Quantum-mechanical backflow and scattering Gandalf Lechner joint - - PowerPoint PPT Presentation

Quantum-mechanical backflow and scattering

Gandalf Lechner

joint work with Henning Bostelmann and Daniela Cadamuro

Backflow

• Quantum physics has some “strange”, counter-intuitive

effects (tunneling, uncertainty relations ....)

• “Backflow” is a (less known) quantum effect of this sort.

History:

• 1969: First description of the effect [Allcock]
• 1998: First quantitative analysis of backflow

[Bracken/Melloy]

• 2003: Backflow as a quantum inequality

[Eveson/Fewster/Verch]

• 2013: Suggestion for experimental observation

[Palmero et. al.]

1

Backflow

• Quantum physics has some “strange”, counter-intuitive

effects (tunneling, uncertainty relations ....)

• “Backflow” is a (less known) quantum effect of this sort.

History:

• 1969: First description of the effect [Allcock]
• 1998: First quantitative analysis of backflow

[Bracken/Melloy]

• 2003: Backflow as a quantum inequality

[Eveson/Fewster/Verch]

• 2013: Suggestion for experimental observation

[Palmero et. al.]

1

Backflow

• Quantum physics has some “strange”, counter-intuitive

effects (tunneling, uncertainty relations ....)

• “Backflow” is a (less known) quantum effect of this sort.

History:

• 1969: First description of the effect [Allcock]
• 1998: First quantitative analysis of backflow

[Bracken/Melloy]

• 2003: Backflow as a quantum inequality

[Eveson/Fewster/Verch]

• 2013: Suggestion for experimental observation

[Palmero et. al.]

1

Momentum and probability currents

Setting: A force-free single particle in one dimension.

• In a probabilistic description, suppose that at time t

0, the particle has momentum 0 (“right-mover”) with probability 1.

• What is the time-dependence of the probability L t that

the position of the particle is 0?

• Analogy (Bracken/Melloy): “Waiting for the bus”
• In classical physics, L t is non-increasing with growing t.
• In quantum physics, not necessarily!
• In this talk, only non-relativistic quantum mechanics

2

Momentum and probability currents

Setting: A force-free single particle in one dimension.

• In a probabilistic description, suppose that at time t = 0,

the particle has momentum > 0 (“right-mover”) with probability 1.

• What is the time-dependence of the probability L t that

the position of the particle is 0?

• Analogy (Bracken/Melloy): “Waiting for the bus”
• In classical physics, L t is non-increasing with growing t.
• In quantum physics, not necessarily!
• In this talk, only non-relativistic quantum mechanics

2

Momentum and probability currents

Setting: A force-free single particle in one dimension.

• In a probabilistic description, suppose that at time t = 0,

the particle has momentum > 0 (“right-mover”) with probability 1.

• What is the time-dependence of the probability L(t) that

the position of the particle is < 0?

• Analogy (Bracken/Melloy): “Waiting for the bus”
• In classical physics, L t is non-increasing with growing t.
• In quantum physics, not necessarily!
• In this talk, only non-relativistic quantum mechanics

2

Momentum and probability currents

Setting: A force-free single particle in one dimension.

• In a probabilistic description, suppose that at time t = 0,

the particle has momentum > 0 (“right-mover”) with probability 1.

• What is the time-dependence of the probability L(t) that

the position of the particle is < 0?

• Analogy (Bracken/Melloy): “Waiting for the bus”
• In classical physics, L t is non-increasing with growing t.
• In quantum physics, not necessarily!
• In this talk, only non-relativistic quantum mechanics

2

Momentum and probability currents

Setting: A force-free single particle in one dimension.

• In a probabilistic description, suppose that at time t = 0,

the particle has momentum > 0 (“right-mover”) with probability 1.

• What is the time-dependence of the probability L(t) that

the position of the particle is < 0?

• Analogy (Bracken/Melloy): “Waiting for the bus”
• In classical physics, L(t) is non-increasing with growing t.
• In quantum physics, not necessarily!
• In this talk, only non-relativistic quantum mechanics

2

Momentum and probability currents

Setting: A force-free single particle in one dimension.

• In a probabilistic description, suppose that at time t = 0,

the particle has momentum > 0 (“right-mover”) with probability 1.

