Sharp asymptotic estimates for eigenvalues of Aharonov ... · Veronica Felli University of...

Sharp asymptotic estimates for eigenvalues ofAharonov-Bohm operators with varying poles

Veronica Felli

University of Milano-Bicocca

Workshop in Nonlinear PDEs

joint works with L. AbatangeloarXiv:1504.00252, arXiv:1505.05280

Veronica Felli AB-operators with varying poles September 10, 2015 1 / 27

Introduction The Aharonov-Bohm operator

The Aharonov-Bohm operator

For a = (a1, a2) ∈ R2 and γ ∈ R \ Z, we consider the vector potential

Aa(x1, x2) = γ

(−(x2 − a2)

(x1 − a1)2 + (x2 − a2)2,

x1 − a1

(x1 − a1)2 + (x2 − a2)2

(x1, x2) ∈ R2 \ a. Aa generates the Aharonov-Bohm magnetic field inR2 with pole a and circulation γ.

The AB magnetic field is produced by an infinitely long thin solenoidintersecting perpendicularly the plane (x1, x2) at the point a, as the radiusof the solenoid goes to zero and the magnetic flux remains constantlyequal to γ. Negletting the irrelevant coordinate along the solenoid, theproblem becomes 2-dimensional.

Introduction The Aharonov-Bohm operator

The Aharonov-Bohm operator

For a = (a1, a2) ∈ R2 and γ ∈ R \ Z, we consider the vector potential

Aa(x1, x2) = γ

(−(x2 − a2)

(x1 − a1)2 + (x2 − a2)2,

x1 − a1

(x1 − a1)2 + (x2 − a2)2

(x1, x2) ∈ R2 \ a. Aa generates the Aharonov-Bohm magnetic field inR2 with pole a and circulation γ.

Aharonov-Bohm effect [Y. Aharonov, D. Bohm, Phys. Rev. (1959)]

The AB magnetic field is a δ-like magnetic field: a quantum particle movingin R2 \ a is affected by the magnetic potential, despite being confined toa region in which the magnetic field is zero.

Introduction The magnetic eigenvalue problem

The magnetic eigenvalue problem

Let us consider the Schrodinger operator with AB vector potential

(i∇+Aa)2u = −∆u+ 2iAa · ∇u+ |Aa|2u.

In a bounded, open and simply connected domain Ω ⊂ R2, ∀ a ∈ Ω, weconsider the eigenvalue problem

(i∇+Aa)2u = λu, in Ω,

u = 0, on ∂Ω,(Ea)

in a weak sense.

The functional setting

For every a ∈ Ω, H1,a(Ω,C) is the completion ofu ∈ H1(Ω,C) ∩ C∞(Ω,C) : u vanishes in a neighborhood of a w.r.t.

‖u‖H1,a(Ω,C) =(‖∇u‖2L2(Ω,C2) + ‖u‖2L2(Ω,C) +

∥∥ u|x−a|

∥∥2

L2(Ω,C)

which is equivalent to(‖(i∇+Aa)u‖2L2(Ω,C2) + ‖u‖2L2(Ω,C)

Hardy type inequality [A. Laptev, T. Weidl (1999)]∫Dr(a)

|(i∇+Aa)u|2 dx ≥(

minj∈Z|j − γ|

)2 ∫Dr(a)

|u(x)|2

|x− a|2dx,

for all r > 0, a ∈ R2 and u ∈ H1,a(Dr(a),C)

Let H1,a0 (Ω,C) =

u ∈ H1

0 (Ω,C) : u|x−a| ∈ L

2(Ω,C)

‖u‖H1,a(Ω,C) =(‖∇u‖2L2(Ω,C2) + ‖u‖2L2(Ω,C) +

∥∥ u|x−a|

∥∥2

L2(Ω,C)

|(i∇+Aa)u|2 dx ≥(

minj∈Z|j − γ|

)2 ∫Dr(a)

|u(x)|2

|x− a|2dx,

Let H1,a0 (Ω,C) =

u ∈ H1

0 (Ω,C) : u|x−a| ∈ L

2(Ω,C)

