

hD).






One can try to generalize both as A = a(3

x ,2

hD,1

x):

(Au)(x) = (2πh)−d

∫∫a(x , p, y)e ih−1〈x−y ,p〉u(y) dy dp (13)

getting in particular, Weyl quantization (called also symmetricquantization)

aw(x , hD)u =

∫∫a(

x + y

2, p)e ih−1〈x−y ,p〉u(y) dy dp; (14)

but while very useful technically (13) brings nothing new; in fact,all quantizations bring essentially the same set of operators.






Really, assuming that a is smooth and starting from (13), we candecompose

a(x , p, y) ∼∑α

1

α!(∂αy a)|y=x(y − x)α.

Plugging into (13), using(y − x)αe ih−1〈x−y ,p〉 = (ih∂p)αe ih−1〈x−y ,p〉 and integrating by parts

we get b′(2

x ,1

hD) with b′ ∼∑

α1α!(−ih)|α|∂αp ∂

αy a|y=x .

Similarly we can arrive to b′′(1

x ,2

hD) or bw(x , hD).From now on we assume that symbols depend on h:a(x , p, h) ∼

∑n≥0 an(x , p)hn; a0 is called principal symbol; it does

not depend on quantization method.







a(x , p, y) ∼∑α

1


Plugging into (13), using(y − x)αe ih−1〈x−y ,p〉 = (ih∂p)αe ih−1〈x−y ,p〉

and integrating by parts

we get b′(2

x ,1



αy a|y=x .


x ,2










a(x , p, y) ∼∑α

1



we get b′(2

x ,1



αy a|y=x .


x ,2










a(x , p, y) ∼∑α

1



we get b′(2

x ,1



αy a|y=x .


x ,2

hD) or bw(x , hD).

From now on we assume that symbols depend on h:a(x , p, h) ∼









a(x , p, y) ∼∑α

1



we get b′(2

x ,1



αy a|y=x .


x ,2








NormsInverse and functionsFunction in the Box

Calculus

Here Op means any method of quantization, Opw is for Weyl only.

1 Op(a + b) = Op(a) + Op(b), Op(λa) = λOp(a);

2 Opw(ab) = 12

(Opw(a) Opw(b) + Opw(b) Opw(a)

)+ O(h2);

3 Op({a, b}) = ih−1[Op(a),Op(b)

]+ O(h2);

4 Opw({a, b}) = ih−1[Opw(a),Opw(b)

]+ O(h3);

5 Opw(a†) =(Opw(a)

)∗;

6 Opw(a ◦ Φ) = F ∗Opw(a)F where(Fu)(p) = (2πh)−d/2

∫e−ih−1〈x ,p〉u(x) dx and

Φ : (x , p)→ (−p, x).

Physicists say: Going top-representation.In two last properties pq-quantization becomes qp- andconversely.






Calculus



2 Opw(ab) = 12


)+ O(h2);

3 Op({a, b}) = ih−1[Op(a),Op(b)

]+ O(h2);


]+ O(h3);


)∗;



Φ : (x , p)→ (−p, x). Physicists say: Going top-representation.

In two last properties pq-quantization becomes qp- andconversely.






Calculus



2 Opw(ab) = 12


)+ O(h2);

3 Op({a, b}) = ih−1[Op(a),Op(b)

]+ O(h2);


]+ O(h3);


)∗;



Φ : (x , p)→ (−p, x). Physicists say: Going top-representation.In two last properties pq-quantization becomes qp- andconversely.






Norms

Must skip it! But it justifies the rest!






Inverse and functions

Let a(x , p) be disjoint from ζ ∈ C. Then we can quatize(a(x , p)− ζ

)−1

and then

Op((a− ζ)−1

)(Op(a)− ζ

)= Op(1) + O(h)

=I + O(h).

One can invert the last thing if h is small enough.Actually, one can define b such that Op(b) = (Op(a)− ζ)−1.This means that (under some assumptions) Spectrum of Op(a) iscontained in Ch-vicinity of the range (set of possible values) of a.Converse statement is true as well: if ζ0 = a(x0, p0) thenSpectrum of Op(a) contains some point of Ch-vicinity of ζ0.








)−1and then

Op((a− ζ)−1

)(Op(a)− ζ

)= Op(1) + O(h)

=I + O(h).









)−1and then

Op((a− ζ)−1

)(Op(a)− ζ

)= Op(1) + O(h)

=I + O(h).

One can invert the last thing if h is small enough.

Actually, one can define b such that Op(b) = (Op(a)− ζ)−1.This means that (under some assumptions) Spectrum of Op(a) iscontained in Ch-vicinity of the range (set of possible values) of a.Converse statement is true as well: if ζ0 = a(x0, p0) thenSpectrum of Op(a) contains some point of Ch-vicinity of ζ0.








