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Approximations of the Neumann Laplacian in nonuniformly collapsing - - PowerPoint PPT Presentation

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Approximations of the Neumann Laplacian in nonuniformly collapsing strips C esar R. de Oliveira UFSCar QMath13 Atlanta C esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 Atlanta 1 / 25 Sources 1 Sources 2 Collapsing

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Approximations of the Neumann Laplacian in nonuniformly collapsing strips

C´ esar R. de Oliveira UFSCar QMath13 – Atlanta

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 1 / 25

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Sources

1 Sources 2 Collapsing regions 3 Effective operator 4 Uniformly collapsing approximations 5 Examples

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 2 / 25

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Sources

Sources

  • R. Bedoya, C. R. de Oliveira & A. A. Verri: Complex Γ-convergence and

magnetic Dirichlet Laplacian in bounded thin tubes. J. Spectr. Theory 4 (2014) 621–642

  • C. R. de Oliveira & A. A. Verri: On the Neumann Laplacian in nonuniformly

collapsing strips. Preprint.

  • L. Friedlander & M. Solomyak: On the spectrum of the Dirichlet Laplacian in a

narrow infinite strip. Amer. Math. Soc. Transl. 225 (2008) 103–116

  • J. K. Hale & G. Raugel: Reaction-diffusion equation in thin domains. J. Math.

pures et appl. 71 (1992) 33–95

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 3 / 25

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Collapsing regions

1 Sources 2 Collapsing regions 3 Effective operator 4 Uniformly collapsing approximations 5 Examples

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 4 / 25

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Collapsing regions

Initial

a

  • ∆

g(x)

Λ

1

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 5 / 25

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Collapsing regions

Initial

a

  • Δ

εg(x)

nonuniformly collapsing

Λε

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 6 / 25

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Collapsing regions

(Un)Bounded region

  • We consider “thick regions” given by functions g : [a, ∞) → (0, ∞) with g(x) → ∞

as x → ∞. Is there an effective operator S = S(g) as ε → 0 ?

  • • A more delicate question: Is there a family of uniformly collapsing regions Qε

whose effective operator coincides with S ?

  • • • Conditions on g:

(c1) C2 function and strictly increasing for large values of x; (c2) j(x) := g′(x)

2g(x) and j′(x) are bounded.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 7 / 25

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Collapsing regions

(Un)Bounded region

  • We consider “thick regions” given by functions g : [a, ∞) → (0, ∞) with g(x) → ∞

as x → ∞. Is there an effective operator S = S(g) as ε → 0 ?

  • • A more delicate question: Is there a family of uniformly collapsing regions Qε

whose effective operator coincides with S ?

  • • • Conditions on g:

(c1) C2 function and strictly increasing for large values of x; (c2) j(x) := g′(x)

2g(x) and j′(x) are bounded.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 7 / 25

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Collapsing regions

(Un)Bounded region

  • We consider “thick regions” given by functions g : [a, ∞) → (0, ∞) with g(x) → ∞

as x → ∞. Is there an effective operator S = S(g) as ε → 0 ?

  • • A more delicate question: Is there a family of uniformly collapsing regions Qε

whose effective operator coincides with S ?

  • • • Conditions on g:

(c1) C2 function and strictly increasing for large values of x; (c2) j(x) := g′(x)

2g(x) and j′(x) are bounded.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 7 / 25

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Collapsing regions

(Un)Bounded region

  • We consider “thick regions” given by functions g : [a, ∞) → (0, ∞) with g(x) → ∞

as x → ∞. Is there an effective operator S = S(g) as ε → 0 ?

  • • A more delicate question: Is there a family of uniformly collapsing regions Qε

whose effective operator coincides with S ?

  • • • Conditions on g:

(c1) C2 function and strictly increasing for large values of x; (c2) j(x) := g′(x)

2g(x) and j′(x) are bounded.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 7 / 25

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Collapsing regions

(Un)Bounded region

  • We consider “thick regions” given by functions g : [a, ∞) → (0, ∞) with g(x) → ∞

as x → ∞. Is there an effective operator S = S(g) as ε → 0 ?

