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Exponential decay estimates for fundamental solutions of Schr odinger-type operators Svitlana Mayboroda, Bruno Poggi University of Minnesota Department of Mathematics AMS Special Session on Regularity of PDEs on Rough Domains Boston,

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SLIDE 1

1

Exponential decay estimates for fundamental solutions of Schr¨

  • dinger-type operators

Svitlana Mayboroda, Bruno Poggi

University of Minnesota Department of Mathematics

AMS Special Session on Regularity of PDEs on Rough Domains Boston, Massachusetts April 21, 2018

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
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SLIDE 2

2

The electric Schr¨

  • dinger operator

By an electric Schr¨

  • dinger operator we mean a second-order linear
  • perator of the form

L := −divA∇ + V, where V is a scalar, real-valued, positive a.e., locally integrable function, and A is a matrix of bounded, measurable complex coefficients satisfying the uniform ellipticity condition λ|ξ|2 ≤ ℜe A(x)ξ, ξ ≡ ℜe

n

  • i,j=1

Aij(x)ξj ¯ ξi and AL∞(Rn) ≤ Λ, (1) for some λ > 0, Λ < ∞ and for all ξ ∈ Cn, x ∈ Rn.

  • The exponential decay of solutions to the Schr¨
  • dinger operator in the

presence of a positive potential is an important property underpinning foundation of quantum physics.

  • However, establishing a precise rate of decay for complicated

potentials is a challenging open problem to this date. (Landis conjecture)

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-3
SLIDE 3

2

The electric Schr¨

  • dinger operator

By an electric Schr¨

  • dinger operator we mean a second-order linear
  • perator of the form

L := −divA∇ + V, where V is a scalar, real-valued, positive a.e., locally integrable function, and A is a matrix of bounded, measurable complex coefficients satisfying the uniform ellipticity condition λ|ξ|2 ≤ ℜe A(x)ξ, ξ ≡ ℜe

n

  • i,j=1

Aij(x)ξj ¯ ξi and AL∞(Rn) ≤ Λ, (1) for some λ > 0, Λ < ∞ and for all ξ ∈ Cn, x ∈ Rn.

  • The exponential decay of solutions to the Schr¨
  • dinger operator in the

presence of a positive potential is an important property underpinning foundation of quantum physics.

  • However, establishing a precise rate of decay for complicated

potentials is a challenging open problem to this date. (Landis conjecture)

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-4
SLIDE 4

2

The electric Schr¨

  • dinger operator

By an electric Schr¨

  • dinger operator we mean a second-order linear
  • perator of the form

L := −divA∇ + V, where V is a scalar, real-valued, positive a.e., locally integrable function, and A is a matrix of bounded, measurable complex coefficients satisfying the uniform ellipticity condition λ|ξ|2 ≤ ℜe A(x)ξ, ξ ≡ ℜe

n

  • i,j=1

Aij(x)ξj ¯ ξi and AL∞(Rn) ≤ Λ, (1) for some λ > 0, Λ < ∞ and for all ξ ∈ Cn, x ∈ Rn.

  • The exponential decay of solutions to the Schr¨
  • dinger operator in the

presence of a positive potential is an important property underpinning foundation of quantum physics.

  • However, establishing a precise rate of decay for complicated

potentials is a challenging open problem to this date. (Landis conjecture)

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
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SLIDE 5

3

Exponential Decay of electric Schr¨

  • dinger operators
  • First results expressing upper estimates on the solutions in terms of a

certain distance associated to V go back to Agmon [1], but not sharp.

  • For eigenfunctions, the decay is governed by the uncertainty principle -

see ADFJM ‘Localization of eigenfunctions via an effective potential’[2]

  • In [9], Shen proved that if V ∈ RH n

2 , then the fundamental solution

Γ to the classical Schr¨

  • dinger operator −∆ + V satisfies the bounds

c1e−ε1d(x,y,V ) |x − y|n−2 ≤ Γ(x, y) ≤ c2e−ε2d(x,y,V ) |x − y|n−2 , (2) where d is a certain distance function depending on V .