‖u‖H1,a(Ω,C) =(‖∇u‖2L2(Ω,C2) + ‖u‖2L2(Ω,C) +

∥∥ u|x−a|

∥∥2

L2(Ω,C)

|(i∇+Aa)u|2 dx ≥(

minj∈Z|j − γ|

)2 ∫Dr(a)

|u(x)|2

|x− a|2dx,

Let H1,a0 (Ω,C) =

u ∈ H1

0 (Ω,C) : u|x−a| ∈ L

2(Ω,C)

The eigenvalues

Classical spectral theory (Ea) admits a sequence of real divergingeigenvalues λakk≥1 with finite multiplicity

λa1 ≤ λa2 ≤ · · · ≤ λaj ≤ . . . (repeated according to their multiplicity)

Problem:

study the behavior of the function

a 7→ λaj

in a neighborhood of a fixed point b ∈ Ω.

Up to a translation, it is not restrictive to consider b = 0 ∈ Ω.

The eigenvalues

Classical spectral theory (Ea) admits a sequence of real divergingeigenvalues λakk≥1 with finite multiplicity

λa1 ≤ λa2 ≤ · · · ≤ λaj ≤ . . . (repeated according to their multiplicity)

Problem:

study the behavior of the function

a 7→ λaj

in a neighborhood of a fixed point b ∈ Ω.

Up to a translation, it is not restrictive to consider b = 0 ∈ Ω.

The case γ = 12 , half-integer circulation

Aa(x) = A0(x− a), where A0(x1, x2) =1

(− x2

x21 + x2

Bonnaillie-Noel, Helffer, Hoffmann-Ostenhof [J. Phys. A (2009)]Noris, Terracini [Indiana Univ. Math. J. (2010)]:nodal domains of eigenfunctions are related to spectral minimalpartitions of the Dirichlet laplacian with points of odd multiplicity

Bonnaillie-Noel, Noris, Nys, Terracini [Analysis and PDE (2014)]Lena [J. Math. Phys. (2015)], Noris, Nys, Terracini [CMP, to appear]Noris, Terracini [Indiana Univ. Math. J. (2010)]:a strong connection between nodal properties of eigenfunctions andthe critical points of the map a 7→ λaj .

The case γ = 12 , half-integer circulation

Aa(x) = A0(x− a), where A0(x1, x2) =1

(− x2

x21 + x2

Bonnaillie-Noel, Helffer, Hoffmann-Ostenhof [J. Phys. A (2009)]Noris, Terracini [Indiana Univ. Math. J. (2010)]:nodal domains of eigenfunctions are related to spectral minimalpartitions of the Dirichlet laplacian with points of odd multiplicity

Bonnaillie-Noel, Noris, Nys, Terracini [Analysis and PDE (2014)]Lena [J. Math. Phys. (2015)], Noris, Nys, Terracini [CMP, to appear]Noris, Terracini [Indiana Univ. Math. J. (2010)]:a strong connection between nodal properties of eigenfunctions andthe critical points of the map a 7→ λaj .

The function a 7→ λaj Regularity

Properties of the map a 7→ λaj

Bonnaillie-Noel, Noris, Nys, Terracini (2014), Lena (2015):

∀ j ≥ 1, the map a 7→ λaj is continuos in Ω and admits a continous

extension on Ω (as a→ ∂Ω, λaj → j-th eigenvalue of −∆ in Ω)

since it constant on ∂Ω, the map a 7→ λaj has an extremal point in Ω.

Improved regularity for simple eigenvalues: if ∃n0 ≥ 1 s.t.

is simple,

then the function a 7→ λan0is analytic in a neighborhood of 0.