)−1and then

Op((a− ζ)−1

)(Op(a)− ζ

)= Op(1) + O(h)

=I + O(h).

One can invert the last thing if h is small enough.Actually, one can define b such that Op(b) = (Op(a)− ζ)−1.

This means that (under some assumptions) Spectrum of Op(a) iscontained in Ch-vicinity of the range (set of possible values) of a.Converse statement is true as well: if ζ0 = a(x0, p0) thenSpectrum of Op(a) contains some point of Ch-vicinity of ζ0.








)−1and then

Op((a− ζ)−1

)(Op(a)− ζ

)= Op(1) + O(h)

=I + O(h).

One can invert the last thing if h is small enough.Actually, one can define b such that Op(b) = (Op(a)− ζ)−1.This means that (under some assumptions) Spectrum of Op(a) iscontained in Ch-vicinity of the range (set of possible values) of a.

Converse statement is true as well: if ζ0 = a(x0, p0) thenSpectrum of Op(a) contains some point of Ch-vicinity of ζ0.








)−1and then

Op((a− ζ)−1

)(Op(a)− ζ

)= Op(1) + O(h)

=I + O(h).







Let a be bounded and C ⊃ D contain range of a. Let f be analyticin D.

Let γ be a closed path of C encircling properly D. Since for h� 1spectrum of A = Op(a) is contained in D

f (A) =1

2πi

∮γ

(ζ − A

)−1dζ (15)

= Op(b) with b0 = 12πi

∮γ(ζ − a0)−1 dζ = f (a0).

For self-adjoint unbounded operators situation is completelydifferent - see next section!






Let a be bounded and C ⊃ D contain range of a. Let f be analyticin D.Let γ be a closed path of C encircling properly D. Since for h� 1spectrum of A = Op(a) is contained in D

f (A) =1

2πi

∮γ

(ζ − A

)−1dζ (15)

= Op(b) with b0 = 12πi

∮γ(ζ − a0)−1 dζ = f (a0).

For self-adjoint unbounded operators situation is completelydifferent - see next section!






Function in the Box

Let us consider a function u. If it is supported in x-boxB ′ = {|x | ≤ γ} then it’s h-Fourier transform Fu is analyticfunction and is not localized.

Let us change notion “supported”. We consider function u,‖u‖ = 1 (norm means L2-norm); we say that u is contained in B ′ if‖u‖{|x |≥γ} ≤ Chs with fixed C and large exponents s.Could u be contained in B ′ = {|x | ≤ γ} and Fu contained inB ′′ = {|p| ≤ ρ}?






Function in the Box

Let us consider a function u. If it is supported in x-boxB ′ = {|x | ≤ γ} then it’s h-Fourier transform Fu is analyticfunction and is not localized.Let us change notion “supported”. We consider function u,‖u‖ = 1 (norm means L2-norm); we say that u is contained in B ′ if‖u‖{|x |≥γ} ≤ Chs with fixed C and large exponents s.

Could u be contained in B ′ = {|x | ≤ γ} and Fu contained inB ′′ = {|p| ≤ ρ}?






Function in the Box

Let us consider a function u. If it is supported in x-boxB ′ = {|x | ≤ γ} then it’s h-Fourier transform Fu is analyticfunction and is not localized.Let us change notion “supported”. We consider function u,‖u‖ = 1 (norm means L2-norm); we say that u is contained in B ′ if‖u‖{|x |≥γ} ≤ Chs with fixed C and large exponents s.Could u be contained in B ′ = {|x | ≤ γ} and Fu contained inB ′′ = {|p| ≤ ρ}?






The answer: yes, if (and only if)

ρ · γ ≥ Csh| log h|. (16)

It is logarithmic uncertainty principle.In quantum mechanics is known uncertainty principle:

ρ0 · γ0 ≥ h. (17)

with quadratic deviations γ0 = ‖xu‖ and ρ0 = ‖pFu‖.







ρ · γ ≥ Csh| log h|. (16)

It is logarithmic uncertainty principle.

In quantum mechanics is known uncertainty principle:

ρ0 · γ0 ≥ h. (17)








ρ · γ ≥ Csh| log h|. (16)

It is logarithmic uncertainty principle.In quantum mechanics is known uncertainty principle:

ρ0 · γ0 ≥ h. (17)






Schrodinger equationOscillatory SolutionsBlow-up!Stationary Phase MethodMaslov Canonical Operator

Schrodinger equation

In Quantum Mechanics evolution is described by non-stationarySchrodinger equation

hDtu = −Hwu (18)

where Hw = Hw(x , t, hD) is a quantum Hamiltonian;

often

Hw =1

2mh2|D|2 + V (x). (19)

and very often H does not depend on t. Assume this (notessential).