  • • A more delicate question: Is there a family of uniformly collapsing regions Qε

whose effective operator coincides with S ?

  • • • Conditions on g:

(c1) C2 function and strictly increasing for large values of x; (c2) j(x) := g′(x)

2g(x) and j′(x) are bounded.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 7 / 25

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Collapsing regions

(Un)Bounded region

The region of interest is Λε :=

  • (x, y) ∈ R2 | 0 < y < εg(x), x ∈ [a, ∞)
  • ,

and the quadratic form (Neumann Laplacian) mε(v) =

  • Λε

|∇v|2dx, dom mε = H1(Λε). After changes of variables, mε(v) is cast as nε(ϕ) :=

  • Q
  • ϕ′ − g′

2g ϕ − y ϕy g′ g

  • 2

+ |ϕy|2 ε2g2

  • dxdy,

where Q := [a, ∞) × (0, 1) is a fixed region. Note that, as ε → 0, nε(ϕ) − → n(ϕ) :=

Q

  • ϕ′ −

g′ 2g ϕ

  • 2

dxdy, if ϕy = 0, ∞, if ϕy = 0. Let Sε and S be the operators associated with nε and n, respectively.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 8 / 25

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Collapsing regions

(Un)Bounded region

The region of interest is Λε :=

  • (x, y) ∈ R2 | 0 < y < εg(x), x ∈ [a, ∞)
  • ,

and the quadratic form (Neumann Laplacian) mε(v) =

  • Λε

|∇v|2dx, dom mε = H1(Λε). After changes of variables, mε(v) is cast as nε(ϕ) :=

  • Q
  • ϕ′ − g′

2g ϕ − y ϕy g′ g

  • 2

+ |ϕy|2 ε2g2

  • dxdy,

where Q := [a, ∞) × (0, 1) is a fixed region. Note that, as ε → 0, nε(ϕ) − → n(ϕ) :=

Q

  • ϕ′ −

g′ 2g ϕ

  • 2

dxdy, if ϕy = 0, ∞, if ϕy = 0. Let Sε and S be the operators associated with nε and n, respectively.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 8 / 25

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Collapsing regions

(Un)Bounded region

The region of interest is Λε :=

  • (x, y) ∈ R2 | 0 < y < εg(x), x ∈ [a, ∞)
  • ,

and the quadratic form (Neumann Laplacian) mε(v) =

  • Λε

|∇v|2dx, dom mε = H1(Λε). After changes of variables, mε(v) is cast as nε(ϕ) :=

  • Q
  • ϕ′ − g′

2g ϕ − y ϕy g′ g

  • 2

+ |ϕy|2 ε2g2

  • dxdy,

where Q := [a, ∞) × (0, 1) is a fixed region. Note that, as ε → 0, nε(ϕ) − → n(ϕ) :=

Q

  • ϕ′ −

g′ 2g ϕ

  • 2

dxdy, if ϕy = 0, ∞, if ϕy = 0. Let Sε and S be the operators associated with nε and n, respectively.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 8 / 25

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Collapsing regions

(Un)Bounded region

The region of interest is Λε :=

  • (x, y) ∈ R2 | 0 < y < εg(x), x ∈ [a, ∞)
  • ,

and the quadratic form (Neumann Laplacian) mε(v) =

  • Λε

|∇v|2dx, dom mε = H1(Λε). After changes of variables, mε(v) is cast as nε(ϕ) :=

  • Q
  • ϕ′ − g′

2g ϕ − y ϕy g′ g

  • 2

+ |ϕy|2 ε2g2

  • dxdy,

where Q := [a, ∞) × (0, 1) is a fixed region. Note that, as ε → 0, nε(ϕ) − → n(ϕ) :=

Q

  • ϕ′ −

g′ 2g ϕ

  • 2

dxdy, if ϕy = 0, ∞, if ϕy = 0. Let Sε and S be the operators associated with nε and n, respectively.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 8 / 25

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Collapsing regions

(Un)Bounded region

Let L :=

  • ϕ(x, y) = w(x)1 | w ∈ L2([a, ∞))
  • .

