Non-accretive Schrödinger operators and exponential decay
of their eigenfunctions
Petr Siegl
Mathematical Institute, University of Bern, Switzerland
http://gemma.ujf.cas.cz/˜siegl/
Based on
[1] D. Krejčiřı́k, N. Raymond, J. Royer, and P. Siegl: Non-accretive Schrödinger
operators and exponential decay of their eigenfunctions
Israel Journal of Mathematics, to appear
arXiv:1605.02437
Schrödinger operators with complex potentials
Main object
• Dirichlet realization of
L = (−i∇ + A)2 + V
• Ω ⊂ Rd open (no additional assumptions)
• V ∈ C 1 (Ω; C ) and A ∈ C 2 (Ω; Rd )
in L2 (Ω)
Schrödinger operators with complex potentials
Main object
• Dirichlet realization of
L = (−i∇ + A)2 + V
in L2 (Ω)
• Ω ⊂ Rd open (no additional assumptions)
• V ∈ C 1 (Ω; C ) and A ∈ C 2 (Ω; Rd )
• restriction on the growth, oscillations and negative Re V :
3
|∇V (x)| + |∇B(x)| = o (|V (x)| + |B(x)|) 2 + 1 ,
(Re V (x))− = o |V (x)| + |B(x)| + 1 ,
• where
B = (Bjk )j,k∈{1,...,d} ,
Bjk := ∂j Ak − ∂k Aj
|x| → ∞
Schrödinger operators with complex potentials
Main object
• Dirichlet realization of
L = (−i∇ + A)2 + V
in L2 (Ω)
• Ω ⊂ Rd open (no additional assumptions)
• V ∈ C 1 (Ω; C ) and A ∈ C 2 (Ω; Rd )
• restriction on the growth, oscillations and negative Re V :
3
|∇V (x)| + |∇B(x)| = o (|V (x)| + |B(x)|) 2 + 1 ,
(Re V (x))− = o |V (x)| + |B(x)| + 1 ,
|x| → ∞
• where
B = (Bjk )j,k∈{1,...,d} ,
Bjk := ∂j Ak − ∂k Aj
Objectives
1. find the Dirichlet realization with ρ(L ) 6= ∅ and describe Dom(L )
2. prove the exponential decay of eigenfunctions of L (due to Im V and B)
Why complex potentials?
• superconductivity1
−∂x2 + i∂y − x2
2
+ iy ,
in
L2 (R2 )
• optics with gains and losses2
2
2
−∆ + (1 + ixy )e−x e−y ,
in
L2 (R2 )
• hydrodynamics3
−
d2
i
+ x2 + f (x) ,
dx2
ε
in
L2 (R)
• open systems4 , quantum resonances5 , damped wave equation6 ,. . .
1
Y. Almog, B. Helffer, and X.-B. Pan. Trans. Amer. Math. Soc. (2013), pp. 1183–1217.
A. Regensburger et al. Phys. Rev. Lett. 107 (2011), p. 233902; J. Yang. Opt. Lett. 39
(2014), pp. 1133–1136.
