Non-accretive Schr odinger operators and exponential decay of their - - PowerPoint PPT Presentation

Non-accretive Schr¨

• dinger operators and exponential decay
• f their eigenfunctions

Petr Siegl Mathematical Institute, University of Bern, Switzerland http://gemma.ujf.cas.cz/˜siegl/ Based on [1] D. Krejˇ ciˇ r´ ık, N. Raymond, J. Royer, and P. Siegl: Non-accretive Schr¨

• dinger
• perators and exponential decay of their eigenfunctions

Israel Journal of Mathematics, to appear arXiv:1605.02437

Schr¨

• dinger operators with complex potentials

Main object

• Dirichlet realization of

L = (−i∇ + A)2 + V in L2(Ω)

• Ω ⊂ Rd open (no additional assumptions)
• V ∈ C1(Ω; C ) and A ∈ C2(Ω; Rd)

Schr¨

• dinger operators with complex potentials

Main object

• Dirichlet realization of

L = (−i∇ + A)2 + V in L2(Ω)

• Ω ⊂ Rd open (no additional assumptions)
• V ∈ C1(Ω; C ) and A ∈ C2(Ω; Rd)
• restriction on the growth, oscillations and negative Re V :

|∇V (x)| + |∇B(x)| = o

• (|V (x)| + |B(x)|)

3 2 + 1

• ,

(Re V (x))− = o

• |V (x)| + |B(x)| + 1
• ,

|x| → ∞

• where

B = (Bjk)j,k∈{1,...,d} , Bjk := ∂jAk − ∂kAj

Schr¨

• dinger operators with complex potentials

Main object

• Dirichlet realization of

L = (−i∇ + A)2 + V in L2(Ω)

• Ω ⊂ Rd open (no additional assumptions)
• V ∈ C1(Ω; C ) and A ∈ C2(Ω; Rd)
• restriction on the growth, oscillations and negative Re V :

|∇V (x)| + |∇B(x)| = o

• (|V (x)| + |B(x)|)

3 2 + 1

• ,

(Re V (x))− = o

• |V (x)| + |B(x)| + 1
• ,

|x| → ∞

• where

B = (Bjk)j,k∈{1,...,d} , Bjk := ∂jAk − ∂kAj Objectives

• 1. find the Dirichlet realization with ρ(L ) = ∅ and describe Dom(L )
• 2. prove the exponential decay of eigenfunctions of L (due to Im V and B)

Why complex potentials?

• superconductivity1

−∂2

x +

i∂y − x22 + iy , in L2(R2)

• optics with gains and losses2

−∆ + (1 + ixy )e−x2e−y2, in L2(R2)

• hydrodynamics3

− d2 dx2 + x2 + i εf(x) , in L2(R)

• open systems4, quantum resonances5, damped wave equation6,. . .
• 1Y. Almog, B. Helffer, and X.-B. Pan. Trans. Amer. Math. Soc. (2013), pp. 1183–1217.
• 2A. Regensburger et al. Phys. Rev. Lett. 107 (2011), p. 233902; J. Yang. Opt. Lett. 39

(2014), pp. 1133–1136.

• 3I. Gallagher, T. Gallay, and F. Nier. Int. Math. Res. Not. IMRN (2009), pp. 2147–2199.
• 4P. Exner. Open quantum systems and Feynman integrals. D. Reidel Publishing Co., 1985.
• 5A. A. Abramov, A. Aslanyan, and E. B. Davies. J. Phys. A: Math. Gen. 34 (2001), p. 57.
• 6J. Sj¨
• strand. Publ. Res. Inst. Math. Sci. 36 (2000), pp. 573–611.