Question:

what is the leading term in the asymptotic expansion of the map a 7→ λan0?

is simple,

Question:

is simple,

Question:

The function a 7→ λaj Rate of convergence of λa → λ0

Let us assume that ∃n0 ≥ 1 such that λ0n0

is simple and denote λ0 = λ0n0

and, for any a ∈ Ω, λa = λan0. Thus λa → λ0.

Let ϕ0 ∈ H1,00 (Ω,C) be an eigenfunction of (i∇+A0)2 associated to λ0

(i∇+A0)2ϕ0 = λ0ϕ0, in Ω,

ϕ0 = 0, on ∂Ω,such that

∫Ω|ϕ0(x)|2 dx = 1.

Theorem [F., Ferrero, Terracini (2011)]

There exists k ∈ N odd s.t.

r−k/2ϕ0(r(cos t, sin t))→ β1eit2√π

cos(k2 t)

+ β2eit2√π

sin(k2 t)

as r → 0+

in C1,τ ([0, 2π],C), with β1, β2 ∈ C, (β1, β2) 6= (0, 0).

[Helffer-Hoffmann-Ostenhof-Hoffmann-Ostenhof-Owen (1999)]: e−it2ϕ0(r(cos t, sin t))

is a multiple of a real-valued function and therefore either β1 = 0 or β2β1

is real.

ϕ0 has at 0 a zero or order k2 for some

odd k ∈ Nϕ0 has got exactly k nodal linesmeeting at 0 and dividing the wholeangle into k equal parts.

Noris, Terracini (2010), Bonnaillie-Noel, Noris, Nys, Terracini (2014):The behavior of the eigenvalue λa is strongly related to the structure ofthe nodal lines of the associated eigenfunction.

rate of convergence of λa to λ0 ←→ order of vanishing of ϕ0 at 0

ϕ0 has at 0 a zero or order k2 for some

odd k ∈ Nϕ0 has got exactly k nodal linesmeeting at 0 and dividing the wholeangle into k equal parts.

Noris, Terracini (2010), Bonnaillie-Noel, Noris, Nys, Terracini (2014):The behavior of the eigenvalue λa is strongly related to the structure ofthe nodal lines of the associated eigenfunction.

rate of convergence of λa to λ0 ←→ order of vanishing of ϕ0 at 0

Theorem [Bonnaillie-Noel, Noris, Nys, Terracini (2014)]

If k ≥ 3, then

|λa − λ0| ≤ C|a|k+12 as a→ 0 for a constant C > 0 independent of a

the Taylor polynomials of the function a 7→ λ0 − λa with center 0 anddegree less or equal than k−1

2 vanish.

The control |a|k+12 is not the optimal one.

Our goal:

to establish the exact order of the asymptotic expansion of λa − λ0

proving that along a suitable direction the exact order is |a|k, where kis the number of nodal lines of ϕ0 at 0

to find the leading term in the Taylor expansion of the eigenvaluefunction, detecting its exact coefficients.

If k ≥ 3, then

2 vanish.

Our goal:

If k ≥ 3, then

2 vanish.

Our goal:

If k ≥ 3, then

2 vanish.

Our goal:

Sharp asymptotics Moving along directions of nodal lines

Moving along directions of nodal lines

Theorem (F.-Abatangelo, 2015)

Let r be the half-line tangent to a nodalline of eigenfunction ϕ0 associated to λ0

ending at 0. Then, as a→ 0 with a ∈ r,

λ0 − λa|a|k

→ −4|β1|2 + |β2|2

(β1, β2) 6= (0, 0) is s.t.

r−k2ϕ0(r(cos t, sin t))→ β1

eit2√π

cos(k2 t)

+ β2eit2√π

sin(k2 t)

as r → 0+

mk < 0 is a negative constant depending only on k.

Moving along directions of nodal lines

Theorem (F.-Abatangelo, 2015)

Let r be the half-line tangent to a nodalline of eigenfunction ϕ0 associated to λ0

ending at 0. Then, as a→ 0 with a ∈ r,

λ0 − λa|a|k

→ −4|β1|2 + |β2|2

(β1, β2) 6= (0, 0) is s.t.

r−k2ϕ0(r(cos t, sin t))→ β1

eit2√π

cos(k2 t)

+ β2eit2√π

sin(k2 t)

as r → 0+

mk < 0 is a negative constant depending only on k.