Schrodinger equation

In Quantum Mechanics evolution is described by non-stationarySchrodinger equation

hDtu = −Hwu (18)

where Hw = Hw(x , t, hD) is a quantum Hamiltonian; often

Hw =1

2mh2|D|2 + V (x). (19)

and very often H does not depend on t. Assume this (notessential).






Oscillatory Solutions

Consider oscillatory solution

u = e ih−1S(x ,t)A(x , t) (20)

with phase S(x , t) and amplitude A(x , t).

To plug it into (18) one needs to understand how operators act onsuch functions:






Oscillatory Solutions

Consider oscillatory solution

u = e ih−1S(x ,t)A(x , t) (20)

with phase S(x , t) and amplitude A(x , t).To plug it into (18) one needs to understand how operators act onsuch functions:






We get

Hw(e ih−1S(x ,t)A(x , t)

)=(H(x ,∇S)A+

h(∑

j

(∂pj H)(x ,∇S)∂xj + K)A +

h2L2A + h3L3A + . . .)e ih−1S(x ,t)

(21)

where

K = − i

2

∑j ,k

(∂2pjpk

H)(x ,∇S)∂2xjxk

S − i

2

∑j

(∂2xjpj

H)(x ,∇S) (22)

and all terms with gained factors h2, h3, . . . appeared withdifferential operators Lk of degrees k .






We get

Hw(e ih−1S(x ,t)A(x , t)

)=(H(x ,∇S)A+

h(∑

j

(∂pj H)(x ,∇S)∂xj + K)A +

h2L2A + h3L3A + . . .)e ih−1S(x ,t) (21)

where

K = − i

2

∑j ,k

(∂2pjpk

H)(x ,∇S)∂2xjxk

S − i

2

∑j

(∂2xjpj

H)(x ,∇S) (22)

and all terms with gained factors h2, h3, . . . appeared withdifferential operators Lk of degrees k .






Now, to satisfy (18) “in principal” we must kill terms with h0

assuming that∂tS + H(x ,∇S) = 0 (23)

which is exactly equation for action (7) from classical mechanics (Ijust assumed that H does not depend on t) and it is called eikonalequation because it first appeared in optics where S is calledeikonal.To kill terms with h we must assume that(

∂t +∑

j

(∂pj H)(x ,∇S)∂xj + K)A = 0 (24)








which is exactly equation for action (7) from classical mechanics (Ijust assumed that H does not depend on t)

and it is called eikonalequation because it first appeared in optics where S is calledeikonal.To kill terms with h we must assume that(

∂t +∑

j

(∂pj H)(x ,∇S)∂xj + K)A = 0 (24)








which is exactly equation for action (7) from classical mechanics (Ijust assumed that H does not depend on t) and it is called eikonalequation because it first appeared in optics where S is calledeikonal.

To kill terms with h we must assume that(∂t +

∑j

(∂pj H)(x ,∇S)∂xj + K)A = 0 (24)








which is exactly equation for action (7) from classical mechanics (Ijust assumed that H does not depend on t) and it is called eikonalequation because it first appeared in optics where S is calledeikonal.To kill terms with h we must assume that(

∂t +∑

j

(∂pj H)(x ,∇S)∂xj + K)A = 0 (24)






and remembering that in view of (23) (∂pj H)(x ,∇S) =dxj

dt alongtrajectories we get equation(d

dt+ K

)A = 0 (25)

which is called transport equation

and in fact means that|A|2 det(DΨt) is constant along trajectories, where Ψt : Rd → Rd

is defined in the following way:we define p = ∇S(x) as t = 0, then move for time t alongHamiltonian trajectories and define Ψt(x) as x-projection of theevolved point (x , p). Basically, it is an energy conservation law.What to do with other powers of h? Assuming as we did silently Hdoes not contain h, we will look at A = A0 + A1h + . . . where A0

is what we defined up to now,








dt+ K

)A = 0 (25)

which is called transport equation and in fact means that|A|2 det(DΨt) is constant along trajectories, where Ψt : Rd → Rd

is defined in the following way:

we define p = ∇S(x) as t = 0, then move for time t alongHamiltonian trajectories and define Ψt(x) as x-projection of theevolved point (x , p). Basically, it is an energy conservation law.What to do with other powers of h? Assuming as we did silently Hdoes not contain h, we will look at A = A0 + A1h + . . . where A0









dt+ K

)A = 0 (25)


is defined in the following way:we define p = ∇S(x) as t = 0, then move for time t alongHamiltonian trajectories and define Ψt(x) as x-projection of theevolved point (x , p).

