Multipoint formulas for inverse scattering at high energies

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HAL Id: hal-02983682

https://hal.archives-ouvertes.fr/hal-02983682

Preprint submitted on 30 Oct 2020

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Multipoint formulas for inverse scattering at high energies

Roman Novikov

To cite this version:

Roman Novikov. Multipoint formulas for inverse scattering at high energies. 2020. �hal-02983682�

(2)

Multipoint formulas for inverse scattering at high energies

R.G. Novikov

CMAP, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau, France;

IEPT RAS, 117997 Moscow, Russia e-mail: novikov@cmap.polytechnique.fr

We consider the Schr¨ odinger equation

− ∆ψ + v(x)ψ = Eψ, x ∈ R ^d , d ≥ 1, E > 0, (1) where v is complex-valued,

v ∈ C c ^∞ ( R ^d ), (2)

where C c ^∞ denotes inﬁnitely smooth compactly supported functions.

For equation (1) we consider the scattering solutions ψ ⁺ (x, k) = e ^ikx + ψ ^sc (x, k), k ∈ R ^d , k ² = E, where ψ ^sc satisﬁes the Sommerfeld radiation condition:

| x | ^(d ⁻ ^1)/2 ( ∂

∂ | x | − i | k | )ψ ^sc (x, k) → 0 as | x | → + ∞ (3) uniformly in x/ | x | . This implies that

ψ ^sc (x, k) = e ⁱ ^| ^k ^|| ^x ^|

| x | ^(d ⁻ ^1)/2 f 1 (k, | k | x

| x | ) + O ( 1

| x | ^(d+1)/2

) , | x | → + ∞ , (4)

where f 1 is the scattering amplitude for equation (1). For more details about deﬁnitions of ψ ⁺ and f 1 , see, e.g., [BSh], [N2] and references therein.

It is convenient to represent f ₁ as follows

f 1 = c(d, | k | )f(θ, ω, E), (θ, ω) ∈ S ^d ⁻ ¹ × S ^d ⁻ ¹ , where (5) c(d, | k | ) = − πi( − 2πi) ^(d ⁻ ^1)/2 | k | ^(d ⁻ ^3)/2 , θ = k/ | k | , ω = x/ | x | .

In order to formulate our results we also use the following notations:

ˆ

v(p) = (2π) ⁻ ^d

∫ R

^d

e ^ipx v(x)dx, p ∈ R ^d , (6)

ω ^⊥ = { p ∈ R ^d : pω = 0 } , ω ∈ S ^d ⁻ ¹ , (7)

θ(p, ω, E) = E ⁻ ^1/2 (p + (E − p ² ) ^1/2 ω), p ∈ ω ^⊥ , ω ∈ S ^d−1 , E ^1/2 > 0, (8)

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R.G. Novikov

α j ( ⃗ ξ) =

j ∏ − 1

i=1

(ξ j − ξ i ) for 1 < j ≤ n, α 1 ( ξ) = 1, ⃗

β _n,j ( ⃗ ξ) =

∏ n

i=j+1

(ξ _i − ξ _j ) for 1 ≤ j < n, β _n,n ( ⃗ ξ) = 1,

(9)

where ⃗ ξ = (ξ 1 , . . . , ξ n ).

Theorem 1. Let v satisfy (2). Then the following formulas hold:

ˆ v(p) =

∑ n

j=1

( − 1) ⁿ ⁻ ^j (s + τ j ) ⁿ ⁻ ¹ f (θ j (s), ω, E j (s))

α j (⃗ τ )β n,j (⃗ τ) + O(s ⁻ ⁿ ) as s → + ∞ , θ j (s) = θ(p, ω, E j (s)), E j (s) = (s + τ j ) ² , s > 0,

⃗

τ = (τ 1 , . . . , τ n ), τ 1 = 0, τ j

1

< τ j

2

for j 1 < j 2 ,

(10a)

ˆ v(p) =

∑ n

j=1

( − 1) ⁿ ⁻ ^j λ ⁿ⁻¹ _j f (θ j (s), ω, E j (s))

α j ( ⃗ λ)β n,j ( ⃗ λ) + O(s ⁻ ⁿ ) as s → + ∞ , θ _j (s) = θ(p, ω, E _j (s)), E _j (s) = (λ _j s) ² , s > 0,

⃗ λ = (λ ₁ , . . . , λ _n ), λ ₁ = 1, λ _j

₁

< λ _j

₂

for j ₁ < j ₂ ,

(10b)

where p ∈ ω ^⊥ , ω ∈ S ^d ⁻ ¹ (and ω, p are ﬁxed).

Formulas (10a), (10b) are explicit asymptotic formulas for ﬁnding the Fourier trans- form ˆ v(p) at ﬁxed p ∈ R ^d , d ≥ 2, from the scattering amplitude f at n points at high energies E 1 , . . . , E n . The precision of these formulas is O(s ⁻ ⁿ ) as s → + ∞ and in this sense is proportional to n. To our knowledge these formulas are new for n ≥ 2. For n = 1, formulas (10a), (10b) are a known variation of the Born formula at high energies for smooth v; see, e.g., Proposition 3.4 of [M] and formula (5.1) of [N1]. (For n = 1, formulas (10a) and (10b) coincide.)