The constant mk

For every odd k ∈ N, ψk(r cos t, r sin t) = rk/2 sin(k2 t), r ≥ 0, t ∈ [0, 2π],

is the unique (up to a multiplicative constant) function which is

vanishing on s0 = [0,+∞)×0harmonic on R2 \ s0 and homogeneous of degree k/2

Let s := [1,+∞)×0 and R2+ = (x1, x2) ∈ R2 : x2 > 0).

Let D1,2s (R2

+) be the completion of C∞c (R2+ \ s) w.r.t. (

∫R2+|∇u|2 dx)1/2.

mk = minu∈D1,2

s (R2+)Jk(u)

where Jk(u) = 12

∫R2+|∇u(x)|2 dx−

∫ 10 u(x1, 0)∂ψk∂x2

(x1, 0) dx1.

mk = Jk(wk) = −12

∫R2+|∇wk(x)|2 dx

= −12

∫ 10∂+ψk∂x2

(x1, 0)wk(x1, 0) dx1 < 0

The constant mk

Let s := [1,+∞)×0 and R2+ = (x1, x2) ∈ R2 : x2 > 0).

Let D1,2s (R2

∫R2+|∇u|2 dx)1/2.

mk = minu∈D1,2

s (R2+)Jk(u)

where Jk(u) = 12

∫R2+|∇u(x)|2 dx−

∫ 10 u(x1, 0)∂ψk∂x2

(x1, 0) dx1.

mk = Jk(wk) = −12

∫R2+|∇wk(x)|2 dx

= −12

∫ 10∂+ψk∂x2

(x1, 0)wk(x1, 0) dx1 < 0

The constant mk

Let s := [1,+∞)×0 and R2+ = (x1, x2) ∈ R2 : x2 > 0).

Let D1,2s (R2

∫R2+|∇u|2 dx)1/2.

mk = minu∈D1,2

s (R2+)Jk(u)

where Jk(u) = 12

∫R2+|∇u(x)|2 dx−

∫ 10 u(x1, 0)∂ψk∂x2

(x1, 0) dx1.

mk = Jk(wk) = −12

∫R2+|∇wk(x)|2 dx

= −12

∫ 10∂+ψk∂x2

(x1, 0)wk(x1, 0) dx1 < 0

Consequences

Analyticity of a 7→ λa

λ0 − λa|a|k

→ 4|β1|2 + |β2|2

as a→ 0 along the half-line opposite to thetangent to a nodal line of ϕ0.

The restriction of the function λ0 − λa on the straight line tangent to anodal line changes sign at 0: is positive on the side of the nodal line andnegative on the opposite side

if λ0 is simple, then 0 cannot be an extremal point of the map a 7→ λa.

Consequences

λ0 − λa|a|k

→ 4|β1|2 + |β2|2

Consequences

λ0 − λa|a|k

→ 4|β1|2 + |β2|2

Consequences

Proposition [Bonnaillie-Noel, Noris, Nys, Terracini (2014)]

Fix any j ∈ N. If 0 is an extremal point of a 7→ λaj , then

(i) either λ0j is not simple

(ii) or the eigenfunction of (i∇+A0)2 associated to λ0j has at 0 a zero

of order k/2 with k ≥ 3 odd.

Our theorem we can exclude alternative (ii): If 0 is an extremalpoint of a 7→ λaj =⇒ λ0

j is not simple.

if λ0 is simple and k ≥ 3, 0 is a saddle point forthe map a 7→ λa.

if λ0 is simple and k = 1, the gradient of the mapa 7→ λa in 0 is different from zero, then 0 is not astationary point, a fortiori not even an extremalpoint (Noris, Terracini (2010)).

Consequences

j is not simple.