Basically, it is an energy conservation law.What to do with other powers of h? Assuming as we did silently Hdoes not contain h, we will look at A = A0 + A1h + . . . where A0









dt+ K

)A = 0 (25)


is defined in the following way:we define p = ∇S(x) as t = 0, then move for time t alongHamiltonian trajectories and define Ψt(x) as x-projection of theevolved point (x , p). Basically, it is an energy conservation law.

What to do with other powers of h? Assuming as we did silently Hdoes not contain h, we will look at A = A0 + A1h + . . . where A0









dt+ K

)A = 0 (25)


is defined in the following way:we define p = ∇S(x) as t = 0, then move for time t alongHamiltonian trajectories and define Ψt(x) as x-projection of theevolved point (x , p). Basically, it is an energy conservation law.What to do with other powers of h?

Assuming as we did silently Hdoes not contain h, we will look at A = A0 + A1h + . . . where A0









dt+ K

)A = 0 (25)


is defined in the following way:we define p = ∇S(x) as t = 0, then move for time t alongHamiltonian trajectories and define Ψt(x) as x-projection of theevolved point (x , p). Basically, it is an energy conservation law.What to do with other powers of h? Assuming as we did silently Hdoes not contain h, we will look at A = A0 + A1h + . . . where A0







A1 is defined from(∂t +

∑j

(∂pj H)(x ,∇S)∂xj + K)A1 + L2A0 = 0

and so on . . .

problem to find a solution of equation (18) withinitial condition u|t=0 = e ih−1S0(x)A0(x) seems to be solved but, infact, fun only begins because happens







∑j

(∂pj H)(x ,∇S)∂xj + K)A1 + L2A0 = 0

and so on . . . problem to find a solution of equation (18) withinitial condition u|t=0 = e ih−1S0(x)A0(x) seems to be solved

but, infact, fun only begins because happens







∑j

(∂pj H)(x ,∇S)∂xj + K)A1 + L2A0 = 0

and so on . . . problem to find a solution of equation (18) withinitial condition u|t=0 = e ih−1S0(x)A0(x) seems to be solved but, infact, fun only begins because happens






Blow-up!

The problem is that S is not defined globally. In fact S isconstracted in the following way: we start from Ψt and along itstrajectories

dS =∑

j

pj dxj . (26)

The trouble is that while Hamiltonian flow Φt and thus map Ψt

are defined globally, at some moment det D(Ψt) can vanish and wecannot restore nicely point of origin and S from x .Not only S becomes non-smooth, amplidude A blows-up to infinity,and the place where it happens is called caustics because it wasobserved in optics first.






Blow-up!


dS =∑

j

pj dxj . (26)


are defined globally, at some moment det D(Ψt) can vanish and wecannot restore nicely point of origin and S from x .

Not only S becomes non-smooth, amplidude A blows-up to infinity,and the place where it happens is called caustics because it wasobserved in optics first.






Blow-up!


dS =∑

j

pj dxj . (26)


are defined globally, at some moment det D(Ψt) can vanish and wecannot restore nicely point of origin and S from x .Not only S becomes non-smooth, amplidude A blows-up to infinity,and the place where it happens is called caustics because it wasobserved in optics first.






And everything seems to be lost beyond the caustics too - becausethere is no justification there!

There are 2 equivalent ways to overcome the obstacle. Both arebased on






And everything seems to be lost beyond the caustics too - becausethere is no justification there!There are 2 equivalent ways to overcome the obstacle. Both arebased on






Stationary Phase Method

Consider oscillatory integral

IN =

∫e ih−1φ(θ)A(θ) dθ (27)

with θ ∈ RN .

Integrating by parts: If φ has no stationary point on supp A thenIN = O(hs) for any s.






Stationary Phase Method

Consider oscillatory integral

IN =

∫e ih−1φ(θ)A(θ) dθ (27)

with θ ∈ RN .Integrating by parts: If φ has no stationary point on supp A thenIN = O(hs) for any s.






Let φ have exactly one stationary point θ0 ∈ supp A and let θ0 beMorse point:

∇φ(θ0) = 0, det Hessφ(θ0) 6= 0. (28)

Then, starting from 1-dimensional integral∫ ∞−∞

e−kθ2/2 dθ =√

2πk−1/2

as Re k > 0 we get






IN = (2πh)N/2e ih−1φ(θ0)eiπ4κ(b0 + hb1 + h2b2 + . . .

)(29)

whereb0 = A(θ0)| det Hessφ(θ0)|−1/2 (30)

and κ = N − 2 sgn Hess(θ0), sgn M is the number of negativeeigenvalues of matrix M.