• What is the time-dependence of the probability L(t) that

the position of the particle is < 0?

• Analogy (Bracken/Melloy): “Waiting for the bus”
• In classical physics, L(t) is non-increasing with growing t.
• In quantum physics, not necessarily!
• In this talk, only non-relativistic quantum mechanics

2

Plots of probability distribution of position with increasing time:

3

Plots of probability distribution of position with increasing time:

3

Plots of probability distribution of position with increasing time:

3

Plots of probability distribution of position with increasing time:

3

Plots of probability distribution of position with increasing time:

3

Plots of probability distribution of position with increasing time:

3

Plots of probability distribution of position with increasing time:

3

Plots of probability distribution of position with increasing time:

3

Plots of probability distribution of position with increasing time:

3

Plots of probability distribution of position with increasing time:

3

Plots of probability distribution of position with increasing time:

3

Plots of probability distribution of position with increasing time:

3

Plots of probability distribution of position with increasing time:

3

Plots of probability distribution of position with increasing time:

3

Plots of probability distribution of position with increasing time:

3

Plots of probability distribution of position with increasing time: .

3

Plots of probability distribution of position with increasing time: .

3

In “bus picture”, this means that the probability L(t) that the bus has not passed already can increase with waiting time t!

4

Mathematical setup

• Normalized wave function ψ ∈ L2(R) has probability density

∣ψ(x)∣2 (for position) and probability current density jψ(x) = i 2 (ψ′(x)ψ(x) − ψ(x)ψ′(x)) .

• j

is purely kinematical. When dynamics with Hamiltonian H is fixed, write je itH x jH t x .

• For example free Hamiltonian H0

p

1 2p2

p . Backflow is the fact that a b , where a) contains only positive momenta, i.e. supp b) j x 0 for all x

5

Mathematical setup

• Normalized wave function ψ ∈ L2(R) has probability density

∣ψ(x)∣2 (for position) and probability current density jψ(x) = i 2 (ψ′(x)ψ(x) − ψ(x)ψ′(x)) .

• jψ is purely kinematical. When dynamics with Hamiltonian H is

fixed, write je−itHψ(x) =∶ jH

ψ(t,x).

• For example free Hamiltonian ̃

(H0ψ)(p) = 1

2p2 ˜

ψ(p). Backflow is the fact that a b , where a) contains only positive momenta, i.e. supp b) j x 0 for all x

5

Mathematical setup

• Normalized wave function ψ ∈ L2(R) has probability density

∣ψ(x)∣2 (for position) and probability current density jψ(x) = i 2 (ψ′(x)ψ(x) − ψ(x)ψ′(x)) .

• jψ is purely kinematical. When dynamics with Hamiltonian H is

fixed, write je−itHψ(x) =∶ jH

ψ(t,x).

• For example free Hamiltonian ̃

(H0ψ)(p) = 1

2p2 ˜

ψ(p). Backflow is the fact that a) /

• ⇒ b), where

a) ψ contains only positive momenta, i.e. supp ˜ ψ ⊂ R+ b) jψ(x) > 0 for all x ∈ R

5

• Current quantum field (quadratic form)

⟨ψ,J(x)ψ⟩ ∶= jψ(x) = i 2 (ψ′(x)ψ(x) − ψ(x)ψ′(x))

• But for test functions f,

J f dxf x j x JH

temp f

dtf t jH t 0 exist as operators.

• Notation: E

projection onto positive/negative spectral subspace of momentum P (“right/left-movers”)

Lemma

For any x , the quadratic form E J x E is unbounded above and below.

• Unboundedness above: high momentum effect.
• Unboundedness below: take “backflow states” as

superposition of high and low (positive) momentum.

6

• Current quantum field (quadratic form)

⟨ψ,J(x)ψ⟩ ∶= jψ(x) = i 2 (ψ′(x)ψ(x) − ψ(x)ψ′(x))

• But for test functions f,

⟨ψ,J(f)ψ⟩ ∶= ∫ dxf(x)jψ(x), ⟨ψ,JH

temp(f)ψ⟩ ∶= ∫ dtf(t)jH ψ(t,0)

exist as operators.

• Notation: E

projection onto positive/negative spectral subspace of momentum P (“right/left-movers”)

Lemma

For any x , the quadratic form E J x E is unbounded above and below.