Consequences

j is not simple.

Sharp asymptotics Idea of the proof

Rotate the axes in such a way that the positive x1-axis is tangent to oneof the k nodal lines of ϕ0 ending at 0 it is not restrictive to assumethat β1 = 0.

For every b ∈ R2, we define θb : R2 \ b → [0, 2π)so that θb(b+ r(cos t, sin t)) = t ∀ r > 0, t ∈ [0, 2π).

∇(θb2

)= Ab, e−i

θb2 (i∇+Ab)

2u = −∆(e−i

θb2 u)

θb(x)

For all a ∈ Ω, let ϕa ∈ H1,a0 (Ω,C) \ 0 be an eigenfunction of (Ea)

associated to the eigenvalue λa such that∫Ω|ϕa(x)|2 dx = 1 and

∫Ωei2

(θ0−θa)(x)ϕa(x)ϕ0(x) dx is positive real.

We have that ϕa → ϕ0 in H1(Ω,C) and in C2loc(Ω \ 0,C) and

(i∇+Aa)ϕa → (i∇+A0)ϕ0 in L2(Ω,C).

Rotate the axes in such a way that the positive x1-axis is tangent to oneof the k nodal lines of ϕ0 ending at 0 it is not restrictive to assumethat β1 = 0.

For every b ∈ R2, we define θb : R2 \ b → [0, 2π)so that θb(b+ r(cos t, sin t)) = t ∀ r > 0, t ∈ [0, 2π).

∇(θb2

)= Ab, e−i

θb2 (i∇+Ab)

2u = −∆(e−i

θb2 u)

θb(x)

For all a ∈ Ω, let ϕa ∈ H1,a0 (Ω,C) \ 0 be an eigenfunction of (Ea)

associated to the eigenvalue λa such that∫Ω|ϕa(x)|2 dx = 1 and

∫Ωei2

(θ0−θa)(x)ϕa(x)ϕ0(x) dx is positive real.

We have that ϕa → ϕ0 in H1(Ω,C) and in C2loc(Ω \ 0,C) and

(i∇+Aa)ϕa → (i∇+A0)ϕ0 in L2(Ω,C).

Blow-up

Theorem

ϕa(|a|x)

|a|k/2→ β2√

πΨk as a = (|a|, 0)→ 0,

in H1,e(DR,C) ∀R > 1, a.e. and in C2loc(R2 \ e,C), where e = (1, 0).

Ψk is the unique function in H1,eloc (R2) satisfying

(i∇+Ae)2Ψk = 0, in R2,

and ∫R2

|(i∇+Ae)(Ψk − ei2θeψk)|2 < +∞.

If wk ∈ D1,2s (R2

+) minimizes the functional Jk, Ψk = eiθe2 (ψk + wk).

Blow-up

Theorem

ϕa(|a|x)

|a|k/2→ β2√

πΨk as a = (|a|, 0)→ 0,

in H1,e(DR,C) ∀R > 1, a.e. and in C2loc(R2 \ e,C), where e = (1, 0).

Ψk is the unique function in H1,eloc (R2) satisfying

(i∇+Ae)2Ψk = 0, in R2,

and ∫R2

|(i∇+Ae)(Ψk − ei2θeψk)|2 < +∞.

If wk ∈ D1,2s (R2

+) minimizes the functional Jk, Ψk = eiθe2 (ψk + wk).

Upper bound for λ0 − λa: the Rayleigh quotient for λ0

For all 1 ≤ j ≤ n0 and a ∈ Ω, let ϕaj ∈ H1,a0 (Ω,C) \ 0 be an

eigenfunction of problem (Ea) associated to the eigenvalue λaj such that∫Ω|ϕaj (x)|2 dx = 1 and

∫Ωϕaj (x)ϕa` (x) dx = 0 if j 6= `.

For j = n0, we choose ϕan0= ϕa.