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-6
SLIDE 6

3

Exponential Decay of electric Schr¨

  • dinger operators
  • First results expressing upper estimates on the solutions in terms of a

certain distance associated to V go back to Agmon [1], but not sharp.

  • For eigenfunctions, the decay is governed by the uncertainty principle -

see ADFJM ‘Localization of eigenfunctions via an effective potential’[2]

  • In [9], Shen proved that if V ∈ RH n

2 , then the fundamental solution

Γ to the classical Schr¨

  • dinger operator −∆ + V satisfies the bounds

c1e−ε1d(x,y,V ) |x − y|n−2 ≤ Γ(x, y) ≤ c2e−ε2d(x,y,V ) |x − y|n−2 , (2) where d is a certain distance function depending on V .

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-7
SLIDE 7

3

Exponential Decay of electric Schr¨

  • dinger operators
  • First results expressing upper estimates on the solutions in terms of a

certain distance associated to V go back to Agmon [1], but not sharp.

  • For eigenfunctions, the decay is governed by the uncertainty principle -

see ADFJM ‘Localization of eigenfunctions via an effective potential’[2]

  • In [9], Shen proved that if V ∈ RH n

2 , then the fundamental solution

Γ to the classical Schr¨

  • dinger operator −∆ + V satisfies the bounds

c1e−ε1d(x,y,V ) |x − y|n−2 ≤ Γ(x, y) ≤ c2e−ε2d(x,y,V ) |x − y|n−2 , (2) where d is a certain distance function depending on V .

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
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SLIDE 8

4

Exponential Decay of electric Schr¨

  • dinger operators
  • A natural question is whether the sharp exponential decay found by

Shen for the fundamental solution to −∆ + V can be extended to the non self-adjoint setting −div A∇ + V .

  • Moreover, we also wondered whether we can obtain exponential decay

results for the fundamental solution to the magnetic Schr¨

  • dinger
  • perator.

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-9
SLIDE 9

4

Exponential Decay of electric Schr¨

  • dinger operators
  • A natural question is whether the sharp exponential decay found by

Shen for the fundamental solution to −∆ + V can be extended to the non self-adjoint setting −div A∇ + V .

  • Moreover, we also wondered whether we can obtain exponential decay

results for the fundamental solution to the magnetic Schr¨

  • dinger
  • perator.

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-10
SLIDE 10

5

The generalized magnetic Schr¨

  • dinger operator

We consider the operator formally given by L = −(∇ − ia)T A(∇ − ia) + V, (3) where a = (a1, . . . , an) is a vector of real-valued L2

loc(Rn) functions, A

and V as before. Denote Da = ∇ − ia, and the magnetic field by B, so that B = curl a. (4)

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
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SLIDE 11

6

Properties of the magnetic Schr¨

  • dinger operator
  • The magnetic Schr¨
  • dinger operator exhibits a property called gauge

invariance: quantitative assumptions should be put on B rather than a.

  • The diamagnetic inequality
  • ∇|u|(x)
  • ≤
  • Dau(x)
  • .

(5)

  • When A ≡ I so that LM := L = (∇ − ia)2 + V , the operator LM is

dominated by the Schr¨

  • dinger operator LE := −∆ + V in the following

sense: for each ε > 0, |(LM + ε)−1f| ≤ (−∆ + ε)−1|f|, for each f ∈ H = L2(Rn). (6) The above is known as the Kato-Simon inequality.

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
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SLIDE 12

6

Properties of the magnetic Schr¨

  • dinger operator
  • The magnetic Schr¨
  • dinger operator exhibits a property called gauge

invariance: quantitative assumptions should be put on B rather than a.

  • The diamagnetic inequality
  • ∇|u|(x)
  • ≤
  • Dau(x)
  • .

(5)

  • When A ≡ I so that LM := L = (∇ − ia)2 + V , the operator LM is

dominated by the Schr¨

  • dinger operator LE := −∆ + V in the following

sense: for each ε > 0, |(LM + ε)−1f| ≤ (−∆ + ε)−1|f|, for each f ∈ H = L2(Rn). (6) The above is known as the Kato-Simon inequality.