Towards the Dirichlet realization: form methods

Simple 1D examples in L2(R) − d2 dx2 + ix3, − d2 dx2 − x2 + ix3, − d2 dx2 − ex2 + iex4

Towards the Dirichlet realization: form methods

Simple 1D examples in L2(R) − d2 dx2 + ix3, − d2 dx2 − x2 + ix3, − d2 dx2 − ex2 + iex4 Lax-Milgram theorem Let

• 1. (V, ·, ·V) be a Hilbert space continuously embedded and dense in H
• 2. ℓ : V × V → C be a continuous sesquilinear form
• 3. ℓ be V-elliptic (V-coercive or coercive )

∃δ > 0, ∀f ∈ V, |ℓ(f, f)| ≥ δfV

Towards the Dirichlet realization: form methods

Simple 1D examples in L2(R) − d2 dx2 + ix3, − d2 dx2 − x2 + ix3, − d2 dx2 − ex2 + iex4 Lax-Milgram theorem Let

• 1. (V, ·, ·V) be a Hilbert space continuously embedded and dense in H
• 2. ℓ : V × V → C be a continuous sesquilinear form
• 3. ℓ be V-elliptic (V-coercive or coercive )

∃δ > 0, ∀f ∈ V, |ℓ(f, f)| ≥ δfV Then the (densely defined) operator L Dom(L ) = {f ∈ V : ∃g ∈ H, ∀v ∈ V, ℓ(f, v) = g, v} , L f = g is bijective from Dom(L ) onto H (= ⇒ ρ(L ) = ∅ ).

Towards the Dirichlet realization

Assumptions Lax-Milgram theorem

• 1. (V, ·, ·V) be a Hilbert space continuously embedded and dense in H
• 2. ℓ : V × V → C be a continuous sesquilinear form
• 3. ℓ be V-elliptic (V-coercive or coercive )

∃δ > 0, ∀f ∈ V, |ℓ(f, f)| ≥ δfV Why it doesn’t work?

Towards the Dirichlet realization

Assumptions Lax-Milgram theorem

• 1. (V, ·, ·V) be a Hilbert space continuously embedded and dense in H
• 2. ℓ : V × V → C be a continuous sesquilinear form
• 3. ℓ be V-elliptic (V-coercive or coercive )

∃δ > 0, ∀f ∈ V, |ℓ(f, f)| ≥ δfV Why it doesn’t work?

• natural candidate for the form of −∂2

x + ix3:

ℓ(f, f) = −f′′ + ix3f, f = f′2 + i

• R

x3|f(x)|2 dx,

Towards the Dirichlet realization

Assumptions Lax-Milgram theorem

• 1. (V, ·, ·V) be a Hilbert space continuously embedded and dense in H
• 2. ℓ : V × V → C be a continuous sesquilinear form
• 3. ℓ be V-elliptic (V-coercive or coercive )

∃δ > 0, ∀f ∈ V, |ℓ(f, f)| ≥ δfV Why it doesn’t work?

• natural candidate for the form of −∂2

x + ix3:

ℓ(f, f) = −f′′ + ix3f, f = f′2 + i

• R

x3|f(x)|2 dx,

• variational space (form domain)

V = Dom(ℓ) = H1(R) ∩ Dom(|x|

3 2 ),

· 2

V = · 2 H1 + |x|

3 2 · 2

Towards the Dirichlet realization

Assumptions Lax-Milgram theorem

• 1. (V, ·, ·V) be a Hilbert space continuously embedded and dense in H
• 2. ℓ : V × V → C be a continuous sesquilinear form
• 3. ℓ be V-elliptic (V-coercive or coercive )

∃δ > 0, ∀f ∈ V, |ℓ(f, f)| ≥ δfV Why it doesn’t work?

• natural candidate for the form of −∂2

x + ix3:

ℓ(f, f) = −f′′ + ix3f, f = f′2 + i

• R

x3|f(x)|2 dx,

• variational space (form domain)

V = Dom(ℓ) = H1(R) ∩ Dom(|x|

3 2 ),

· 2

V = · 2 H1 + |x|

3 2 · 2

!!! no coercivity

Towards the Dirichlet realization: new Lax-Milgram

Generalized Lax-Milgram theorem of Almog-Helffer7 Let

• 1. (V, ·, ·V) be a Hilbert space continuously embedded and dense in H
• 2. ℓ : V × V → C be a continuous sesquilinear form
• 7Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466.
• 8A. S. Bonnet-Ben Dhia, P. Ciarlet Jr., and C. M. Zw¨
• lf. J. Comput. Appl. Math. 234

(2010), pp. 1912–1919; A. F. M. ter Elst, M. Sauter, and H. Vogt. J. Funct. Anal. 269 (2015),

• pp. 705–744; L. Grubiˇ

si´ c et al. Mathematika 59 (2013), pp. 169–189.