Maslov Canonical Operator

Near caustics we go to p-representation

Fu(p) = (2πh)−d/2

∫e ih−1(S(x)−〈p,x〉)A(x , h) dx (31)

where I skip t;

due to S.P.M.

Fu(p) = e ih−1(S(p)B(p, h) (32)

as long as det Hess S 6= 0 with S(p) = S(x(p))− 〈x(p), p〉 andx(p) is defined from p = ∇S(x).Look at this geometrically: consider manifoldΛ0 = {(x , p) : p = ∇S(x)}. It is Lagrangian manifold i.e. it isd-dimensional and symplectic form

∑j dxj ∧ dpj restricted to it is

0.








Fu(p) = (2πh)−d/2

∫e ih−1(S(x)−〈p,x〉)A(x , h) dx (31)

where I skip t; due to S.P.M.

Fu(p) = e ih−1(S(p)B(p, h) (32)

as long as det Hess S 6= 0 with S(p) = S(x(p))− 〈x(p), p〉 andx(p) is defined from p = ∇S(x).

Look at this geometrically: consider manifoldΛ0 = {(x , p) : p = ∇S(x)}. It is Lagrangian manifold i.e. it isd-dimensional and symplectic form


0.








Fu(p) = (2πh)−d/2

∫e ih−1(S(x)−〈p,x〉)A(x , h) dx (31)


Fu(p) = e ih−1(S(p)B(p, h) (32)

as long as det Hess S 6= 0 with S(p) = S(x(p))− 〈x(p), p〉 andx(p) is defined from p = ∇S(x).Look at this geometrically: consider manifoldΛ0 = {(x , p) : p = ∇S(x)}.

It is Lagrangian manifold i.e. it isd-dimensional and symplectic form


0.








Fu(p) = (2πh)−d/2

∫e ih−1(S(x)−〈p,x〉)A(x , h) dx (31)


Fu(p) = e ih−1(S(p)B(p, h) (32)

as long as det Hess S 6= 0 with S(p) = S(x(p))− 〈x(p), p〉 andx(p) is defined from p = ∇S(x).Look at this geometrically: consider manifoldΛ0 = {(x , p) : p = ∇S(x)}. It is Lagrangian manifold i.e. it isd-dimensional and symplectic form


0.Victor Ivrii Quantize!





As t changes Λ0 evolves into Lagrangian manifolld Λt = Φt(Λ0).

As long as x are good coordinates on Λt , S(x) exists. Thencaustics come. If d = 1 and x is no more good local coordinate, pis and in p-representation everything is well as long as p is a goodcoordinate; then we switch back to x etc.For d ≥ 2 one should consider coordinate systems (xI , pI ) with

I ⊂ {1, . . . , d} and complementary set I and mixed representations(partial Fourier transforms).This way we build “correct” representations of u locally on Λt andthen join them.Note that S(x) and its modifications SI (xI , pI ) are defineduniquely by Hamiltonian dynamics (we still skip t).






As t changes Λ0 evolves into Lagrangian manifolld Λt = Φt(Λ0).As long as x are good coordinates on Λt , S(x) exists. Thencaustics come.

If d = 1 and x is no more good local coordinate, pis and in p-representation everything is well as long as p is a goodcoordinate; then we switch back to x etc.For d ≥ 2 one should consider coordinate systems (xI , pI ) with







As t changes Λ0 evolves into Lagrangian manifolld Λt = Φt(Λ0).As long as x are good coordinates on Λt , S(x) exists. Thencaustics come. If d = 1 and x is no more good local coordinate, pis

and in p-representation everything is well as long as p is a goodcoordinate; then we switch back to x etc.For d ≥ 2 one should consider coordinate systems (xI , pI ) with







As t changes Λ0 evolves into Lagrangian manifolld Λt = Φt(Λ0).As long as x are good coordinates on Λt , S(x) exists. Thencaustics come. If d = 1 and x is no more good local coordinate, pis and in p-representation everything is well as long as p is a goodcoordinate;

then we switch back to x etc.For d ≥ 2 one should consider coordinate systems (xI , pI ) with







As t changes Λ0 evolves into Lagrangian manifolld Λt = Φt(Λ0).As long as x are good coordinates on Λt , S(x) exists. Thencaustics come. If d = 1 and x is no more good local coordinate, pis and in p-representation everything is well as long as p is a goodcoordinate; then we switch back to x etc.