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-13
SLIDE 13

6

Properties of the magnetic Schr¨

  • dinger operator
  • The magnetic Schr¨
  • dinger operator exhibits a property called gauge

invariance: quantitative assumptions should be put on B rather than a.

  • The diamagnetic inequality
  • ∇|u|(x)
  • ≤
  • Dau(x)
  • .

(5)

  • When A ≡ I so that LM := L = (∇ − ia)2 + V , the operator LM is

dominated by the Schr¨

  • dinger operator LE := −∆ + V in the following

sense: for each ε > 0, |(LM + ε)−1f| ≤ (−∆ + ε)−1|f|, for each f ∈ H = L2(Rn). (6) The above is known as the Kato-Simon inequality.

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
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SLIDE 14

7

Existence of a fundamental solution to LM?

  • Some authors (Ben Ali [3], Kurata and Sugano [7]) showed properties
  • f the fundamental solution to LM under ad-hoc smoothness

assumptions on the magnetic potential a.

  • On the other hand, the natural setting to make sense of LM in the

weak sense requires only that a ∈ L2

loc(Rn), V ∈ L1 loc(Rn).

  • Through a smooth approximation method, we establish the existence
  • f an integral kernel to LM in the above context.

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-15
SLIDE 15

7

Existence of a fundamental solution to LM?

  • Some authors (Ben Ali [3], Kurata and Sugano [7]) showed properties
  • f the fundamental solution to LM under ad-hoc smoothness

assumptions on the magnetic potential a.

  • On the other hand, the natural setting to make sense of LM in the

weak sense requires only that a ∈ L2

loc(Rn), V ∈ L1 loc(Rn).

  • Through a smooth approximation method, we establish the existence
  • f an integral kernel to LM in the above context.

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-16
SLIDE 16

7

Existence of a fundamental solution to LM?

  • Some authors (Ben Ali [3], Kurata and Sugano [7]) showed properties
  • f the fundamental solution to LM under ad-hoc smoothness

assumptions on the magnetic potential a.

  • On the other hand, the natural setting to make sense of LM in the

weak sense requires only that a ∈ L2

loc(Rn), V ∈ L1 loc(Rn).

  • Through a smooth approximation method, we establish the existence
  • f an integral kernel to LM in the above context.

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
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SLIDE 17

8

Some more history

  • Before Shen obtained the sharp exponential decay result for the

fundamental solution to −∆ + V , he first obtained a polynomial decay result in [11].

  • He later also obtained polynomial decay results for the magnetic

Schr¨

  • dinger operator LM in [12].
  • In [6], Kurata obtained exponential decay results for −div A∇ + V

and LM by integrating certain heat kernel estimates. He obtained the bound |Γ(x, y)| ≤ Ce−ε(1+m(x,V +|B|)|x−y|)

2 2k0+3

|x − y|n−2 for a.e. x, y ∈ Rn, which is not sharp.

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-18
SLIDE 18

8

Some more history

  • Before Shen obtained the sharp exponential decay result for the

fundamental solution to −∆ + V , he first obtained a polynomial decay result in [11].

  • He later also obtained polynomial decay results for the magnetic

Schr¨

  • dinger operator LM in [12].
  • In [6], Kurata obtained exponential decay results for −div A∇ + V

and LM by integrating certain heat kernel estimates. He obtained the bound |Γ(x, y)| ≤ Ce−ε(1+m(x,V +|B|)|x−y|)

2 2k0+3

|x − y|n−2 for a.e. x, y ∈ Rn, which is not sharp.

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-19
SLIDE 19

8

Some more history

  • Before Shen obtained the sharp exponential decay result for the

fundamental solution to −∆ + V , he first obtained a polynomial decay result in [11].

  • He later also obtained polynomial decay results for the magnetic

Schr¨

  • dinger operator LM in [12].
  • In [6], Kurata obtained exponential decay results for −div A∇ + V

and LM by integrating certain heat kernel estimates. He obtained the bound |Γ(x, y)| ≤ Ce−ε(1+m(x,V +|B|)|x−y|)

2 2k0+3

|x − y|n−2 for a.e. x, y ∈ Rn, which is not sharp.