Towards the Dirichlet realization: new Lax-Milgram

Generalized Lax-Milgram theorem of Almog-Helffer7 Let

• 1. (V, ·, ·V) be a Hilbert space continuously embedded and dense in H
• 2. ℓ : V × V → C be a continuous sesquilinear form

3. A-H coercivity: ∃Φ1, Φ2 bounded linear maps on V and H and ∃δ > 0, ∀f ∈ V, |ℓ(f, f)| + |ℓ(Φ1f, f)| ≥ δf2

V,

|ℓ(f, f)| + |ℓ(f, Φ2f)| ≥ δf2

V.

• 7Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466.
• 8A. S. Bonnet-Ben Dhia, P. Ciarlet Jr., and C. M. Zw¨
• lf. J. Comput. Appl. Math. 234

(2010), pp. 1912–1919; A. F. M. ter Elst, M. Sauter, and H. Vogt. J. Funct. Anal. 269 (2015),

• pp. 705–744; L. Grubiˇ

si´ c et al. Mathematika 59 (2013), pp. 169–189.

Towards the Dirichlet realization: new Lax-Milgram

Generalized Lax-Milgram theorem of Almog-Helffer7 Let

• 1. (V, ·, ·V) be a Hilbert space continuously embedded and dense in H
• 2. ℓ : V × V → C be a continuous sesquilinear form

3. A-H coercivity: ∃Φ1, Φ2 bounded linear maps on V and H and ∃δ > 0, ∀f ∈ V, |ℓ(f, f)| + |ℓ(Φ1f, f)| ≥ δf2

V,

|ℓ(f, f)| + |ℓ(f, Φ2f)| ≥ δf2

V.

Then the (densely defined) operator L Dom(L ) = {f ∈ V : ∃g ∈ H, ∀v ∈ V, ℓ(f, v) = g, v} , L f = g is bijective from Dom(L ) onto H (= ⇒ ρ(L ) = ∅ ).

• 7Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466.
• 8A. S. Bonnet-Ben Dhia, P. Ciarlet Jr., and C. M. Zw¨
• lf. J. Comput. Appl. Math. 234

(2010), pp. 1912–1919; A. F. M. ter Elst, M. Sauter, and H. Vogt. J. Funct. Anal. 269 (2015),

• pp. 705–744; L. Grubiˇ

si´ c et al. Mathematika 59 (2013), pp. 169–189.

Towards the Dirichlet realization: new Lax-Milgram

Generalized Lax-Milgram theorem of Almog-Helffer7 Let

• 1. (V, ·, ·V) be a Hilbert space continuously embedded and dense in H
• 2. ℓ : V × V → C be a continuous sesquilinear form

3. A-H coercivity: ∃Φ1, Φ2 bounded linear maps on V and H and ∃δ > 0, ∀f ∈ V, |ℓ(f, f)| + |ℓ(Φ1f, f)| ≥ δf2

V,

|ℓ(f, f)| + |ℓ(f, Φ2f)| ≥ δf2

V.

Then the (densely defined) operator L Dom(L ) = {f ∈ V : ∃g ∈ H, ∀v ∈ V, ℓ(f, v) = g, v} , L f = g is bijective from Dom(L ) onto H (= ⇒ ρ(L ) = ∅ ).

• similar recent results8
• 7Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466.
• 8A. S. Bonnet-Ben Dhia, P. Ciarlet Jr., and C. M. Zw¨
• lf. J. Comput. Appl. Math. 234

(2010), pp. 1912–1919; A. F. M. ter Elst, M. Sauter, and H. Vogt. J. Funct. Anal. 269 (2015),

• pp. 705–744; L. Grubiˇ

si´ c et al. Mathematika 59 (2013), pp. 169–189.

A-H-L-M: how does it help?