• Unboundedness above: high momentum effect.
• Unboundedness below: take “backflow states” as

superposition of high and low (positive) momentum.

6

• Current quantum field (quadratic form)

⟨ψ,J(x)ψ⟩ ∶= jψ(x) = i 2 (ψ′(x)ψ(x) − ψ(x)ψ′(x))

• But for test functions f,

⟨ψ,J(f)ψ⟩ ∶= ∫ dxf(x)jψ(x), ⟨ψ,JH

temp(f)ψ⟩ ∶= ∫ dtf(t)jH ψ(t,0)

exist as operators.

• Notation: E± = projection onto positive/negative spectral

subspace of momentum P (“right/left-movers”)

Lemma

For any x , the quadratic form E J x E is unbounded above and below.

• Unboundedness above: high momentum effect.
• Unboundedness below: take “backflow states” as

superposition of high and low (positive) momentum.

6

• Current quantum field (quadratic form)

⟨ψ,J(x)ψ⟩ ∶= jψ(x) = i 2 (ψ′(x)ψ(x) − ψ(x)ψ′(x))

• But for test functions f,

⟨ψ,J(f)ψ⟩ ∶= ∫ dxf(x)jψ(x), ⟨ψ,JH

temp(f)ψ⟩ ∶= ∫ dtf(t)jH ψ(t,0)

exist as operators.

• Notation: E± = projection onto positive/negative spectral

subspace of momentum P (“right/left-movers”)

Lemma

For any x ∈ R, the quadratic form E+J(x)E+ is unbounded above and below.

• Unboundedness above: high momentum effect.
• Unboundedness below: take “backflow states” as

superposition of high and low (positive) momentum.

6

Bounds on backflow

Backflow occurs, but is “bounded/small” in a specific sense:

• Amount of probability flowing across x

0 in time interval 0 T : inf

T 0 dtjP2 2 t 0

T 1 E 0 038 [Bracken/Melloy 98, Eveson/Fewster/Verch 03, ... ] no analytic formula for this “backflow constant” 3 8 exists.

• Spatial extent: For f

0, inf E J f E inf dxf x j x 1 E cf [Eveson/Fewster/Verch 03]

• Examples of “quantum inequalities” [Fewster]
• Known results are either about interaction-free situation, or of

purely kinematical nature.

7

Bounds on backflow

Backflow occurs, but is “bounded/small” in a specific sense:

• Amount of probability flowing across x = 0 in time interval [0,T]:

inf{∫

T 0 dtjP2/2 ψ

(t,0) ∶ T > 0, ∥ψ∥ = 1, ψ = E+ψ} ≈ −0.038 [Bracken/Melloy 98, Eveson/Fewster/Verch 03, ... ] no analytic formula for this “backflow constant” ≈ −3.8% exists.

• Spatial extent: For f

0, inf E J f E inf dxf x j x 1 E cf [Eveson/Fewster/Verch 03]

• Examples of “quantum inequalities” [Fewster]
• Known results are either about interaction-free situation, or of

purely kinematical nature.

7

Bounds on backflow

Backflow occurs, but is “bounded/small” in a specific sense:

• Amount of probability flowing across x = 0 in time interval [0,T]:

inf{∫

T 0 dtjP2/2 ψ

(t,0) ∶ T > 0, ∥ψ∥ = 1, ψ = E+ψ} ≈ −0.038 [Bracken/Melloy 98, Eveson/Fewster/Verch 03, ... ] no analytic formula for this “backflow constant” ≈ −3.8% exists.

• Spatial extent: For f ≥ 0,

infσ(E+J(f)E+) = inf{∫ dxf(x)jψ(x) ∶ ∥ψ∥ = 1, ψ = E+ψ} ≥ −cf > −∞ [Eveson/Fewster/Verch 03]

• Examples of “quantum inequalities” [Fewster]
• Known results are either about interaction-free situation, or of

purely kinematical nature.

7

Bounds on backflow

Backflow occurs, but is “bounded/small” in a specific sense:

• Amount of probability flowing across x = 0 in time interval [0,T]:

inf{∫

T 0 dtjP2/2 ψ

(t,0) ∶ T > 0, ∥ψ∥ = 1, ψ = E+ψ} ≈ −0.038 [Bracken/Melloy 98, Eveson/Fewster/Verch 03, ... ] no analytic formula for this “backflow constant” ≈ −3.8% exists.