3
I. Gallagher, T. Gallay, and F. Nier. Int. Math. Res. Not. IMRN (2009), pp. 2147–2199.
4
P. Exner. Open quantum systems and Feynman integrals. D. Reidel Publishing Co., 1985.
5
A. A. Abramov, A. Aslanyan, and E. B. Davies. J. Phys. A: Math. Gen. 34 (2001), p. 57.
6
J. Sjöstrand. Publ. Res. Inst. Math. Sci. 36 (2000), pp. 573–611.
2
Towards the Dirichlet realization: form methods
Simple 1D examples in L2 (R)
−
d2
+ ix3 ,
dx2
−
d2
− x2 + ix3 ,
dx2
−
2
4
d2
− ex + iex
dx2
Towards the Dirichlet realization: form methods
Simple 1D examples in L2 (R)
−
d2
+ ix3 ,
dx2
−
d2
− x2 + ix3 ,
dx2
−
2
4
d2
− ex + iex
dx2
Lax-Milgram theorem
Let
1. (V, h·, ·iV ) be a Hilbert space continuously embedded and dense in H
2. ` : V × V → C be a continuous sesquilinear form
3. ` be V-elliptic (V-coercive or coercive )
∃δ > 0,
∀f ∈ V,
|`(f, f )| ≥ δkf kV
Towards the Dirichlet realization: form methods
Simple 1D examples in L2 (R)
−
d2
+ ix3 ,
dx2
−
d2
− x2 + ix3 ,
dx2
−
2
4
d2
− ex + iex
dx2
Lax-Milgram theorem
Let
1. (V, h·, ·iV ) be a Hilbert space continuously embedded and dense in H
2. ` : V × V → C be a continuous sesquilinear form
3. ` be V-elliptic (V-coercive or coercive )
∃δ > 0,
∀f ∈ V,
|`(f, f )| ≥ δkf kV
Then the (densely defined) operator L
Dom(L ) = {f ∈ V : ∃g ∈ H, ∀v ∈ V, `(f, v) = hg, vi} ,
Lf = g
is bijective from Dom(L ) onto H (=⇒ ρ(L ) 6= ∅ ).
Towards the Dirichlet realization
Assumptions Lax-Milgram theorem
1. (V, h·, ·iV ) be a Hilbert space continuously embedded and dense in H
2. ` : V × V → C be a continuous sesquilinear form
3. ` be V-elliptic (V-coercive or coercive )
∃δ > 0,
Why it doesn’t work?
∀f ∈ V,
|`(f, f )| ≥ δkf kV
Towards the Dirichlet realization
Assumptions Lax-Milgram theorem
1. (V, h·, ·iV ) be a Hilbert space continuously embedded and dense in H
2. ` : V × V → C be a continuous sesquilinear form
3. ` be V-elliptic (V-coercive or coercive )
∃δ > 0,
∀f ∈ V,
|`(f, f )| ≥ δkf kV
Why it doesn’t work?
2 + ix3 :
• natural candidate for the form of −∂x
`(f, f ) = h−f 00 + ix3 f, f i = kf 0 k2 + i
Z
R
x3 |f (x)|2 dx,
Towards the Dirichlet realization
Assumptions Lax-Milgram theorem
1. (V, h·, ·iV ) be a Hilbert space continuously embedded and dense in H
2. ` : V × V → C be a continuous sesquilinear form
3. ` be V-elliptic (V-coercive or coercive )
∃δ > 0,
∀f ∈ V,
|`(f, f )| ≥ δkf kV
Why it doesn’t work?
2 + ix3 :
• natural candidate for the form of −∂x
`(f, f ) = h−f 00 + ix3 f, f i = kf 0 k2 + i
Z
x3 |f (x)|2 dx,
R
• variational space (form domain)
3
V = Dom(`) = H 1 (R) ∩ Dom(|x| 2 ),
3
k · k2V = k · k2H 1 + k|x| 2 · k2
Towards the Dirichlet realization
Assumptions Lax-Milgram theorem
1. (V, h·, ·iV ) be a Hilbert space continuously embedded and dense in H
2. ` : V × V → C be a continuous sesquilinear form
3. ` be V-elliptic (V-coercive or coercive )
∃δ > 0,
∀f ∈ V,
|`(f, f )| ≥ δkf kV
Why it doesn’t work?
2 + ix3 :
• natural candidate for the form of −∂x
`(f, f ) = h−f 00 + ix3 f, f i = kf 0 k2 + i
Z
x3 |f (x)|2 dx,
R
• variational space (form domain)
3
V = Dom(`) = H 1 (R) ∩ Dom(|x| 2 ),
!!! no coercivity
3
k · k2V = k · k2H 1 + k|x| 2 · k2
Towards the Dirichlet realization: new Lax-Milgram
Generalized Lax-Milgram theorem of Almog-Helffer7
Let
1. (V, h·, ·iV ) be a Hilbert space continuously embedded and dense in H
2. ` : V × V → C be a continuous sesquilinear form
7
Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466.
A. S. Bonnet-Ben Dhia, P. Ciarlet Jr., and C. M. Zwölf. 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. Grubišić et al. Mathematika 59 (2013), pp. 169–189.
8
Towards the Dirichlet realization: new Lax-Milgram
Generalized Lax-Milgram theorem of Almog-Helffer7
Let
1. (V, h·, ·iV ) be a Hilbert space continuously embedded and dense in H
2. ` : V × V → C be a continuous sesquilinear form
3. A-H coercivity: ∃Φ1 , Φ2 bounded linear maps on V and H and
∃δ > 0,
7
∀f ∈ V,
|`(f, f )| + |`(Φ1 f, f )| ≥ δkf k2V ,
|`(f, f )| + |`(f, Φ2 f )| ≥ δkf k2V .
Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466.
A. S. Bonnet-Ben Dhia, P. Ciarlet Jr., and C. M. Zwölf. 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. Grubišić et al. Mathematika 59 (2013), pp. 169–189.
8
Towards the Dirichlet realization: new Lax-Milgram
Generalized Lax-Milgram theorem of Almog-Helffer7
Let
1. (V, h·, ·iV ) be a Hilbert space continuously embedded and dense in H
2. ` : 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,
|`(f, f )| + |`(Φ1 f, f )| ≥ δkf k2V ,
|`(f, f )| + |`(f, Φ2 f )| ≥ δkf k2V .
Then the (densely defined) operator L
Dom(L ) = {f ∈ V : ∃g ∈ H, ∀v ∈ V, `(f, v) = hg, vi} ,
Lf = g
is bijective from Dom(L ) onto H (=⇒ ρ(L ) 6= ∅ ).
7
Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466.
A. S. Bonnet-Ben Dhia, P. Ciarlet Jr., and C. M. Zwölf. 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. Grubišić et al. Mathematika 59 (2013), pp. 169–189.
8
Towards the Dirichlet realization: new Lax-Milgram
Generalized Lax-Milgram theorem of Almog-Helffer7
Let
1. (V, h·, ·iV ) be a Hilbert space continuously embedded and dense in H
2. ` : 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,
|`(f, f )| + |`(Φ1 f, f )| ≥ δkf k2V ,
|`(f, f )| + |`(f, Φ2 f )| ≥ δkf k2V .
Then the (densely defined) operator L
Dom(L ) = {f ∈ V : ∃g ∈ H, ∀v ∈ V, `(f, v) = hg, vi} ,
Lf = g
is bijective from Dom(L ) onto H (=⇒ ρ(L ) 6= ∅ ).
• similar recent results8
7
Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466.
A. S. Bonnet-Ben Dhia, P. Ciarlet Jr., and C. M. Zwölf. 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. Grubišić et al. Mathematika 59 (2013), pp. 169–189.
8
A-H-L-M: how does it help?
A-H-L-M assumptions
1. (V, h·, ·iV ) be a Hilbert space continuously embedded and dense in H
2. ` : V × V → C be a continuous sesquilinear form
3. A-H coercivity: ∃Φ1 , Φ2 bounded linear maps on V and H and
∃δ > 0,
Back to ix3 example
∀f ∈ V,
|`(f, f )| + |`(Φ1 f, f )| ≥ δkf k2V ,
|`(f, f )| + |`(f, Φ2 f )| ≥ δkf k2V .
A-H-L-M: how does it help?
A-H-L-M assumptions
1. (V, h·, ·iV ) be a Hilbert space continuously embedded and dense in H
2. ` : 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,
|`(f, f )| + |`(Φ1 f, f )| ≥ δkf k2V ,
|`(f, f )| + |`(f, Φ2 f )| ≥ δkf k2V .
Back to ix3 example
R
2 + ix3 : `(f, f ) = kf 0 k2 + i
• the form of −∂x
x3 |f (x)|2 dx,
R
A-H-L-M: how does it help?
A-H-L-M assumptions
1. (V, h·, ·iV ) be a Hilbert space continuously embedded and dense in H
2. ` : 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,
|`(f, f )| + |`(Φ1 f, f )| ≥ δkf k2V ,
|`(f, f )| + |`(f, Φ2 f )| ≥ δkf k2V .
Back to ix3 example
R
2 + ix3 : `(f, f ) = kf 0 k2 + i
• the form of −∂x
x3 |f (x)|2 dx,
R
• variational space (form domain)
3
V = Dom(`) = H 1 (R) ∩ Dom(|x| 2 ),
3
k · k2V = k · k2H 1 + k|x| 2 · k2
A-H-L-M: how does it help?
A-H-L-M assumptions
1. (V, h·, ·iV ) be a Hilbert space continuously embedded and dense in H
2. ` : 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,
|`(f, f )| + |`(Φ1 f, f )| ≥ δkf k2V ,
|`(f, f )| + |`(f, Φ2 f )| ≥ δkf k2V .