For d ≥ 2 one should consider coordinate systems (xI , pI ) with







As t changes Λ0 evolves into Lagrangian manifolld Λt = Φt(Λ0).As long as x are good coordinates on Λt , S(x) exists. Thencaustics come. If d = 1 and x is no more good local coordinate, pis and in p-representation everything is well as long as p is a goodcoordinate; then we switch back to x etc.For d ≥ 2 one should consider coordinate systems (xI , pI ) with

I ⊂ {1, . . . , d} and complementary set I

and mixed representations(partial Fourier transforms).This way we build “correct” representations of u locally on Λt andthen join them.Note that S(x) and its modifications SI (xI , pI ) are defineduniquely by Hamiltonian dynamics (we still skip t).







I ⊂ {1, . . . , d} and complementary set I and mixed representations(partial Fourier transforms).

This way we build “correct” representations of u locally on Λt andthen join them.Note that S(x) and its modifications SI (xI , pI ) are defineduniquely by Hamiltonian dynamics (we still skip t).







I ⊂ {1, . . . , d} and complementary set I and mixed representations(partial Fourier transforms).This way we build “correct” representations of u locally on Λt andthen join them.

Note that S(x) and its modifications SI (xI , pI ) are defineduniquely by Hamiltonian dynamics (we still skip t).






So are A0,I (xI , pI ) but some factor.

Actually A0,I define a

half-density on Λ in the sense that A0,I | det Dλ(xI , pI )|1/2 are equal

where λ are some coordinate on Λ.This mysterious factor is e

iπ4κ with some integer κ. To understand

it consider x-representation just before caustic and just after it. Inboth cases ν = N − 2 sgn Hess S = d − 2 sgn Dxp where N is the

same. So passing through caustic we acquire e−iπ2ν where ν is an

increment of sgn Dxp. This leads to topological Maslov index andMaslov bundle.To unsderstand it better, consider the following static picture (theyare important in some problems) as d = 1:






So are A0,I (xI , pI ) but some factor. Actually A0,I define a


where λ are some coordinate on Λ.

This mysterious factor is eiπ4κ with some integer κ. To understand












iπ4κ with some integer κ.

To understandit consider x-representation just before caustic and just after it. Inboth cases ν = N − 2 sgn Hess S = d − 2 sgn Dxp where N is the












it consider x-representation just before caustic and just after it.

Inboth cases ν = N − 2 sgn Hess S = d − 2 sgn Dxp where N is the













same.

So passing through caustic we acquire e−iπ2ν where ν is an













increment of sgn Dxp.

This leads to topological Maslov index andMaslov bundle.To unsderstand it better, consider the following static picture (theyare important in some problems) as d = 1:












increment of sgn Dxp. This leads to topological Maslov index andMaslov bundle.

To unsderstand it better, consider the following static picture (theyare important in some problems) as d = 1:






A

B

C

D

Since dpdx > 0 on arcs AB and CD and dp

dx < 0 on arcs BC and DA,passing from AB to BC and from CD to DA adds 1 to Maslovindex.

So, Maslov index of the closed path ABCD is 2. Obviously,Maslov index is important only mod 4.






A

B

C

D


dx < 0 on arcs BC and DA,passing from AB to BC and from CD to DA adds 1 to Maslovindex. So, Maslov index of the closed path ABCD is 2.

Obviously,Maslov index is important only mod 4.






A

B

C

D


dx < 0 on arcs BC and DA,passing from AB to BC and from CD to DA adds 1 to Maslovindex. So, Maslov index of the closed path ABCD is 2. Obviously,Maslov index is important only mod 4.





Lagrangian distributionsFourier Integral OperatorsHeisenberg approach

Lagrangian distributions

I don’t want your stupidrepresentations-schmuresentations!I want u(x)!

Maslov: So what?

Duistermaat-Hormander: OK!You want it - you’ll get it!






So, converting from representations we get (the sum of) oscillatoryintegral

Ih = (2πh)−N/2

∫e ih−1φ(x ,θ)A(x , θ) dθ (33)

(compare to (27)).

Obviously only vicinity of manifold

Cφ = {(x , θ) : ∇θφ(x , θ) = 0} (34)

but instead of condition rank∇2θφ = d we assume that

rank∇θ x∇θφ = d . (35)






So, converting from representations we get (the sum of) oscillatoryintegral

Ih = (2πh)−N/2

∫e ih−1φ(x ,θ)A(x , θ) dθ (33)

(compare to (27)).Obviously only vicinity of manifold

Cφ = {(x , θ) : ∇θφ(x , θ) = 0} (34)

but instead of condition rank∇2θφ = d we assume that

rank∇θ x∇θφ = d . (35)






Λφ = {(x ,∇xφ(x , θ)) : (x , θ) ∈ Cφ} (36)

is a smooth Lagrangian manifold and

Ih(x) is L2-bounded.It is proven that Λφ rather than φ,N is important:if Λ′φ = Λφ then for each A(x , θ) exists A′(x , θ′) such that Ih ≡ I ′hmod O(hs). Inverse is also true.We call Ih Lagrangian distribution with Lagrangian manifoldΛ = Λφ.Further, PDO a(x , hD) applied to Lagrangian distribution givesLagrangian distribution with amplitude a(x ,∇xφ)A(modulo O(h)).