• Spatial extent: For f ≥ 0,

infσ(E+J(f)E+) = inf{∫ dxf(x)jψ(x) ∶ ∥ψ∥ = 1, ψ = E+ψ} ≥ −cf > −∞ [Eveson/Fewster/Verch 03]

• Examples of “quantum inequalities” [Fewster]
• Known results are either about interaction-free situation, or of

purely kinematical nature.

7

Bounds on backflow

Backflow occurs, but is “bounded/small” in a specific sense:

• Amount of probability flowing across x = 0 in time interval [0,T]:

inf{∫

T 0 dtjP2/2 ψ

(t,0) ∶ T > 0, ∥ψ∥ = 1, ψ = E+ψ} ≈ −0.038 [Bracken/Melloy 98, Eveson/Fewster/Verch 03, ... ] no analytic formula for this “backflow constant” ≈ −3.8% exists.

• Spatial extent: For f ≥ 0,

infσ(E+J(f)E+) = inf{∫ dxf(x)jψ(x) ∶ ∥ψ∥ = 1, ψ = E+ψ} ≥ −cf > −∞ [Eveson/Fewster/Verch 03]

• Examples of “quantum inequalities” [Fewster]
• Known results are either about interaction-free situation, or of

purely kinematical nature.

7

Backflow and interaction (scattering)

• What can we say about backflow in an interacting system?
• For example, take Hamiltonian H = 1

2P2 + V(X) with potential V.

• For non-constant V, the space E+H of right-movers is not

preserved by time evolution e−itH.

• What means “backflow” in such a situation?
• To compare with free system, consider states with right-moving

asymptotics (say, incoming).

• sketches. What are the effects of reflection and transmission?
• Reflection = “classical backflow” (amplifies backflow). Is this

still a “small effect”?

• Interesting special case: Reflection-less potentials as

non-relativistic analogues of integrable QFTs. Do they have backflow?

8

Backflow and interaction (scattering)

• What can we say about backflow in an interacting system?
• For example, take Hamiltonian H = 1

2P2 + V(X) with potential V.

• For non-constant V, the space E+H of right-movers is not

preserved by time evolution e−itH.

• What means “backflow” in such a situation?
• To compare with free system, consider states with right-moving

asymptotics (say, incoming).

• sketches. What are the effects of reflection and transmission?
• Reflection = “classical backflow” (amplifies backflow). Is this

still a “small effect”?

• Interesting special case: Reflection-less potentials as

non-relativistic analogues of integrable QFTs. Do they have backflow?

8

Backflow and interaction (scattering)

• What can we say about backflow in an interacting system?
• For example, take Hamiltonian H = 1

2P2 + V(X) with potential V.

• For non-constant V, the space E+H of right-movers is not

preserved by time evolution e−itH.

• What means “backflow” in such a situation?
• To compare with free system, consider states with right-moving

asymptotics (say, incoming).

• sketches. What are the effects of reflection and transmission?
• Reflection = “classical backflow” (amplifies backflow). Is this

still a “small effect”?

• Interesting special case: Reflection-less potentials as

non-relativistic analogues of integrable QFTs. Do they have backflow?

8

Potential scattering in quantum mechanics

• Time-dependent setting: H = 1

2P2 + V(X) full Hamiltonian, H0 = 1 2P2 free

• Hamiltonian. Møller operator:

ΩV = s-lim

t→−∞ eitHe−itH0 ,

• exists under suitable regularity and short range assumptions on V
• maps “free” solution eikx of Schrödinger equation to “interacting”

solution with incoming asymptotics eikx.

Theorem

Let V L1 satisfy dx 1 x V x (“short range”). Then a)

V exists.

b)

1 2 2 x k x

V x

k x

k2

k x , k

0, has unique solution

k x

eikx RV k e ikx

• 1

x TV k eikx

• 1

x c) For C0 , have x

1 2

dk

k x

k

9

Potential scattering in quantum mechanics

• Time-dependent setting: H = 1

2P2 + V(X) full Hamiltonian, H0 = 1 2P2 free

• Hamiltonian. Møller operator:

ΩV = s-lim

t→−∞ eitHe−itH0 ,

• exists under suitable regularity and short range assumptions on V
• maps “free” solution eikx of Schrödinger equation to “interacting”

solution with incoming asymptotics eikx.