Theorem (1) (by Kato-Robinson Theorem) For all f ∈ L2(Q) one has, as ε → 0,

  • S−1

ε f − (S−1 ⊕ 0)f

  • −

→ 0 , where 0 is the null operator on L⊥.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 9 / 25

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Collapsing regions

(Un)Bounded region

Let L :=

  • ϕ(x, y) = w(x)1 | w ∈ L2([a, ∞))
  • .

Theorem (1) (by Kato-Robinson Theorem) For all f ∈ L2(Q) one has, as ε → 0,

  • S−1

ε f − (S−1 ⊕ 0)f

  • −

→ 0 , where 0 is the null operator on L⊥.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 9 / 25

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Effective operator

1 Sources 2 Collapsing regions 3 Effective operator 4 Uniformly collapsing approximations 5 Examples

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 10 / 25

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Effective operator

(Un)Bounded region

The goal now is to characterize S: for this we need (c2), i.e., bounded j = g′

2g and j′.

Theorem (2) For g as above, we have (Sw)(x) := −w′′(x) + ̺(x)w(x), with ̺(x) := j2(x) + j′(x) and a Robin condition at the end point a, that is, dom S = {w ∈ H2([a, ∞)) | j(a) w(a) = w′(a)}.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 11 / 25

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Effective operator

(Un)Bounded region

The goal now is to characterize S: for this we need (c2), i.e., bounded j = g′

2g and j′.

Theorem (2) For g as above, we have (Sw)(x) := −w′′(x) + ̺(x)w(x), with ̺(x) := j2(x) + j′(x) and a Robin condition at the end point a, that is, dom S = {w ∈ H2([a, ∞)) | j(a) w(a) = w′(a)}.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 11 / 25

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Effective operator

(Un)Bounded region

The goal now is to characterize S: for this we need (c2), i.e., bounded j = g′

2g and j′.

Theorem (2) For g as above, we have (Sw)(x) := −w′′(x) + ̺(x)w(x), with ̺(x) := j2(x) + j′(x) and a Robin condition at the end point a, that is, dom S = {w ∈ H2([a, ∞)) | j(a) w(a) = w′(a)}.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 11 / 25

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Uniformly collapsing approximations

1 Sources 2 Collapsing regions 3 Effective operator 4 Uniformly collapsing approximations 5 Examples

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 12 / 25

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Uniformly collapsing approximations

Diverging region

Second main goal: finding uniformly collapsing regions Qε whose effective operator coincides with S.

a

g(x) gε(x) εgε(x)

zε 1/ε

α (0<α<1) Qε

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 13 / 25

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Uniformly collapsing approximations

Uniformly collapsing approximations

Pick bounded functions gε : [a, +∞) → R as in the previous figure, which converges pointwise to g with collapsing εg (nonuniformly) and εgε (uniformly). Recall that Qε denotes the region below εgε(x). Consider the Neumann quadratic form fε(ψ) =

  • Qε

|∇ψ|2 dxdy, dom fε = H1(Qε). Set Q := [a, ∞) × (0, 1). After changes of variables, we pass to hε(ψ) =

  • Q
  • ψ′ − g′

ε

2gε ψ − y g′

ε

gε ψy

  • 2

+ |ψy|2 ε2g2

ε

  • dxdy,

(1) dom hε = H1(Q) ⊂ L2(Q). Denote by Hε the associated operator whose behavior we are interested in understanding as ε → 0.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 14 / 25

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Uniformly collapsing approximations

Uniformly collapsing approximations

Pick bounded functions gε : [a, +∞) → R as in the previous figure, which converges pointwise to g with collapsing εg (nonuniformly) and εgε (uniformly). Recall that Qε denotes the region below εgε(x). Consider the Neumann quadratic form fε(ψ) =

  • Qε

|∇ψ|2 dxdy, dom fε = H1(Qε). Set Q := [a, ∞) × (0, 1). After changes of variables, we pass to hε(ψ) =