Formulas (10a) and (10b) follow from Proposition 3.4 of [M] about the asymptotic expansion of f (θ, ω, s ² ) as s → + ∞ , for s(θ − (θω)ω) = p, and from applying to this expansion Theorems 3.1 and 3.2 of [N2].

Acknowledgements

The author is partially supported by the PRC n 2795 CNRS/RFBR: Probl` emes in- verses et int´ egrabilit´ e.

References

[BSh] F.A. Berezin, M.A. Shubin, The Schr¨ odinger Equation, Vol. 66 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, 1991.

[ M] R.B. Melrose, Geometric Scattering Theory. Stanford Lectures. Cambridge University Press, 1995.

2

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Multipoint formulas for scattered far ﬁeld

[ N1] R.G. Novikov, Multidimensional inverse scattering for the Schr¨ odinger equation, Book series: Springer Proceedings in Mathematics and Statistics. Title of volume: Math- ematical Analysis, its Applications and Computation - ISAAC 2019, Aveiro, Por- tugal, July 29-August 2; Editors: P. Cerejeiras, M. Reissig (to appear), e-preprint:

https://hal.archives-ouvertes.fr/hal-02465839v1

[ N2] R.G. Novikov, Multipoint formulas for scattered far ﬁeld in multidimensions, Inverse Problems 36(9), 095001 (2020)

3

Multipoint formulas for inverse scattering at high energies

HAL Id: hal-02983682

https://hal.archives-ouvertes.fr/hal-02983682

Preprint submitted on 30 Oct 2020

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Multipoint formulas for inverse scattering at high energies

Roman Novikov

To cite this version:

Roman Novikov. Multipoint formulas for inverse scattering at high energies. 2020. �hal-02983682�

Multipoint formulas for inverse scattering at high energies

R.G. Novikov

CMAP, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau, France;

IEPT RAS, 117997 Moscow, Russia e-mail: novikov@cmap.polytechnique.fr

We consider the Schr¨ odinger equation

− ∆ψ + v(x)ψ = Eψ, x ∈ R d , d ≥ 1, E > 0, (1) where v is complex-valued,

v ∈ C c ∞ ( R d ), (2)

where C c ∞ denotes inﬁnitely smooth compactly supported functions.

For equation (1) we consider the scattering solutions ψ + (x, k) = e ikx + ψ sc (x, k), k ∈ R d , k 2 = E, where ψ sc satisﬁes the Sommerfeld radiation condition:

| x | (d − 1)/2 ( ∂

∂ | x | − i | k | )ψ sc (x, k) → 0 as | x | → + ∞ (3) uniformly in x/ | x | . This implies that

ψ sc (x, k) = e i | k || x |

| x | (d − 1)/2 f 1 (k, | k | x

| x | ) + O ( 1

| x | (d+1)/2

) , | x | → + ∞ , (4)

where f 1 is the scattering amplitude for equation (1). For more details about deﬁnitions of ψ + and f 1 , see, e.g., [BSh], [N2] and references therein.

It is convenient to represent f 1 as follows

f 1 = c(d, | k | )f(θ, ω, E), (θ, ω) ∈ S d − 1 × S d − 1 , where (5) c(d, | k | ) = − πi( − 2πi) (d − 1)/2 | k | (d − 3)/2 , θ = k/ | k | , ω = x/ | x | .

In order to formulate our results we also use the following notations:

ˆ

v(p) = (2π) − d

∫ R

e ipx v(x)dx, p ∈ R d , (6)

ω ⊥ = { p ∈ R d : pω = 0 } , ω ∈ S d − 1 , (7)

θ(p, ω, E) = E − 1/2 (p + (E − p 2 ) 1/2 ω), p ∈ ω ⊥ , ω ∈ S d−1 , E 1/2 > 0, (8)

R.G. Novikov

α j ( ⃗ ξ) =

j ∏ − 1

i=1

(ξ j − ξ i ) for 1 < j ≤ n, α 1 ( ξ) = 1, ⃗

β n,j ( ⃗ ξ) =

∏ n

i=j+1

(ξ i − ξ j ) for 1 ≤ j < n, β n,n ( ⃗ ξ) = 1,

(9)

where ⃗ ξ = (ξ 1 , . . . , ξ n ).