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
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SLIDE 20

9

The Reverse H¨

  • lder class RHp

To state our results, we need just a few more definitions. We say that w ∈ Lp

loc(Rn), with w > 0 a.e., belongs to the Reverse

H¨

  • lder class RHp = RHp(Rn) if there exists a constant C so that for

any ball B ⊂ Rn,

  • B

wp 1/p ≤ C

B

w. (7)

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
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SLIDE 21

10

The Fefferman-Phong-Shen maximal function m(x, w)

For a function w ∈ RHp, p ≥ n

2 , define the maximal function m(x, w)

by 1 m(x, w) := sup

r>0

  • r :

1 rn−2 ˆ

B(x,r)

w ≤ 1

  • ,

(8) and the distance function d(x, y, w) = inf

γ 1

ˆ m(γ(t), w)|γ′(t)| dt, (9) where γ : [0, 1] → Rn is absolutely continuous and γ(0) = x, γ(1) = y.

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-22
SLIDE 22

10

The Fefferman-Phong-Shen maximal function m(x, w)

For a function w ∈ RHp, p ≥ n

2 , define the maximal function m(x, w)

by 1 m(x, w) := sup

r>0

  • r :

1 rn−2 ˆ

B(x,r)

w ≤ 1

  • ,

(8) and the distance function d(x, y, w) = inf

γ 1

ˆ m(γ(t), w)|γ′(t)| dt, (9) where γ : [0, 1] → Rn is absolutely continuous and γ(0) = x, γ(1) = y.

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-23
SLIDE 23

11

m(x, w) and the uncertainty principle

The function m measures the sum of the contributions of the kinetic energy ℜeADafDaf and potential energy V |f|2, and is intimately related to the uncertainty principle through the following estimate which is often known as the Fefferman-Phong inequality: Suppose that a ∈ L2

loc(Rn)n, and moreover assume (12) (next slide).

Then, for all u ∈ C1

c (Rn),

ˆ

Rn m2(x, V + |B|)|u|2 dx ≤ C

ˆ

Rn(|Dau|2 + V |u|2) dx,

(10) where C depends on the constants c, c′ from (12) and on V + |B|RH n

2 . Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-24
SLIDE 24

11

m(x, w) and the uncertainty principle

The function m measures the sum of the contributions of the kinetic energy ℜeADafDaf and potential energy V |f|2, and is intimately related to the uncertainty principle through the following estimate which is often known as the Fefferman-Phong inequality: Suppose that a ∈ L2

loc(Rn)n, and moreover assume (12) (next slide).

Then, for all u ∈ C1

c (Rn),

ˆ

Rn m2(x, V + |B|)|u|2 dx ≤ C

ˆ

Rn(|Dau|2 + V |u|2) dx,

(10) where C depends on the constants c, c′ from (12) and on V + |B|RH n

2 . Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-25
SLIDE 25

12

L2 exponential decay

Theorem 1 (Mayboroda, P. 2018)

For any operator L given by (3) and for any f ∈ L2(Rn) with compact support, there exist constants ˜ d, ε, C > 0 such that ˆ

  • x∈Rn|d(x,supp f,V +|B|)≥ ˜

d

m (·, V + |B|)2 |u|2e2εd(·,supp f,V +|B|) ≤ C ˆ

Rn |f|2

1 m(x, V + |B|)2 , (11) where u := L−1f (in a weak sense), provided that A is an elliptic matrix with complex bounded measurable coefficients, and i) either a = 0 and V ∈ RHn/2, ii) or, more generally, a ∈ L2

loc(Rn), V > 0 a.e. on Rn, and

   V + |B| ∈ RHn/2, 0 ≤ V ≤ c m(·, V + |B|)2, |∇B| ≤ c′ m(·, V + |B|)3. (12)

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-26
SLIDE 26

12

L2 exponential decay

Theorem 1 (Mayboroda, P. 2018)

For any operator L given by (3) and for any f ∈ L2(Rn) with compact support, there exist constants ˜ d, ε, C > 0 such that ˆ