Λφ = {(x ,∇xφ(x , θ)) : (x , θ) ∈ Cφ} (36)

is a smooth Lagrangian manifold and Ih(x) is L2-bounded.

It is proven that Λφ rather than φ,N is important:if Λ′φ = Λφ then for each A(x , θ) exists A′(x , θ′) such that Ih ≡ I ′hmod O(hs). Inverse is also true.We call Ih Lagrangian distribution with Lagrangian manifoldΛ = Λφ.Further, PDO a(x , hD) applied to Lagrangian distribution givesLagrangian distribution with amplitude a(x ,∇xφ)A(modulo O(h)).






Λφ = {(x ,∇xφ(x , θ)) : (x , θ) ∈ Cφ} (36)

is a smooth Lagrangian manifold and Ih(x) is L2-bounded.It is proven that Λφ rather than φ,N is important:if Λ′φ = Λφ then for each A(x , θ) exists A′(x , θ′) such that Ih ≡ I ′hmod O(hs).

Inverse is also true.We call Ih Lagrangian distribution with Lagrangian manifoldΛ = Λφ.Further, PDO a(x , hD) applied to Lagrangian distribution givesLagrangian distribution with amplitude a(x ,∇xφ)A(modulo O(h)).






Λφ = {(x ,∇xφ(x , θ)) : (x , θ) ∈ Cφ} (36)

is a smooth Lagrangian manifold and Ih(x) is L2-bounded.It is proven that Λφ rather than φ,N is important:if Λ′φ = Λφ then for each A(x , θ) exists A′(x , θ′) such that Ih ≡ I ′hmod O(hs). Inverse is also true.

We call Ih Lagrangian distribution with Lagrangian manifoldΛ = Λφ.Further, PDO a(x , hD) applied to Lagrangian distribution givesLagrangian distribution with amplitude a(x ,∇xφ)A(modulo O(h)).






Λφ = {(x ,∇xφ(x , θ)) : (x , θ) ∈ Cφ} (36)

is a smooth Lagrangian manifold and Ih(x) is L2-bounded.It is proven that Λφ rather than φ,N is important:if Λ′φ = Λφ then for each A(x , θ) exists A′(x , θ′) such that Ih ≡ I ′hmod O(hs). Inverse is also true.We call Ih Lagrangian distribution with Lagrangian manifoldΛ = Λφ.

Further, PDO a(x , hD) applied to Lagrangian distribution givesLagrangian distribution with amplitude a(x ,∇xφ)A(modulo O(h)).






Λφ = {(x ,∇xφ(x , θ)) : (x , θ) ∈ Cφ} (36)

is a smooth Lagrangian manifold and Ih(x) is L2-bounded.It is proven that Λφ rather than φ,N is important:if Λ′φ = Λφ then for each A(x , θ) exists A′(x , θ′) such that Ih ≡ I ′hmod O(hs). Inverse is also true.We call Ih Lagrangian distribution with Lagrangian manifoldΛ = Λφ.Further, PDO a(x , hD) applied to Lagrangian distribution givesLagrangian distribution with amplitude a(x ,∇xφ)A(modulo O(h)).







If we want to consider general solutions to

hDtu = −Hwu (18)

we need to decompose u0 = u|t=0 into plane waves

u0(x) = (2πh)−d

∫∫e ih−1〈x−y ,ξ〉u(y) dξ dy , (37)

construct Lagrangian distribution solutions Ih(x , t, ξ) with initialdata e ih−1〈x−y ,ξ〉 and plug them into

u(x , t) = (2πh)−d

∫∫Ih(x , t, ξ)u(y) dξ dy , (38)








hDtu = −Hwu (18)


u0(x) = (2πh)−d

∫∫e ih−1〈x−y ,ξ〉u(y) dξ dy , (37)

construct Lagrangian distribution solutions Ih(x , t, ξ) with initialdata e ih−1〈x−y ,ξ〉

and plug them into

u(x , t) = (2πh)−d

∫∫Ih(x , t, ξ)u(y) dξ dy , (38)








hDtu = −Hwu (18)


u0(x) = (2πh)−d

∫∫e ih−1〈x−y ,ξ〉u(y) dξ dy , (37)

construct Lagrangian distribution solutions Ih(x , t, ξ) with initialdata e ih−1〈x−y ,ξ〉 and plug them into

u(x , t) = (2πh)−d

∫∫Ih(x , t, ξ)u(y) dξ dy , (38)






arriving to

u(x) = Fu0 =

∫K(x , y)u(y) dy (39)

with Schwartz kernel

K(x , y) = (2πh)−d−N/2

∫e ih−1φ(x ,y ,θ)A(x , y , θ) dθ (40)

which is Lagrangian distribution with Lagrangian manifold

Λ ={

(x , y ,∇xφ(x , y , θ),∇xφ(x , y , θ)), (x , y , θ) ∈ Cφ}, (41)