Theorem

Let V ∈ L1(R) satisfy ∫R dx (1 + ∣x∣)∣V(x)∣ < ∞ (“short range”). Then a) ΩV exists. b) − 1

2∂2 xϕk(x) + V(x)ϕk(x) = k2 ϕk(x), k > 0, has unique solution

ϕk(x) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ eikx + RV(k)e−ikx + o(1) x ≪ 0 TV(k)eikx + o(1) x ≫ 0 c) For ˜ ψ ∈ C∞

0 (R+), have (Ωψ)(x) = 1 √ 2π ∫ ∞

dk ϕk(x) ˜ ψ(k).

9

Backflow and scattering

The backflow in a scattering situation is measured by the operator, f ≥ 0, ⟨ψ,E+Ω∗

VJ(f)ΩVE+ψ⟩ = ∫ dxf(x)jΩVE+ψ(x).

and the quadratic form E+Ω∗

VJ(x)ΩVE+.

Theorem: Backflow exists in any (short-range) potential

Given arbitrary V L1 and arbitrary x , the quadratic form E

VJ x VE is unbounded above and below.

Idea of proof: Comparison with free case (“perturbation theory”). Relies on known estimates on the difference between

k x and its

asymptotics (e.g. [Deift/Trubowitz 79]).

10

Backflow and scattering

The backflow in a scattering situation is measured by the operator, f ≥ 0, ⟨ψ,E+Ω∗

VJ(f)ΩVE+ψ⟩ = ∫ dxf(x)jΩVE+ψ(x).

and the quadratic form E+Ω∗

VJ(x)ΩVE+.

Theorem: Backflow exists in any (short-range) potential

Given arbitrary V ∈ L1+(R) and arbitrary x ∈ R, the quadratic form E+Ω∗

VJ(x)ΩVE+ is unbounded above and below.

Idea of proof: Comparison with free case (“perturbation theory”). Relies on known estimates on the difference between

k x and its

asymptotics (e.g. [Deift/Trubowitz 79]).

10

Backflow and scattering

The backflow in a scattering situation is measured by the operator, f ≥ 0, ⟨ψ,E+Ω∗

VJ(f)ΩVE+ψ⟩ = ∫ dxf(x)jΩVE+ψ(x).

and the quadratic form E+Ω∗

VJ(x)ΩVE+.

Theorem: Backflow exists in any (short-range) potential

Given arbitrary V ∈ L1+(R) and arbitrary x ∈ R, the quadratic form E+Ω∗

VJ(x)ΩVE+ is unbounded above and below.

Idea of proof: Comparison with free case (“perturbation theory”). Relies on known estimates on the difference between φk(x) and its asymptotics (e.g. [Deift/Trubowitz 79]).

10

Backflow and scattering

To show that the averaged current E+Ω∗J(f)ΩE+ is bounded below, expand E+Ω∗J(f)ΩE+ = E+Ω∗E+J(f)E+ΩE+ + E+Ω∗E+J(f)(i + P)−1E−(i + P)(Ω − 1)E+ + E+(Ω∗ − 1)(i + P)E−(i + P)−1J(f)ΩE+.

• We have J f

i P

1

• Relevant term to estimate is P

1 .

11

Backflow and scattering

To show that the averaged current E+Ω∗J(f)ΩE+ is bounded below, expand E+Ω∗J(f)ΩE+ = E+Ω∗E+J(f)E+ΩE+ + E+Ω∗E+J(f)(i + P)−1E−(i + P)(Ω − 1)E+ + E+(Ω∗ − 1)(i + P)E−(i + P)−1J(f)ΩE+.

• We have ∥J(f)(i + P)−1∥ < ∞
• Relevant term to estimate is P(Ω − 1).

11

Mathematical question (work in progress)

Let H0,H be selfadjoint on a Hilbert space H, such that Møller

• perator Ω exists. For which (unbounded) functions g (e.g.

g(λ) = λα, g(λ) = eλ...) do we have ∥g(H0)(Ω − 1)∥ < ∞ ? Examples:

• L2

, H0 P, H P V X with V C0 . Then 1 H2 1 is unbounded for every 0.