A-H-L-M assumptions

• 1. (V, ·, ·V) be a Hilbert space continuously embedded and dense in H
• 2. ℓ : V × V → C be a continuous sesquilinear form

3. A-H coercivity: ∃Φ1, Φ2 bounded linear maps on V and H and ∃δ > 0, ∀f ∈ V, |ℓ(f, f)| + |ℓ(Φ1f, f)| ≥ δf2

V,

|ℓ(f, f)| + |ℓ(f, Φ2f)| ≥ δf2

V.

Back to ix3 example

A-H-L-M: how does it help?

A-H-L-M assumptions

• 1. (V, ·, ·V) be a Hilbert space continuously embedded and dense in H
• 2. ℓ : V × V → C be a continuous sesquilinear form

3. A-H coercivity: ∃Φ1, Φ2 bounded linear maps on V and H and ∃δ > 0, ∀f ∈ V, |ℓ(f, f)| + |ℓ(Φ1f, f)| ≥ δf2

V,

|ℓ(f, f)| + |ℓ(f, Φ2f)| ≥ δf2

V.

Back to ix3 example

• the form of −∂2

x + ix3: ℓ(f, f) = f′2 + i R x3|f(x)|2 dx,

A-H-L-M: how does it help?

A-H-L-M assumptions

• 1. (V, ·, ·V) be a Hilbert space continuously embedded and dense in H
• 2. ℓ : V × V → C be a continuous sesquilinear form

3. A-H coercivity: ∃Φ1, Φ2 bounded linear maps on V and H and ∃δ > 0, ∀f ∈ V, |ℓ(f, f)| + |ℓ(Φ1f, f)| ≥ δf2

V,

|ℓ(f, f)| + |ℓ(f, Φ2f)| ≥ δf2

V.

Back to ix3 example

• the form of −∂2

x + ix3: ℓ(f, f) = f′2 + i R x3|f(x)|2 dx,

• variational space (form domain)

V = Dom(ℓ) = H1(R) ∩ Dom(|x|

3 2 ),

· 2

V = · 2 H1 + |x|

3 2 · 2

A-H-L-M: how does it help?

A-H-L-M assumptions

• 1. (V, ·, ·V) be a Hilbert space continuously embedded and dense in H
• 2. ℓ : V × V → C be a continuous sesquilinear form

3. A-H coercivity: ∃Φ1, Φ2 bounded linear maps on V and H and ∃δ > 0, ∀f ∈ V, |ℓ(f, f)| + |ℓ(Φ1f, f)| ≥ δf2

V,

|ℓ(f, f)| + |ℓ(f, Φ2f)| ≥ δf2

V.

Back to ix3 example

• the form of −∂2

x + ix3: ℓ(f, f) = f′2 + i R x3|f(x)|2 dx,

• variational space (form domain)

V = Dom(ℓ) = H1(R) ∩ Dom(|x|

3 2 ),

· 2

V = · 2 H1 + |x|

3 2 · 2

• A-H coercivity: Φ1 = Φ2 :=

Im V |V |+1 · = x3 |x|3+1 ·

A-H-L-M: how does it help?

A-H-L-M assumptions

• 1. (V, ·, ·V) be a Hilbert space continuously embedded and dense in H
• 2. ℓ : V × V → C be a continuous sesquilinear form

3. A-H coercivity: ∃Φ1, Φ2 bounded linear maps on V and H and ∃δ > 0, ∀f ∈ V, |ℓ(f, f)| + |ℓ(Φ1f, f)| ≥ δf2

V,

|ℓ(f, f)| + |ℓ(f, Φ2f)| ≥ δf2

V.

Back to ix3 example

• the form of −∂2

x + ix3: ℓ(f, f) = f′2 + i R x3|f(x)|2 dx,

• variational space (form domain)

V = Dom(ℓ) = H1(R) ∩ Dom(|x|

3 2 ),

· 2

V = · 2 H1 + |x|

3 2 · 2

• A-H coercivity: Φ1 = Φ2 :=

Im V |V |+1 · = x3 |x|3+1 ·

ℓ(Φ1f, f) = i

• R

|x|3 |x|3 + 1 |x|3|f(x)|2 dx + . . .