  • Q
  • ψ′ − g′

ε

2gε ψ − y g′

ε

gε ψy

  • 2

+ |ψy|2 ε2g2

ε

  • dxdy,

(1) dom hε = H1(Q) ⊂ L2(Q). Denote by Hε the associated operator whose behavior we are interested in understanding as ε → 0.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 14 / 25

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Uniformly collapsing approximations

Uniformly collapsing approximations

Pick bounded functions gε : [a, +∞) → R as in the previous figure, which converges pointwise to g with collapsing εg (nonuniformly) and εgε (uniformly). Recall that Qε denotes the region below εgε(x). Consider the Neumann quadratic form fε(ψ) =

  • Qε

|∇ψ|2 dxdy, dom fε = H1(Qε). Set Q := [a, ∞) × (0, 1). After changes of variables, we pass to hε(ψ) =

  • Q
  • ψ′ − g′

ε

2gε ψ − y g′

ε

gε ψy

  • 2

+ |ψy|2 ε2g2

ε

  • dxdy,

(1) dom hε = H1(Q) ⊂ L2(Q). Denote by Hε the associated operator whose behavior we are interested in understanding as ε → 0.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 14 / 25

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Uniformly collapsing approximations

Uniformly collapsing approximations

Pick bounded functions gε : [a, +∞) → R as in the previous figure, which converges pointwise to g with collapsing εg (nonuniformly) and εgε (uniformly). Recall that Qε denotes the region below εgε(x). Consider the Neumann quadratic form fε(ψ) =

  • Qε

|∇ψ|2 dxdy, dom fε = H1(Qε). Set Q := [a, ∞) × (0, 1). After changes of variables, we pass to hε(ψ) =

  • Q
  • ψ′ − g′

ε

2gε ψ − y g′

ε

gε ψy

  • 2

+ |ψy|2 ε2g2

ε

  • dxdy,

(1) dom hε = H1(Q) ⊂ L2(Q). Denote by Hε the associated operator whose behavior we are interested in understanding as ε → 0.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 14 / 25

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Uniformly collapsing approximations

Uniformly collapsing regions

  • First a reduction of dimension. Consider again the subspace

L =

  • w(x) 1 | w ∈ L2([a, ∞))
  • ,

the one-dimensional quadratic form tε(w) := hε(w 1) = ∞

a

  • w′ − g′

ε

2gε w

  • 2dx ,

dom tε = H1([a, ∞)), (2) and denote by Tε the associated operator. Under the above conditions: Theorem (3)(based on Friedlander & Solomyak method) For g as above, one has

  • Hε

−1 −

  • T −1

ε

⊕ 0

  • −

→ 0, ε → 0 , where 0 is the null operator on the subspace L⊥.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 15 / 25

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Uniformly collapsing approximations

Uniformly collapsing regions

  • First a reduction of dimension. Consider again the subspace

L =

  • w(x) 1 | w ∈ L2([a, ∞))
  • ,

the one-dimensional quadratic form tε(w) := hε(w 1) = ∞

a

  • w′ − g′

ε

2gε w

  • 2dx ,

dom tε = H1([a, ∞)), (2) and denote by Tε the associated operator. Under the above conditions: Theorem (3)(based on Friedlander & Solomyak method) For g as above, one has

  • Hε

−1 −

  • T −1

ε

⊕ 0

  • −

→ 0, ε → 0 , where 0 is the null operator on the subspace L⊥.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 15 / 25

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Uniformly collapsing approximations

Uniformly collapsing regions

  • First a reduction of dimension. Consider again the subspace

L =

  • w(x) 1 | w ∈ L2([a, ∞))
  • ,

the one-dimensional quadratic form tε(w) := hε(w 1) = ∞

a

  • w′ − g′

ε

2gε w

  • 2dx ,

dom tε = H1([a, ∞)), (2) and denote by Tε the associated operator. Under the above conditions: Theorem (3)(based on Friedlander & Solomyak method) For g as above, one has

  • Hε

−1 −

  • T −1

ε

⊕ 0

  • −

→ 0, ε → 0 , where 0 is the null operator on the subspace L⊥.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 15 / 25

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Uniformly collapsing approximations

Uniformly collapsing regions

  • Tε is already unidimensional. The next task is the limit of Tε.