• L2

n , H0

, H H0 integral operator with kernel K

n n and suppK compact. Then g H0

1 for all functions g. (similar to [GL/Verch 15]) In general – between these two extremes – a lot of relevant information seems to be encoded in the boundary behavior of the resolvents of the Hamiltonians (LAP).

12

Mathematical question (work in progress)

Let H0,H be selfadjoint on a Hilbert space H, such that Møller

• perator Ω exists. For which (unbounded) functions g (e.g.

g(λ) = λα, g(λ) = eλ...) do we have ∥g(H0)(Ω − 1)∥ < ∞ ? Examples:

• H = L2(R), H0 = P, H = P + V(X) with V ∈ C∞

0 (R). Then

(1 + H2

0)ε(Ω − 1) is unbounded for every ε > 0.

• L2

n , H0

, H H0 integral operator with kernel K

n n and suppK compact. Then g H0

1 for all functions g. (similar to [GL/Verch 15]) In general – between these two extremes – a lot of relevant information seems to be encoded in the boundary behavior of the resolvents of the Hamiltonians (LAP).

12

Mathematical question (work in progress)

Let H0,H be selfadjoint on a Hilbert space H, such that Møller

• perator Ω exists. For which (unbounded) functions g (e.g.

g(λ) = λα, g(λ) = eλ...) do we have ∥g(H0)(Ω − 1)∥ < ∞ ? Examples:

• H = L2(R), H0 = P, H = P + V(X) with V ∈ C∞

0 (R). Then

(1 + H2

0)ε(Ω − 1) is unbounded for every ε > 0.

• H = L2(Rn), H0 = −∆, H = H0+ integral operator with kernel

K ∈ S(Rn × Rn) and supp ˜ K compact. Then ∥g(H0)(Ω − 1)∥ < ∞ for all functions g. (similar to [GL/Verch 15]) In general – between these two extremes – a lot of relevant information seems to be encoded in the boundary behavior of the resolvents of the Hamiltonians (LAP).

12

Mathematical question (work in progress)

Let H0,H be selfadjoint on a Hilbert space H, such that Møller

• perator Ω exists. For which (unbounded) functions g (e.g.

g(λ) = λα, g(λ) = eλ...) do we have ∥g(H0)(Ω − 1)∥ < ∞ ? Examples:

• H = L2(R), H0 = P, H = P + V(X) with V ∈ C∞

0 (R). Then

(1 + H2

0)ε(Ω − 1) is unbounded for every ε > 0.

• H = L2(Rn), H0 = −∆, H = H0+ integral operator with kernel

K ∈ S(Rn × Rn) and supp ˜ K compact. Then ∥g(H0)(Ω − 1)∥ < ∞ for all functions g. (similar to [GL/Verch 15]) In general – between these two extremes – a lot of relevant information seems to be encoded in the boundary behavior of the resolvents of the Hamiltonians (LAP).

12

Backflow and scattering

In concrete 1d QM case (i.e. H = L2(R), H0 = 1

2P2, H = H0 + V(X)), use

integral form of Schrödinger eqn (Lippmann-Schwinger), φk(x) = TV(k)eikx + ∫ dyV(y)Gk(x − y)φk(y) and estimate (P(Ω − TV)E+ψ)(x) = 1 √ 2πi d dx ∫

∞

dk∫R dyV(y)Gk(x − y)φk(y) ˜ ψ(k) using known asymptotics of the solutions φk(x)

Theorem: Backflow is bounded in any (short-range) potential

Let V ∈ L1+(R) and f ≥ 0. Then E+Ω∗

VJ(f)ΩVE+ is bounded below.

• Reflection processes do not destroy boundedness of backflow.
• Heuristic explanation: Unboundedness below could only occur

at high momentum, but for high momentum, reflection processes are supressed sufficiently well.

13

Backflow and scattering

In concrete 1d QM case (i.e. H = L2(R), H0 = 1

2P2, H = H0 + V(X)), use

integral form of Schrödinger eqn (Lippmann-Schwinger), φk(x) = TV(k)eikx + ∫ dyV(y)Gk(x − y)φk(y) and estimate (P(Ω − TV)E+ψ)(x) = 1 √ 2πi d dx ∫

∞

dk∫R dyV(y)Gk(x − y)φk(y) ˜ ψ(k) using known asymptotics of the solutions φk(x)

Theorem: Backflow is bounded in any (short-range) potential

Let V ∈ L1+(R) and f ≥ 0. Then E+Ω∗

VJ(f)ΩVE+ is bounded below.