  • x∈Rn|d(x,supp f,V +|B|)≥ ˜

d

m (·, V + |B|)2 |u|2e2εd(·,supp f,V +|B|) ≤ C ˆ

Rn |f|2

1 m(x, V + |B|)2 , (11) where u := L−1f (in a weak sense), provided that A is an elliptic matrix with complex bounded measurable coefficients, and i) either a = 0 and V ∈ RHn/2, ii) or, more generally, a ∈ L2

loc(Rn), V > 0 a.e. on Rn, and

   V + |B| ∈ RHn/2, 0 ≤ V ≤ c m(·, V + |B|)2, |∇B| ≤ c′ m(·, V + |B|)3. (12)

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-27
SLIDE 27

13

L2 exponential decay for the resolvent

An analogous estimate holds for the resolvent operator (I + t2L)−1, t > 0: ˆ

  • x∈Rn|d(x,supp f,Bt)≥ ˜

d

  • m
  • ·, Bt

2 (I + t2L)−1f

  • 2 e2εd(·,supp f,Bt)

≤ C ˆ

Rn

|f|2m

  • ·, Bt

2 . where B := V + |B| + 1

t2 .

  • In other words, L−1f decays as e−εd(·,supp f,V +|B|) away from the

support of f and the resolvent decays as e−εd(·,supp f,V +|B|+ 1

t2 ). Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-28
SLIDE 28

13

L2 exponential decay for the resolvent

An analogous estimate holds for the resolvent operator (I + t2L)−1, t > 0: ˆ

  • x∈Rn|d(x,supp f,Bt)≥ ˜

d

  • m
  • ·, Bt

2 (I + t2L)−1f

  • 2 e2εd(·,supp f,Bt)

≤ C ˆ

Rn

|f|2m

  • ·, Bt

2 . where B := V + |B| + 1

t2 .

  • In other words, L−1f decays as e−εd(·,supp f,V +|B|) away from the

support of f and the resolvent decays as e−εd(·,supp f,V +|B|+ 1

t2 ). Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-29
SLIDE 29

14

  • L2 exponential decay for the resolvents has appeared in the literature,

see Germinet and Klein [5], but purely in terms of

1 t2 . Also in many

  • ther sources.
  • The estimate (11) (for the operator L−1) is entirely new and is a

consequence of the decay afforded by our assumptions on V and B.

  • Our results are in the nature of best possible under the very general

assumptions.

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-30
SLIDE 30

14

  • L2 exponential decay for the resolvents has appeared in the literature,

see Germinet and Klein [5], but purely in terms of

1 t2 . Also in many

  • ther sources.
  • The estimate (11) (for the operator L−1) is entirely new and is a

consequence of the decay afforded by our assumptions on V and B.

  • Our results are in the nature of best possible under the very general

assumptions.

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-31
SLIDE 31

14

  • L2 exponential decay for the resolvents has appeared in the literature,

see Germinet and Klein [5], but purely in terms of

1 t2 . Also in many

  • ther sources.
  • The estimate (11) (for the operator L−1) is entirely new and is a

consequence of the decay afforded by our assumptions on V and B.

  • Our results are in the nature of best possible under the very general

assumptions.

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-32
SLIDE 32

15

A Moser estimate

Definition 2

We say that the operator L has a Moser estimate if for each ball B ⊂ Rn and each function u which solves Lu = 0 in the weak sense on 2B, it follows that u ∈ L∞(B) and uL∞(cB) ≤ C

  • 2B

|u|2 1

2 ,

(13) where c, C are independent of B and u.

  • The electric Schr¨
  • dinger operators with real matrix A, and the

magnetic Schr¨

  • dinger operator both have Moser estimates.

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-33
SLIDE 33

15

A Moser estimate

Definition 2

We say that the operator L has a Moser estimate if for each ball B ⊂ Rn and each function u which solves Lu = 0 in the weak sense on 2B, it follows that u ∈ L∞(B) and uL∞(cB) ≤ C

  • 2B

|u|2 1

2 ,

(13) where c, C are independent of B and u.

  • The electric Schr¨
  • dinger operators with real matrix A, and the

magnetic Schr¨

  • dinger operator both have Moser estimates.