Courant-Fisher minimax characterization of the eigenvalue λ0, we havethat

λ0 =min

u∈F\0

∫Ω |(i∇+A0)u|2dx∫

Ω |u|2 dx: F is a subspace of H1,0

0 (Ω,C),dimF = n0

Upper bound for λ0 − λa: the Rayleigh quotient for λ0

For all 1 ≤ j ≤ n0 and a ∈ Ω, let ϕaj ∈ H1,a0 (Ω,C) \ 0 be an

eigenfunction of problem (Ea) associated to the eigenvalue λaj such that∫Ω|ϕaj (x)|2 dx = 1 and

∫Ωϕaj (x)ϕa` (x) dx = 0 if j 6= `.

For j = n0, we choose ϕan0= ϕa.

Courant-Fisher minimax characterization of the eigenvalue λ0, we havethat

λ0 =min

u∈F\0

∫Ω |(i∇+A0)u|2dx∫

Ω |u|2 dx: F is a subspace of H1,0

0 (Ω,C),dimF = n0

For R > 2 and a = (|a|, 0) with |a| small, define vj,R,a as follows:

vj,R,a =

vextj,R,a, in Ω \DR|a|,

vintj,R,a, in DR|a|,j = 1, . . . , n0,

vextj,R,a := ei2

(θ0−θa)ϕaj in Ω\DR|a|, (i∇+A0)2vextj,R,a = λajvextj,R,a in Ω\DR|a|,

(i∇+A0)2vintj,R,a = 0, in DR|a|,

vintj,R,a = ei2

(θ0−θa)ϕaj , on ∂DR|a|.

spanv1,R,a, . . . , vn0,R,a)

= n0 After a Gram-Schmidtnormalization, use vj,R,a as test functions in the Rayleigh quotient

λ0−λa ≤∫DR|a|

∣∣(i∇+A0)vintn0,R,a

∣∣2dx−∫DR|a|

∣∣(i∇+Aa)ϕa∣∣2dx+o(|a|k)

as |a| → 0.

vj,R,a =

vintj,R,a, in DR|a|,j = 1, . . . , n0,

vextj,R,a := ei2

vintj,R,a = ei2

spanv1,R,a, . . . , vn0,R,a)

After a Gram-Schmidtnormalization, use vj,R,a as test functions in the Rayleigh quotient

as |a| → 0.

vj,R,a =

vintj,R,a, in DR|a|,j = 1, . . . , n0,

vextj,R,a := ei2

vintj,R,a = ei2

spanv1,R,a, . . . , vn0,R,a)

= n0 After a Gram-Schmidtnormalization, use vj,R,a as test functions in the Rayleigh quotient

as |a| → 0.

Blow-up and let R→ +∞

lim sup|a|→0

λ0 − λa|a|k

≤ |β2|2

πk√π(ξ(1)−

√π)

ξ(r) := 1√π

∫ 2π

i2θe(r cos t,r sin t)Ψk(r cos t, r sin t) sin

(k2 t)dt.

In a similar way we obtain a lower bound for λ0 − λa estimating theRayleigh quotient for λa

lim inf|a|→0

λ0 − λa|a|k

≥ |β2|2

πk√π(ξ(1)−

√π)

lim|a|→0

λ0 − λa|a|k

=|β2|2

πk√π(ξ(1)−

√π)

ξ(1)−√π =

1√π

∫ 2π

0wk(cos t, sin t) sin

(k2 t)dt = ω(1)

ω(r) :=1√π

∫ 2π

0wk(r cos t, r sin t) sin

(k2 t)dt.

wk ∈ D1,2s (R2

+) minimizes Jk . . .

ξ(1)−√π = ω(1) = − 4

k√πmk > 0

ξ(1)−√π =

1√π

∫ 2π

0wk(cos t, sin t) sin

(k2 t)dt = ω(1)

ω(r) :=1√π

∫ 2π

0wk(r cos t, r sin t) sin

(k2 t)dt.

wk ∈ D1,2s (R2

+) minimizes Jk . . .