• Reflection processes do not destroy boundedness of backflow.
• Heuristic explanation: Unboundedness below could only occur

at high momentum, but for high momentum, reflection processes are supressed sufficiently well.

13

Examples and Numerics

Numerical results on specific potentials:

14

Examples and Numerics

Numerical results on specific potentials:

• Textbook example V(x) =square well potential.
• “backflow on the left ≥ free backflow ≥ backflow on the right”

14

Examples and Numerics

Numerical results on specific potentials:

• Transparent Pöschl-Teller potential V(x) = − ℓ(ℓ+1)

2 cosh2 x, ℓ ∈ N.

• “backflow on the left ≥ free backflow ≥ backflow on the right”

14

Examples and Numerics

Numerical results on specific potentials:

• Transparent Pöschl-Teller potential V(x) = − ℓ(ℓ+1)

2 cosh2 x, ℓ ∈ N.

• “backflow on the left ≥ free backflow ≥ backflow on the right”

14

Transparent potentials and integrable QFT

• A non-relativistic analogue of the elastic 2-body S-matrix of an

integrable QFT is the transmission coefficient TV of a transparent potential V

• In Pöschl-Teller potential V x

1 2 cosh2 x, have

TV k

n 1

k in k in S msinh i msinh i

Question

Can integrability of a QFT be characterized in terms of a quantum inequality?

15

Transparent potentials and integrable QFT

• A non-relativistic analogue of the elastic 2-body S-matrix of an

integrable QFT is the transmission coefficient TV of a transparent potential V

• In Pöschl-Teller potential V(x) = − ℓ(ℓ+1)

2 cosh2 x, have

TV(k) =

ℓ

∏

n=1

k + in k − in ↝ S(θ) = ∏

ν

msinhθ + iν msinhθ − iν

Question

Can integrability of a QFT be characterized in terms of a quantum inequality?

15

Transparent potentials and integrable QFT

• A non-relativistic analogue of the elastic 2-body S-matrix of an

integrable QFT is the transmission coefficient TV of a transparent potential V

• In Pöschl-Teller potential V(x) = − ℓ(ℓ+1)

2 cosh2 x, have

TV(k) =

ℓ

∏

n=1

k + in k − in ↝ S(θ) = ∏

ν

msinhθ + iν msinhθ − iν

Question

Can integrability of a QFT be characterized in terms of a quantum inequality?

15

Conclusion/Outlook

• In generic scattering situations, backflow always exists

and is always spatially bounded.

• Generalizations: Higher dimensions, multi-particle

systems, other PDEs, quantum field theory, ... require general analysis of bounds on Møller operators.

• Transparent potentials have special “backflow profiles”

(left and right asympotics “free”)

• As a QFT analogue, can integrability of a QFT be

characterized in terms of a quantum inequality?

16

Conclusion/Outlook

• In generic scattering situations, backflow always exists

and is always spatially bounded.

• Generalizations: Higher dimensions, multi-particle

systems, other PDEs, quantum field theory, ... require general analysis of bounds on Møller operators.

• Transparent potentials have special “backflow profiles”

(left and right asympotics “free”)

• As a QFT analogue, can integrability of a QFT be

characterized in terms of a quantum inequality?

16

Conclusion/Outlook

• In generic scattering situations, backflow always exists

and is always spatially bounded.

• Generalizations: Higher dimensions, multi-particle

systems, other PDEs, quantum field theory, ... require general analysis of bounds on Møller operators.

• Transparent potentials have special “backflow profiles”

(left and right asympotics “free”)

• As a QFT analogue, can integrability of a QFT be

characterized in terms of a quantum inequality?

16

Conclusion/Outlook

• In generic scattering situations, backflow always exists

and is always spatially bounded.

• Generalizations: Higher dimensions, multi-particle

systems, other PDEs, quantum field theory, ... require general analysis of bounds on Møller operators.

• Transparent potentials have special “backflow profiles”

(left and right asympotics “free”)

• As a QFT analogue, can integrability of a QFT be

characterized in terms of a quantum inequality?

16