Cφ = {(x , y , θ),∇θφ(x , y , θ) = 0}. (42)

for symplectic form∑

dxj ∧ dpj −∑

dyj ∧ dqj . Note extra factor(2πh)−d !






arriving to

u(x) = Fu0 =

∫K(x , y)u(y) dy (39)


K(x , y) = (2πh)−d−N/2

∫e ih−1φ(x ,y ,θ)A(x , y , θ) dθ (40)


Λ ={


Cφ = {(x , y , θ),∇θφ(x , y , θ) = 0}. (42)


dxj ∧ dpj −∑

dyj ∧ dqj .

Note extra factor(2πh)−d !






arriving to

u(x) = Fu0 =

∫K(x , y)u(y) dy (39)


K(x , y) = (2πh)−d−N/2

∫e ih−1φ(x ,y ,θ)A(x , y , θ) dθ (40)


Λ ={


Cφ = {(x , y , θ),∇θφ(x , y , θ) = 0}. (42)


dxj ∧ dpj −∑

dyj ∧ dqj . Note extra factor(2πh)−d !






Assuming that

rank∇x ,θ∇θφ = rank∇y ,θ∇θ = d (x , y , θ) ∈ Cφ (43)

Λφ becomes a graph of symplectomorphism Φ : (y , q)→ (x , p); Λφis called then Canonical graph.






Under this assumption

1 Operator norm of F is bounded,

2 F1F2 corresponds to composition Φ1 ◦ Φ2 ofsymplectomorphisms,

3 F∗ corresponds to inverse symplectomorphism Φ−1,

4 F−1 if exists corresponds to inverse symplectomorphism Φ−1,

5 For given Φ we can chose unitary F ,

6 F is PDO iff Φ = Id ,

7 If A is PDO then F−1AF is also PDO with principal symbolA0 ◦ Φ.

8 So, basically FIOs are quantum analogues ofsymplectomorphisms.






Heisenberg approach

I am confused with all your oscillatoryintegralsI want something simpler!

Heisenberg to the rescue!Heisenberg: wave functions do not evolve!Observables do!






Heisenberg approach

I am confused with all your oscillatoryintegrals

I want something simpler!







Heisenberg approach








Heisenberg approach


Heisenberg to the rescue!

Heisenberg: wave functions do not evolve!Observables do!






Heisenberg approach


Heisenberg to the rescue!Heisenberg: wave functions do not evolve!

Observables do!






Heisenberg approach








According to our equation hDtu = −Hwu

u = e−ih−1tHwu0, u0 = u|t=0 (44)

and applying observable (i.e. operator) A we get

u = Ae−ih−1tHwu0 = e−ih−1tHw

Atu0, (45)

At = e ih−1tHwAe−ih−1tHw

. (46)

Then∂tA = ih−1[Hw,At ]. (47)






Assuming that At = awt is PDO and replacing [Hw, aw

t ] by−ih{H, a}w

we get∂ta + {H, at} = 0 (48)

which means exactly that at is conserved along trajectories ofHamiltonian system.

We can fix error in [Hw, awt ] ' −ih{H, a}w by taking in

account higher powers of h in at =∑

n≥0 an,thn - Trust me!

and justify assumption that At is PDO if A was - Trust me!

So we did everything without oscillatory integrals,

but one needs to know both approaches!






Assuming that At = awt is PDO and replacing [Hw, aw

t ] by−ih{H, a}w we get

∂ta + {H, at} = 0 (48)

which means exactly that at is conserved along trajectories ofHamiltonian system.

We can fix error in [Hw, awt ] ' −ih{H, a}w by taking in

account higher powers of h in at =∑

n≥0 an,thn - Trust me!

and justify assumption that At is PDO if A was - Trust me!

So we did everything without oscillatory integrals,

but one needs to know both approaches!






An End


Victor Ivriiweyl.math.toronto.edu/.../preprints/GradTalk2.pdf · 2017-08-05 · Why to bother?...

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Transcript of Victor Ivriiweyl.math.toronto.edu/.../preprints/GradTalk2.pdf · 2017-08-05 · Why to bother?...