ξ(1)−√π = ω(1) = − 4

k√πmk > 0

λ0 − λa|a|k

→ |β2|2

πk√π(ξ(1)−

√π) = −4

|β2|2

as a→ 0 tangentially to any of the k nodal lines of ϕ0.

The Taylor polynomials of the function a 7→ λ0 − λa with center 0 anddegree strictly smaller than k vanish, since they vanish on the kindependent directions corresponding to the nodal lines of ϕ0

λ0 − λa = P (a) + o(|a|k), as |a| → 0+,

for some

P 6≡ 0, P (a) = P (a1, a2) =

k∑j=0

cjak−j1 aj2

homogeneous polynomial of degree k.

λ0 − λa|a|k

→ |β2|2

πk√π(ξ(1)−

√π) = −4

|β2|2

as a→ 0 tangentially to any of the k nodal lines of ϕ0.

The Taylor polynomials of the function a 7→ λ0 − λa with center 0 anddegree strictly smaller than k vanish, since they vanish on the kindependent directions corresponding to the nodal lines of ϕ0

λ0 − λa = P (a) + o(|a|k), as |a| → 0+,

for some

P 6≡ 0, P (a) = P (a1, a2) =

k∑j=0

cjak−j1 aj2

homogeneous polynomial of degree k.

Sharp asymptotics Moving along any direction

Moving along any direction

For α ∈ [0, 2π), p = (cosα, sinα)and a = |a|p = |a|(cosα, sinα)

ϕa(|a|x)

|a|k/2→ β2√

πΨp as a = |a|p→ 0,

in H1,p(DR,C) for every R > 1 andin C2

loc(R2 \ p,C).

Ψp is the unique function in H1,ploc (R2,C) satisfying

(i∇+Ap)2Ψp = 0, in R2,∫R2\Dr

∣∣(i∇+Ap)(Ψp − ei2

(θp−θp0 )ei2θ0ψk)

∣∣2 dx < +∞, ∀ r > 1.

θp(p+r(cos t, sin t)) = t, θp0 (r(cos t, sin t)) = t, ∀ r > 0, t ∈ [α, α+ 2π)

Moving along any direction

For α ∈ [0, 2π), p = (cosα, sinα)and a = |a|p = |a|(cosα, sinα)

ϕa(|a|x)

|a|k/2→ β2√

πΨp as a = |a|p→ 0,

in H1,p(DR,C) for every R > 1 andin C2

loc(R2 \ p,C).

Ψp is the unique function in H1,ploc (R2,C) satisfying

(i∇+Ap)2Ψp = 0, in R2,∫R2\Dr

∣∣(i∇+Ap)(Ψp − ei2

(θp−θp0 )ei2θ0ψk)

∣∣2 dx < +∞, ∀ r > 1.

θp(p+r(cos t, sin t)) = t, θp0 (r(cos t, sin t)) = t, ∀ r > 0, t ∈ [α, α+ 2π)

For α ∈ [0, 2π), p = (cosα, sinα) and a = |a|p = |a|(cosα, sinα)

lim|a|→0

λ0 − λa|a|k

=|β2|2

πk√πf(α),

wheref : [0, 2π)→ R, f(α) = (ξp(1)−

√π),

ξp(r) :=1√π

∫ 2π

(θp−θp0 )(r cos t,r sin t)Ψp(r cos t, r sin t)e−it2 sin

(k2 t)dt.

We already know that f(0) = ξe(1)−√π = ξ(1)−

√π = − 4

k√πmk > 0

For α ∈ [0, 2π), p = (cosα, sinα) and a = |a|p = |a|(cosα, sinα)

lim|a|→0

λ0 − λa|a|k

=|β2|2

πk√πf(α),

wheref : [0, 2π)→ R, f(α) = (ξp(1)−

√π),

ξp(r) :=1√π

∫ 2π

(θp−θp0 )(r cos t,r sin t)Ψp(r cos t, r sin t)e−it2 sin

(k2 t)dt.