Theorem (4)(based on Bedoya, deO & Verri) Let g : [a, ∞) → R be as above. Then: (A) The sequence Tε converges in the strong resolvent sense to S. (B) If j(x) = g′(x)

2g(x) vanishes as x → ∞, then

  • T −1

ε

− S−1 − → 0. Recall: Sw = −w′′ + ̺(x)w, with ̺ = j2 + j′, and b.c. j(a)w(a) = w′(a).

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 16 / 25

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Uniformly collapsing approximations

Uniformly collapsing regions

  • Tε is already unidimensional. The next task is the limit of Tε.

Theorem (4)(based on Bedoya, deO & Verri) Let g : [a, ∞) → R be as above. Then: (A) The sequence Tε converges in the strong resolvent sense to S. (B) If j(x) = g′(x)

2g(x) vanishes as x → ∞, then

  • T −1

ε

− S−1 − → 0. Recall: Sw = −w′′ + ̺(x)w, with ̺ = j2 + j′, and b.c. j(a)w(a) = w′(a).

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 16 / 25

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Uniformly collapsing approximations

Uniformly collapsing regions

  • Tε is already unidimensional. The next task is the limit of Tε.

Theorem (4)(based on Bedoya, deO & Verri) Let g : [a, ∞) → R be as above. Then: (A) The sequence Tε converges in the strong resolvent sense to S. (B) If j(x) = g′(x)

2g(x) vanishes as x → ∞, then

  • T −1

ε

− S−1 − → 0. Recall: Sw = −w′′ + ̺(x)w, with ̺ = j2 + j′, and b.c. j(a)w(a) = w′(a).

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 16 / 25

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SLIDE 34

Uniformly collapsing approximations

Uniformly collapsing regions

  • Tε is already unidimensional. The next task is the limit of Tε.

Theorem (4)(based on Bedoya, deO & Verri) Let g : [a, ∞) → R be as above. Then: (A) The sequence Tε converges in the strong resolvent sense to S. (B) If j(x) = g′(x)

2g(x) vanishes as x → ∞, then

  • T −1

ε

− S−1 − → 0. Recall: Sw = −w′′ + ̺(x)w, with ̺ = j2 + j′, and b.c. j(a)w(a) = w′(a).

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 16 / 25

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Uniformly collapsing approximations

Uniformly collapsing regions

In summary: through such uniformly collapsing Qε we have recovered S (initially found from Kato-Robinson) as the effective operator. Especially in case j(x) = g′(x) 2g(x) → 0, x → ∞, there is a norm convergence

  • Hε

−1 −

  • S−1 ⊕ 0
  • → 0.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 17 / 25

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SLIDE 36

Uniformly collapsing approximations

Uniformly collapsing regions

In summary: through such uniformly collapsing Qε we have recovered S (initially found from Kato-Robinson) as the effective operator. Especially in case j(x) = g′(x) 2g(x) → 0, x → ∞, there is a norm convergence

  • Hε

−1 −

  • S−1 ⊕ 0
  • → 0.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 17 / 25

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SLIDE 37

Examples

1 Sources 2 Collapsing regions 3 Effective operator 4 Uniformly collapsing approximations 5 Examples

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 18 / 25

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Examples

Examples

Class I. [Power law] Take g(x) = γxβ, γ, β > 0, for x ≥ 1. Then a = 1 and j(x) = β/(2x) vanishes at infinity. So, as ε → 0, there is a norm resolvent convergence (in uniformly collapsing regions) to the effective operator (Sw)(x) = −w′′(x) + β(β − 2) 4x2 w(x) , β 2 w(1) = w′(1) .

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 19 / 25

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SLIDE 39

Examples

Examples

Class I. [Power law] Take g(x) = γxβ, γ, β > 0, for x ≥ 1. Then a = 1 and j(x) = β/(2x) vanishes at infinity. So, as ε → 0, there is a norm resolvent convergence (in uniformly collapsing regions) to the effective operator (Sw)(x) = −w′′(x) + β(β − 2) 4x2 w(x) , β 2 w(1) = w′(1) .