Dirichlet realization

Theorem [KRRS16] Let assumptions on V and A be satisfied. Then the (densely defined) operator L associated by the A-H-L-M theorem with the form ℓ(f, f) = (−i∇ + A)f2 +

• Ω

V |f|2dx, Dom(ℓ) = V =

• f ∈ H1

A,0(Ω) : (|V | + |B| + 1)

1 2 f ∈ L2(Ω)

• ,

has a non-empty resolvent set.

Remarks on the assumption and separation property

|∇V (x)| + |∇B(x)| = o

• (|V (x)| + |B(x)|)

3 2 + 1

• ,

|x| → ∞ The power 3/2

• 9E. C. Titchmarsh. Eigenfunction expansions associated with second-order differential
• equations. Oxford: Clarendon Press, 1948.
• 10W. D. Evans and A. Zettl. Proc. Roy. Soc. Edinburgh Sect. A 80 (1978), pp. 151–162;
• W. N. Everitt and M. Giertz. Proc. Roy. Soc. Edinburgh Sect. A 79 (1978), pp. 257–265.

Remarks on the assumption and separation property

|∇V (x)| + |∇B(x)| = o

• (|V (x)| + |B(x)|)

3 2 + 1

• ,

|x| → ∞ The power 3/2

• appears in various places, e.g. Titchmarsh9
• optimal for the separation property of the domain in the self-adjoint case10
• 9E. C. Titchmarsh. Eigenfunction expansions associated with second-order differential
• equations. Oxford: Clarendon Press, 1948.
• 10W. D. Evans and A. Zettl. Proc. Roy. Soc. Edinburgh Sect. A 80 (1978), pp. 151–162;
• W. N. Everitt and M. Giertz. Proc. Roy. Soc. Edinburgh Sect. A 79 (1978), pp. 257–265.

Remarks on the assumption and separation property

|∇V (x)| + |∇B(x)| = o

• (|V (x)| + |B(x)|)

3 2 + 1

• ,

|x| → ∞ The power 3/2

• appears in various places, e.g. Titchmarsh9
• optimal for the separation property of the domain in the self-adjoint case10

Theorem [KRRS16] Let assumptions on V and A be satisfied. Then the separation property holds Dom(L ) = Dom((−i∇ + A)2) ∩ Dom(V ) and L · 2 + · 2 ∼ (−i∇ + A)2 · 2 + V · 2 + · 2

• 9E. C. Titchmarsh. Eigenfunction expansions associated with second-order differential
• equations. Oxford: Clarendon Press, 1948.
• 10W. D. Evans and A. Zettl. Proc. Roy. Soc. Edinburgh Sect. A 80 (1978), pp. 151–162;
• W. N. Everitt and M. Giertz. Proc. Roy. Soc. Edinburgh Sect. A 79 (1978), pp. 257–265.

Decay of eigenfunctions: classical setting

Single-well potentials (in L2(R))

• harmonic oscillator: −∂2

x + x2 in L2(R) and Hermite functions

|hn(x)| = Cn e− x2

2

Hn(x)

Decay of eigenfunctions: classical setting

Single-well potentials (in L2(R))

• harmonic oscillator: −∂2

x + x2 in L2(R) and Hermite functions

|hn(x)| = Cn e− x2

2

Hn(x)

Decay of eigenfunctions: classical setting

Single-well potentials (in L2(R))

• harmonic oscillator: −∂2

x + x2 in L2(R) and Hermite functions

|hn(x)| = Cn e− x2

2

Hn(x)

• Liouville-Green approximation for single-well potentials (V (xn) = λn)

|ψn(x)| ≤ Cn (V (x) − λn)

1 4

exp

• −

x

xn

(V (t) − λn)

1 2 dt

• ,

|x| ≥ xn + δ

Decay of eigenfunctions: classical setting

Single-well potentials (in L2(R))

• harmonic oscillator: −∂2

x + x2 in L2(R) and Hermite functions

|hn(x)| = Cn e− x2

2

Hn(x)

• Liouville-Green approximation for single-well potentials (V (xn) = λn)

|ψn(x)| ≤ Cn (V (x) − λn)

1 4

exp

• −

x

xn

(V (t) − λn)

1 2 dt

• ,

|x| ≥ xn + δ

• the eigenfunctions satisfy

eW ψn ∈ L2(R) with the weight W = W(V ) (for single wells: V

1 2 )