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-34
SLIDE 34

16

Upper bound exponential decay

Theorem 3 (Mayboroda, P. 2018)

Suppose a ∈ L2

loc(Rn), A is an elliptic matrix with complex, bounded

coefficients, V ∈ L1

loc(Rn), and that L is an operator for which there

exists a fundamental solution bounded above by a multiple of |x − y|2−n. Moreover, if a ≡ 0, assume V ∈ RH n

2 ; otherwise assume

(12). Then there exists ε > 0 and a constant C > 0, depending on L, such that

  • B(x,

1 m(x,V +|B|) )

|Γ(z, y)|2 dz 1

2 ≤ Ce−εd(x,y,V +|B|)

|x − y|n−2 for all x, y ∈ Rn. (14) If L has a Moser estimate, then |Γ(x, y)| ≤ Ce−εd(x,y,V +|B|) |x − y|n−2 for all x, y ∈ Rn. (15)

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-35
SLIDE 35

17

Upper bound exponential decay

Theorem 3 establishes in particular the upper bound exponential decay for both electric Schr¨

  • dinger operators −div A∇ + V , and the magnetic

Schr¨

  • dinger operator (∇ − ia)2 + V .

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-36
SLIDE 36

18

A scale-invariant Harnack inequality

Definition 4

We say that the operator L satisfying assumptions (12) has the m-scale invariant Harnack Inequality if whenever B = B(x0, r), r <

c m(x0,V +|B|), x0 ∈ Rn, the following property holds. For any u

which solves Lu = 0 in the weak sense on 2B, sup

x∈B

|u(x)| ≤ C inf

x∈B |u(x)|,

(16) with the constant C > 0 independent of B.

  • The electric Schr¨
  • dinger operators with real matrix A have the

m-scale invariant Harnack inequality.

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
slide-37
SLIDE 37

18

A scale-invariant Harnack inequality

Definition 4

We say that the operator L satisfying assumptions (12) has the m-scale invariant Harnack Inequality if whenever B = B(x0, r), r <

c m(x0,V +|B|), x0 ∈ Rn, the following property holds. For any u

which solves Lu = 0 in the weak sense on 2B, sup

x∈B

|u(x)| ≤ C inf

x∈B |u(x)|,

(16) with the constant C > 0 independent of B.

  • The electric Schr¨
  • dinger operators with real matrix A have the

m-scale invariant Harnack inequality.

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
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19

Lower bound exponential decay

Theorem 5 (Mayboroda, P. 2018)

Suppose that a ∈ L2

loc(Rn), A is an elliptic matrix with complex,

bounded coefficients, V ∈ L1

loc(Rn), and that L, L0 := L − V , L∗ 0 are

  • perators for which there exist fundamental solutions Γ ≡ ΓV , Γ0 Γ∗

0.

Assume that ΓV , Γ0, Γ∗

0 are bounded above by a multiple of |x − y|2−n,

and that Γ0 is bounded below by a multiple of |x − y|2−n. Suppose that L has a Moser estimate, and that L satisfies the m-scale invariant Harnack Inequality. Moreover, if a ≡ 0, assume that V ∈ RH n

2 ;

  • therwise assume (12). Then there exist constants c and ε2 depending
  • n λ, Λ, V + |B|RH n

2 , n and the constants from (12) such that

|Γ(x, y)| ≥ ce−ε2d(x,y,V +|B|) |x − y|n−2 . (17)

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
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20

Exponential decay of the real electric Schr¨

  • dinger
  • perators

If ΓE is the fundamental solution to an electric Schr¨

  • dinger operator

with real matrix A and V ∈ RH n

2 , then the last few theorems imply

that c1e−ε1d(x,y,V ) |x − y|n−2 ≤ Γ(x, y) ≤ c2e−ε2d(x,y,V ) |x − y|n−2 . (18)

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
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21

Thanks for listening!

Thank you. :)

  • S. Mayboroda is supported in part by the NSF INSPIRE Award DMS

1344235, NSF CAREER Award DMS 1220089, and Simons Fellowship. Both authors would like to thank the Mathematical Sciences Research Institute (NSF grant DMS 1440140) for support and hospitality.

Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
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Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators
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Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨

  • dinger operators