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 19 / 25

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SLIDE 40

Examples

Examples

Note that for g(x) = γxβ the effective potential ̺(x) = β(β−2)

4x2

: does not depend on γ; vanishes for β = 2 and is proportional to x−2 for all values of β; is negative for 0 < β < 2 and positive for β > 2.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 20 / 25

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SLIDE 41

Examples

Examples

Note that for g(x) = γxβ the effective potential ̺(x) = β(β−2)

4x2

: does not depend on γ; vanishes for β = 2 and is proportional to x−2 for all values of β; is negative for 0 < β < 2 and positive for β > 2.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 20 / 25

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SLIDE 42

Examples

Examples

Note that for g(x) = γxβ the effective potential ̺(x) = β(β−2)

4x2

: does not depend on γ; vanishes for β = 2 and is proportional to x−2 for all values of β; is negative for 0 < β < 2 and positive for β > 2.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 20 / 25

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Examples

Examples

Class II. [Exponential of a power] For x ≥ 1, consider g(x) = γ exβ, γ, β > 0. Now j(x) =

β 2 x1−β : it is bounded only if β ≤ 1 and vanishes at infinity if β < 1.

The effective operator in this case is (Sw)(x) = (Sβw)(x) := −w′′(x) + ̺β(x)w(x) , β 2 w(1) = w′(1), with ̺β(x) := 1

4

  • β2

x2(1−β) − 2β(1−β) x2−β

  • .

By Theorem 4, if 0 < β < 1, one has (in Qε) norm resolvent convergence to the effective operator, whereas for β = 1 we have strong convergence.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 21 / 25

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Examples

Examples

Class II. [Exponential of a power] For x ≥ 1, consider g(x) = γ exβ, γ, β > 0. Now j(x) =

β 2 x1−β : it is bounded only if β ≤ 1 and vanishes at infinity if β < 1.

The effective operator in this case is (Sw)(x) = (Sβw)(x) := −w′′(x) + ̺β(x)w(x) , β 2 w(1) = w′(1), with ̺β(x) := 1

4

  • β2

x2(1−β) − 2β(1−β) x2−β

  • .

By Theorem 4, if 0 < β < 1, one has (in Qε) norm resolvent convergence to the effective operator, whereas for β = 1 we have strong convergence.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 21 / 25

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SLIDE 45

Examples

Examples

Class II. [Exponential of a power] For x ≥ 1, consider g(x) = γ exβ, γ, β > 0. Now j(x) =

β 2 x1−β : it is bounded only if β ≤ 1 and vanishes at infinity if β < 1.

The effective operator in this case is (Sw)(x) = (Sβw)(x) := −w′′(x) + ̺β(x)w(x) , β 2 w(1) = w′(1), with ̺β(x) := 1

4

  • β2

x2(1−β) − 2β(1−β) x2−β

  • .

By Theorem 4, if 0 < β < 1, one has (in Qε) norm resolvent convergence to the effective operator, whereas for β = 1 we have strong convergence.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 21 / 25

slide-46
SLIDE 46

Examples

Examples

Class II. [Exponential of a power] For x ≥ 1, consider g(x) = γ exβ, γ, β > 0. Now j(x) =

β 2 x1−β : it is bounded only if β ≤ 1 and vanishes at infinity if β < 1.

The effective operator in this case is (Sw)(x) = (Sβw)(x) := −w′′(x) + ̺β(x)w(x) , β 2 w(1) = w′(1), with ̺β(x) := 1

4

  • β2

x2(1−β) − 2β(1−β) x2−β

  • .