We already know that f(0) = ξe(1)−√π = ξ(1)−

√π = − 4

k√πmk > 0

Symmetry properties of f(α)

Rotation of 2πk :

R1(x1, x2) =

(cos 2π

k − sin 2πk

sin 2πk cos 2π

)Reflexion through the x1-axis:

R2(x) = R2(x1, x2) = (x1,−x2),

Problem:

how does the limit profile Ψp change when the above trasformationsact on p?

how does the coefficient ξp(1) and hence f(α) change when theabove trasformations act on p?

Symmetry properties of f(α)

Rotation of 2πk :

R1(x1, x2) =

(cos 2π

k − sin 2πk

sin 2πk cos 2π

)Reflexion through the x1-axis:

R2(x) = R2(x1, x2) = (x1,−x2),

Problem:

how does the limit profile Ψp change when the above trasformationsact on p?

how does the coefficient ξp(1) and hence f(α) change when theabove trasformations act on p?

The function ψk (to which the limit profile is asymptotic at ∞ up tosuitable phases) is invariant (up to phases) under the actions R1,R2:

(θ0R1)(ψk R1) = −eiπk e

i2θ0ψk, ψk R2 = ψk.

ΨR−11 (p) = −e−i

πk(Ψp R1

), ΨR2(p) = −eiθR2(p)

(Ψp R2

ξR−11 (p)(1) = ξp(1), ξR2(p)(1) = ξp(1),

f(α) = f(α− 2π

)f(α) = f(2π − α)

ΨR−11 (p) = −e−i

πk(Ψp R1

), ΨR2(p) = −eiθR2(p)

(Ψp R2

ξR−11 (p)(1) = ξp(1), ξR2(p)(1) = ξp(1),

f(α) = f(α− 2π

)f(α) = f(2π − α)

ΨR−11 (p) = −e−i

πk(Ψp R1

), ΨR2(p) = −eiθR2(p)

(Ψp R2

ξR−11 (p)(1) = ξp(1), ξR2(p)(1) = ξp(1),

f(α) = f(α− 2π

)f(α) = f(2π − α)

ΨR−11 (p) = −e−i

πk(Ψp R1

), ΨR2(p) = −eiθR2(p)

(Ψp R2

ξR−11 (p)(1) = ξp(1), ξR2(p)(1) = ξp(1),

f(α) = f(α− 2π

)f(α) = f(2π − α)

g : R→ R, g(α) := P (cosα, sinα)

g(α) =∑k

j=0 cj(cosα)k−j(sinα)j

g(α) = |β2|2√πkf(α) ⇒ g(α) = g

(α+ 2π

), g(α) = g(2π − α) ∀α ∈ R

c0 = g(0) = −4 |β2|2

π mk > 0

g(α) = c0 cos(kα) for all α ∈ R.

Theorem [F.-Abatangelo (2015)]

If α ∈ [0, 2π), then

λ0 − λa|a|k

→ C0 cos(kα) as a→ 0 with a = |a|(cosα, sinα),

with C0 = −4 |β2|2

π mk.

The leading term in the Taylor expansion of a 7→ λ0 − λa is then given by

P (|a|(cosα, sinα)) = C0|a|k cos(kα).

Hence P (a1, a2) = C0<((a1 + i a2)k

)and then

the polynomial P is harmonic.

Theorem [F.-Abatangelo (2015)]

If α ∈ [0, 2π), then

λ0 − λa|a|k

→ C0 cos(kα) as a→ 0 with a = |a|(cosα, sinα),

with C0 = −4 |β2|2

π mk.

The leading term in the Taylor expansion of a 7→ λ0 − λa is then given by

P (|a|(cosα, sinα)) = C0|a|k cos(kα).

Hence P (a1, a2) = C0<((a1 + i a2)k

)and then

the polynomial P is harmonic.

Sharp asymptotic estimates for eigenvalues of Aharonov ... · Veronica Felli University of...

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