Weighted coercivity and eigenfunction decay

Theorem [KRRS16]

• eigenvalue λ ∈ σp(L ) ∩ Λ(V, B),

eigenfunction ψ: L ψ = λψ

Weighted coercivity and eigenfunction decay

Theorem [KRRS16]

• eigenvalue λ ∈ σp(L ) ∩ Λ(V, B),

eigenfunction ψ: L ψ = λψ

• weight W such that

|∇W|2 ≤ (γ1(|V | + |B| + 1) − Re(λ) − | Im(λ)| − γ2)+

Weighted coercivity and eigenfunction decay

Theorem [KRRS16]

• eigenvalue λ ∈ σp(L ) ∩ Λ(V, B),

eigenfunction ψ: L ψ = λψ

• weight W such that

|∇W|2 ≤ (γ1(|V | + |B| + 1) − Re(λ) − | Im(λ)| − γ2)+ Then e

1−ε 3

W ψ ∈ L2(Ω) ,

ε ∈ (0, 1).

Weighted coercivity and eigenfunction decay

Theorem [KRRS16]

• eigenvalue λ ∈ σp(L ) ∩ Λ(V, B),

eigenfunction ψ: L ψ = λψ

• weight W such that

|∇W|2 ≤ (γ1(|V | + |B| + 1) − Re(λ) − | Im(λ)| − γ2)+ Then e

1−ε 3

W ψ ∈ L2(Ω) ,

ε ∈ (0, 1).

• the same conclusion holds for generalized eigenfunctions (root vectors)

Weighted coercivity and eigenfunction decay

Theorem [KRRS16]

• eigenvalue λ ∈ σp(L ) ∩ Λ(V, B),

eigenfunction ψ: L ψ = λψ

• weight W such that

|∇W|2 ≤ (γ1(|V | + |B| + 1) − Re(λ) − | Im(λ)| − γ2)+ Then e

1−ε 3

W ψ ∈ L2(Ω) ,

ε ∈ (0, 1).

• the same conclusion holds for generalized eigenfunctions (root vectors)
• weight optimal (possibly up to constants) for complex polynomial potentials11

Weighted coercivity and eigenfunction decay

Theorem [KRRS16]

• eigenvalue λ ∈ σp(L ) ∩ Λ(V, B),

eigenfunction ψ: L ψ = λψ

• weight W such that

|∇W|2 ≤ (γ1(|V | + |B| + 1) − Re(λ) − | Im(λ)| − γ2)+ Then e

1−ε 3

W ψ ∈ L2(Ω) ,

ε ∈ (0, 1).

• the same conclusion holds for generalized eigenfunctions (root vectors)
• weight optimal (possibly up to constants) for complex polynomial potentials11

Theorem [KRRS16] For every W ∈ W 1,∞(Ω; R) and all f ∈ C∞

0 (Ω), we have

Re ℓ(f, e2W f) + Im ℓ(f, Φe2W f) ≥ 1 2 (−i∇ + A)eW f2 +

• Ω

|eW f|2

• (Im V )2 +

1 12d |B|2

|V | + |B| + 1 + Re V − 9 |∇Φ|2 + |∇Ψ|2 + |∇W|2 dx

Summary of results

Main assumption |∇V (x)| + |∇B(x)| = o

• (|V (x)| + |B(x)|)

3 2 + 1

• ,

(Re V (x))− = o

• |V (x)| + |B(x)| + 1
• ,

|x| → ∞

Summary of results

Main assumption |∇V (x)| + |∇B(x)| = o

• (|V (x)| + |B(x)|)

3 2 + 1

• ,

(Re V (x))− = o

• |V (x)| + |B(x)| + 1
• ,

|x| → ∞ Results

• Dirichlet realization of L = (−i∇ + A)2 + V in L2(Ω) with ρ(L ) = ∅

Summary of results

Main assumption |∇V (x)| + |∇B(x)| = o

• (|V (x)| + |B(x)|)

3 2 + 1

• ,

(Re V (x))− = o

• |V (x)| + |B(x)| + 1
• ,

|x| → ∞ Results

• Dirichlet realization of L = (−i∇ + A)2 + V in L2(Ω) with ρ(L ) = ∅
• separation property of Dom(L )