Theorem 1. Let v satisfy (2). Then the following formulas hold:

ˆ v(p) =

∑ n

j=1

( − 1) n − j (s + τ j ) n − 1 f (θ j (s), ω, E j (s))

α j (⃗ τ )β n,j (⃗ τ) + O(s − n ) as s → + ∞ , θ j (s) = θ(p, ω, E j (s)), E j (s) = (s + τ j ) 2 , s > 0,

⃗

τ = (τ 1 , . . . , τ n ), τ 1 = 0, τ j

< τ j

for j 1 < j 2 ,

(10a)

ˆ v(p) =

∑ n

j=1

( − 1) n − j λ n−1 j f (θ j (s), ω, E j (s))

α j ( ⃗ λ)β n,j ( ⃗ λ) + O(s − n ) as s → + ∞ , θ j (s) = θ(p, ω, E j (s)), E j (s) = (λ j s) 2 , s > 0,

⃗ λ = (λ 1 , . . . , λ n ), λ 1 = 1, λ j

< λ j

for j 1 < j 2 ,

(10b)

where p ∈ ω ⊥ , ω ∈ S d − 1 (and ω, p are ﬁxed).

Formulas (10a) and (10b) follow from Proposition 3.4 of [M] about the asymptotic expansion of f (θ, ω, s 2 ) as s → + ∞ , for s(θ − (θω)ω) = p, and from applying to this expansion Theorems 3.1 and 3.2 of [N2].

Acknowledgements

The author is partially supported by the PRC n 2795 CNRS/RFBR: Probl` emes in- verses et int´ egrabilit´ e.

References

[BSh] F.A. Berezin, M.A. Shubin, The Schr¨ odinger Equation, Vol. 66 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, 1991.

[ M] R.B. Melrose, Geometric Scattering Theory. Stanford Lectures. Cambridge University Press, 1995.

2

Multipoint formulas for scattered far ﬁeld

https://hal.archives-ouvertes.fr/hal-02465839v1

[ N2] R.G. Novikov, Multipoint formulas for scattered far ﬁeld in multidimensions, Inverse Problems 36(9), 095001 (2020)

3

− ∆ψ + v(x)ψ = Eψ, x ∈ R ^d , d ≥ 1, E > 0, (1) where v is complex-valued,

v ∈ C c ^∞ ( R ^d ), (2)

where C c ^∞ denotes inﬁnitely smooth compactly supported functions.

For equation (1) we consider the scattering solutions ψ ⁺ (x, k) = e ^ikx + ψ ^sc (x, k), k ∈ R ^d , k ² = E, where ψ ^sc satisﬁes the Sommerfeld radiation condition:

| x | ^(d ⁻ ^1)/2 ( ∂

∂ | x | − i | k | )ψ ^sc (x, k) → 0 as | x | → + ∞ (3) uniformly in x/ | x | . This implies that

ψ ^sc (x, k) = e ⁱ ^| ^k ^|| ^x ^|

| x | ^(d ⁻ ^1)/2 f 1 (k, | k | x

| x | ^(d+1)/2

where f 1 is the scattering amplitude for equation (1). For more details about deﬁnitions of ψ ⁺ and f 1 , see, e.g., [BSh], [N2] and references therein.

It is convenient to represent f ₁ as follows

f 1 = c(d, | k | )f(θ, ω, E), (θ, ω) ∈ S ^d ⁻ ¹ × S ^d ⁻ ¹ , where (5) c(d, | k | ) = − πi( − 2πi) ^(d ⁻ ^1)/2 | k | ^(d ⁻ ^3)/2 , θ = k/ | k | , ω = x/ | x | .

v(p) = (2π) ⁻ ^d

e ^ipx v(x)dx, p ∈ R ^d , (6)

ω ^⊥ = { p ∈ R ^d : pω = 0 } , ω ∈ S ^d ⁻ ¹ , (7)

θ(p, ω, E) = E ⁻ ^1/2 (p + (E − p ² ) ^1/2 ω), p ∈ ω ^⊥ , ω ∈ S ^d−1 , E ^1/2 > 0, (8)

β _n,j ( ⃗ ξ) =

(ξ _i − ξ _j ) for 1 ≤ j < n, β _n,n ( ⃗ ξ) = 1,

( − 1) ⁿ ⁻ ^j (s + τ j ) ⁿ ⁻ ¹ f (θ j (s), ω, E j (s))

α j (⃗ τ )β n,j (⃗ τ) + O(s ⁻ ⁿ ) as s → + ∞ , θ j (s) = θ(p, ω, E j (s)), E j (s) = (s + τ j ) ² , s > 0,

( − 1) ⁿ ⁻ ^j λ ⁿ⁻¹ _j f (θ j (s), ω, E j (s))

α j ( ⃗ λ)β n,j ( ⃗ λ) + O(s ⁻ ⁿ ) as s → + ∞ , θ _j (s) = θ(p, ω, E _j (s)), E _j (s) = (λ _j s) ² , s > 0,

⃗ λ = (λ ₁ , . . . , λ _n ), λ ₁ = 1, λ _j

< λ _j

for j ₁ < j ₂ ,

where p ∈ ω ^⊥ , ω ∈ S ^d ⁻ ¹ (and ω, p are ﬁxed).

Formulas (10a) and (10b) follow from Proposition 3.4 of [M] about the asymptotic expansion of f (θ, ω, s ² ) as s → + ∞ , for s(θ − (θω)ω) = p, and from applying to this expansion Theorems 3.1 and 3.2 of [N2].