By Theorem 4, if 0 < β < 1, one has (in Qε) norm resolvent convergence to the effective operator, whereas for β = 1 we have strong convergence.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 21 / 25

slide-47
SLIDE 47

Examples

Examples

For “regions” g(x) = γ exβ, the effective potential ̺β(x) = 1

4

  • β2

x2(1−β) − 2β(1−β) x2−β

  • :

does not depend on γ; for 0 < β < 1, it is bounded and vanishes at ∞. Furthermore, it is negative in a neighborhood of 1 and positive for large values of x; for β = 1 (the exponentially thick region), it is constant and equals to 1/4, and so the transition point from norm to strong resolvent approximations.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 22 / 25

slide-48
SLIDE 48

Examples

Examples

For “regions” g(x) = γ exβ, the effective potential ̺β(x) = 1

4

  • β2

x2(1−β) − 2β(1−β) x2−β

  • :

does not depend on γ; for 0 < β < 1, it is bounded and vanishes at ∞. Furthermore, it is negative in a neighborhood of 1 and positive for large values of x; for β = 1 (the exponentially thick region), it is constant and equals to 1/4, and so the transition point from norm to strong resolvent approximations.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 22 / 25

slide-49
SLIDE 49

Examples

Examples

For “regions” g(x) = γ exβ, the effective potential ̺β(x) = 1

4

  • β2

x2(1−β) − 2β(1−β) x2−β

  • :

does not depend on γ; for 0 < β < 1, it is bounded and vanishes at ∞. Furthermore, it is negative in a neighborhood of 1 and positive for large values of x; for β = 1 (the exponentially thick region), it is constant and equals to 1/4, and so the transition point from norm to strong resolvent approximations.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 22 / 25

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SLIDE 50

Examples

Examples

If time permits. Final remarks:

  • The condition that j(x) is bounded implies that g(x) ≤ γ eκx.

In the borderline case g(x) = γ eκx one has the effective potential ̺(x) = κ2

4 .

  • For g(x) = x3 + 1

2 sin(x3) x

, x ≥ 1, it follows that j(x) vanishes at infinity and ̺(x) is bounded but oscillates wildly for large x.

  • Naturally, a spectral analysis should be undertaken ...

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 23 / 25

slide-51
SLIDE 51

Examples

Examples

If time permits. Final remarks:

  • The condition that j(x) is bounded implies that g(x) ≤ γ eκx.

In the borderline case g(x) = γ eκx one has the effective potential ̺(x) = κ2

4 .

  • For g(x) = x3 + 1

2 sin(x3) x

, x ≥ 1, it follows that j(x) vanishes at infinity and ̺(x) is bounded but oscillates wildly for large x.

  • Naturally, a spectral analysis should be undertaken ...

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 23 / 25

slide-52
SLIDE 52

Examples

Examples

If time permits. Final remarks:

  • The condition that j(x) is bounded implies that g(x) ≤ γ eκx.

In the borderline case g(x) = γ eκx one has the effective potential ̺(x) = κ2

4 .

  • For g(x) = x3 + 1

2 sin(x3) x

, x ≥ 1, it follows that j(x) vanishes at infinity and ̺(x) is bounded but oscillates wildly for large x.

  • Naturally, a spectral analysis should be undertaken ...

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 23 / 25

slide-53
SLIDE 53

Examples

Examples

If time permits. Final remarks:

  • The condition that j(x) is bounded implies that g(x) ≤ γ eκx.

In the borderline case g(x) = γ eκx one has the effective potential ̺(x) = κ2

4 .

  • For g(x) = x3 + 1

2 sin(x3) x

, x ≥ 1, it follows that j(x) vanishes at infinity and ̺(x) is bounded but oscillates wildly for large x.

  • Naturally, a spectral analysis should be undertaken ...

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 23 / 25

slide-54
SLIDE 54

Examples

Examples

If time permits. Final remarks:

  • The condition that j(x) is bounded implies that g(x) ≤ γ eκx.

In the borderline case g(x) = γ eκx one has the effective potential ̺(x) = κ2

4 .

  • For g(x) = x3 + 1

2 sin(x3) x

, x ≥ 1, it follows that j(x) vanishes at infinity and ̺(x) is bounded but oscillates wildly for large x.

  • Naturally, a spectral analysis should be undertaken ...

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 23 / 25

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SLIDE 55

Examples C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 24 / 25

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SLIDE 56

Examples

Thanks

Thank you.

C´ esar R. de Oliveira UFSCar Nonuniformly Collapsing QMath13 – Atlanta 25 / 25