Dom(L ) = Dom((−i∇ + A)2) ∩ Dom(V )

Summary of results

Main assumption |∇V (x)| + |∇B(x)| = o

• (|V (x)| + |B(x)|)

3 2 + 1

• ,

(Re V (x))− = o

• |V (x)| + |B(x)| + 1
• ,

|x| → ∞ Results

• Dirichlet realization of L = (−i∇ + A)2 + V in L2(Ω) with ρ(L ) = ∅
• separation property of Dom(L )

Dom(L ) = Dom((−i∇ + A)2) ∩ Dom(V )

• decay of eigenfunctions W = W(V, B): eW ψ ∈ L2(Ω)

Summary of results

Main assumption |∇V (x)| + |∇B(x)| = o

• (|V (x)| + |B(x)|)

3 2 + 1

• ,

(Re V (x))− = o

• |V (x)| + |B(x)| + 1
• ,

|x| → ∞ Results

• Dirichlet realization of L = (−i∇ + A)2 + V in L2(Ω) with ρ(L ) = ∅
• separation property of Dom(L )

Dom(L ) = Dom((−i∇ + A)2) ∩ Dom(V )

• decay of eigenfunctions W = W(V, B): eW ψ ∈ L2(Ω)
• e.g. if |V (x)| + |B(x)| → ∞ as |x| → ∞, then

W(x) ≥ γ|x| − M

2017 CIRM conference

CIRM conference on

Mathematical aspects of the physics with non-self-adjoint operators

5 – 9 June 2017 Marseille, France

Marseille, colonie grecque 1869 by Pierre Puvis de Chavannes Mus´ ee des beaux-arts de Marseille

http://www.ujf.cas.cz/NSAatCIRM

Invited speakers:

Wolfgang Arendt (Ulm) Anne Sophie Bonnet-BenDhia (Paris) Lyonell Boulton (Edinburgh) Nicolas Burq (Orsay) Cristina Cˆ amara (Lisbon) A.F.M. ter Elst (Auckland) Luca Fanelli (Rome) Eduard Feireisl (Prague) Didier Felbacq (Montpellier) Eva A. Gallardo Guti´ errez (Madrid) Ilya Goldsheid (London) Bernard Helffer (Orsay) Patrick Joly (Paris) Martin Kolb (Paderborn) Vadim Kostrykin (Mainz) Stanislas Kupin (Bordeaux) Yehuda Pinchover (Haifa) Zdenˇ ek Strakoˇ s (Prague) Christiane Tretter (Bern)

Organisers:

David Krejˇ ciˇ r´ ık (Prague) Petr Siegl (Bern)

Advisory board:

Guy Bouchitt´ e (Toulon) Fritz Gesztesy (Columbia) Alain Joye (Grenoble) Luis Vega (Bilbao)

The conference is made possible by the kind financial support from and organised at:

Centre International de Rencontres Math´ ematiques

Corollaries of results

Domain truncations [B¨

• gli-S-Tretter’15]12
• L = −∆ + V in L2(R), |V (x)| → ∞ as |x| → ∞ (⇒ compact resolvent of L )
• 12S. B¨
• gli, P Siegl, and C. Tretter. arXiv: 1512.01826. 2015.
• 13Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466.

Corollaries of results

Domain truncations [B¨

• gli-S-Tretter’15]12
• L = −∆ + V in L2(R), |V (x)| → ∞ as |x| → ∞ (⇒ compact resolvent of L )
• Ln = −∆ + V in L2(Bn(0)) with Dirichlet b.c.
• 12S. B¨
• gli, P Siegl, and C. Tretter. arXiv: 1512.01826. 2015.
• 13Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466.

Corollaries of results

Domain truncations [B¨

• gli-S-Tretter’15]12
• L = −∆ + V in L2(R), |V (x)| → ∞ as |x| → ∞ (⇒ compact resolvent of L )
• Ln = −∆ + V in L2(Bn(0)) with Dirichlet b.c.
• the approximation is spectrally exact
• 1. Every λ ∈ σ(L ) is approximated: there is {λn}, λn ∈ σ(Ln), such that

λn → λ as n → ∞.

• 2. No pollution: if {λn}, λn ∈ σ(Ln), has an accumulation point λ, then

λ ∈ σ(L ).

• 12S. B¨
• gli, P Siegl, and C. Tretter. arXiv: 1512.01826. 2015.
• 13Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466.

Corollaries of results

Domain truncations [B¨

• gli-S-Tretter’15]12
• L = −∆ + V in L2(R), |V (x)| → ∞ as |x| → ∞ (⇒ compact resolvent of L )
• Ln = −∆ + V in L2(Bn(0)) with Dirichlet b.c.
• the approximation is spectrally exact
• 1. Every λ ∈ σ(L ) is approximated: there is {λn}, λn ∈ σ(Ln), such that

λn → λ as n → ∞.

• 2. No pollution: if {λn}, λn ∈ σ(Ln), has an accumulation point λ, then

λ ∈ σ(L ).

• the rate estimate (for a simple λ ∈ σ(L ), L ψ = λψ)

|λ − λn| ≤

• ψ ↾ Rd \ Bn(0)
• ψ
• 12S. B¨
• gli, P Siegl, and C. Tretter. arXiv: 1512.01826. 2015.
• 13Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466.

Corollaries of results

Domain truncations [B¨

• gli-S-Tretter’15]12
• L = −∆ + V in L2(R), |V (x)| → ∞ as |x| → ∞ (⇒ compact resolvent of L )
• Ln = −∆ + V in L2(Bn(0)) with Dirichlet b.c.
• the approximation is spectrally exact
• 1. Every λ ∈ σ(L ) is approximated: there is {λn}, λn ∈ σ(Ln), such that

λn → λ as n → ∞.

• 2. No pollution: if {λn}, λn ∈ σ(Ln), has an accumulation point λ, then

λ ∈ σ(L ).

• the rate estimate (for a simple λ ∈ σ(L ), L ψ = λψ)

|λ − λn| ≤

• ψ ↾ Rd \ Bn(0)
• ψ

Corrolaries of [KRRS16]

• exponential convergence rate of |λ − λn|
• 12S. B¨
• gli, P Siegl, and C. Tretter. arXiv: 1512.01826. 2015.
• 13Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466.

Corollaries of results

Domain truncations [B¨

• gli-S-Tretter’15]12
• L = −∆ + V in L2(R), |V (x)| → ∞ as |x| → ∞ (⇒ compact resolvent of L )
• Ln = −∆ + V in L2(Bn(0)) with Dirichlet b.c.
• the approximation is spectrally exact
• 1. Every λ ∈ σ(L ) is approximated: there is {λn}, λn ∈ σ(Ln), such that

λn → λ as n → ∞.

• 2. No pollution: if {λn}, λn ∈ σ(Ln), has an accumulation point λ, then

λ ∈ σ(L ).

• the rate estimate (for a simple λ ∈ σ(L ), L ψ = λψ)

|λ − λn| ≤

• ψ ↾ Rd \ Bn(0)
• ψ

Corrolaries of [KRRS16]

• exponential convergence rate of |λ − λn|
• extension of results of Almog-Helffer on the completeness of eigensystem13
• 12S. B¨
• gli, P Siegl, and C. Tretter. arXiv: 1512.01826. 2015.
• 13Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466.

Examples

L = −∂2

x + ix3

• all eigenvalues are real14
• 1

2 3 4 sn 10 5 5 10 15 20 ReΛ

• 1

2 3 4 sn 10 5 5 10 ImΛ

• 14K. C. Shin. Comm. Math. Phys. 229 (2002), pp. 543–564.

Examples

L = −∂2

x + ix3

• all eigenvalues are real14
• 1

2 3 4 sn 10 5 5 10 15 20 ReΛ

• 1

2 3 4 sn 10 5 5 10 ImΛ

• the first eigenvalue and the rate (Dirichlet BC)
• 1

2 3 4 sn 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 ReΛ

• 1

2 3 4 sn 15 10 5 logΛ1Λn1

• 14K. C. Shin. Comm. Math. Phys. 229 (2002), pp. 543–564.