Back to ix3 example
R
2 + ix3 : `(f, f ) = kf 0 k2 + i
• the form of −∂x
x3 |f (x)|2 dx,
R
• variational space (form domain)
3
V = Dom(`) = H 1 (R) ∩ Dom(|x| 2 ),
•
A-H coercivity: Φ1 = Φ2 :=
Im V
·
|V |+1
=
3
k · k2V = k · k2H 1 + k|x| 2 · k2
x3
·
|x|3 +1
A-H-L-M: how does it help?
A-H-L-M assumptions
1. (V, h·, ·iV ) be a Hilbert space continuously embedded and dense in H
2. ` : 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, f )| + |`(Φ1 f, f )| ≥ δkf k2V ,
∀f ∈ V,
|`(f, f )| + |`(f, Φ2 f )| ≥ δkf k2V .
Back to ix3 example
R
2 + ix3 : `(f, f ) = kf 0 k2 + i
• the form of −∂x
x3 |f (x)|2 dx,
R
• variational space (form domain)
3
V = Dom(`) = H 1 (R) ∩ Dom(|x| 2 ),
•
A-H coercivity: Φ1 = Φ2 :=
Im V
·
|V |+1
Z
`(Φ1 f, f ) = i
R
=
3
k · k2V = k · k2H 1 + k|x| 2 · k2
x3
·
|x|3 +1
|x|3
|x|3 |f (x)|2 dx + . . .
|x|3 + 1
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
`(f, f ) = k(−i∇ + A)f k2 +
Z
V |f |2 dx,
Ω
Dom(`) = V =
n
1
1
f ∈ HA,0
(Ω) : (|V | + |B| + 1) 2 f ∈ L2 (Ω)
has a non-empty resolvent set.
o
,
Remarks on the assumption and separation property
3
|∇V (x)| + |∇B(x)| = o (|V (x)| + |B(x)|) 2 + 1 ,
|x| → ∞
The power 3/2
9
E. C. Titchmarsh. Eigenfunction expansions associated with second-order differential
equations. Oxford: Clarendon Press, 1948.
10
W. 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
3
|∇V (x)| + |∇B(x)| = o (|V (x)| + |B(x)|) 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
9
E. C. Titchmarsh. Eigenfunction expansions associated with second-order differential
equations. Oxford: Clarendon Press, 1948.
10
W. 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
3
|∇V (x)| + |∇B(x)| = o (|V (x)| + |B(x)|) 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
kL · k2 + k · k2 ∼ k(−i∇ + A)2 · k2 + kV · k2 + k · k2
9
E. C. Titchmarsh. Eigenfunction expansions associated with second-order differential
equations. Oxford: Clarendon Press, 1948.
10
W. 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))
2 + x2 in L2 (R) and Hermite functions
• harmonic oscillator: −∂x
|hn (x)| = Cn e−
x2
2
Hn (x)
Decay of eigenfunctions: classical setting
Single-well potentials (in L2 (R))
2 + x2 in L2 (R) and Hermite functions
• harmonic oscillator: −∂x
|hn (x)| = Cn e−
x2
2
Hn (x)
Decay of eigenfunctions: classical setting
Single-well potentials (in L2 (R))
2 + x2 in L2 (R) and Hermite functions
• harmonic oscillator: −∂x
|hn (x)| = Cn e−
x2
2
Hn (x)
• Liouville-Green approximation for single-well potentials (V (xn ) = λn )
|ψn (x)| ≤
Z
Cn
1
(V (x) − λn ) 4
exp
x
1
(V (t) − λn ) 2 dt
−
xn
,
|x| ≥ xn + δ
Decay of eigenfunctions: classical setting
Single-well potentials (in L2 (R))
2 + x2 in L2 (R) and Hermite functions
• harmonic oscillator: −∂x
|hn (x)| = Cn e−
x2
2
Hn (x)
• Liouville-Green approximation for single-well potentials (V (xn ) = λn )
|ψn (x)| ≤
Z
Cn
1
exp
(V (x) − λn ) 4
x
1
(V (t) − λn ) 2 dt
−
,
xn
• the eigenfunctions satisfy
eW ψn ∈ L2 (R)
with the weight W = W (V )
(for single wells:
R
V
1
2
)
|x| ≥ xn + δ
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
W
3
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
W
3
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
W
3
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]
eigenfunction ψ: L ψ = λψ
• eigenvalue λ ∈ σp (L ) ∩ Λ(V, B),
• weight W such that
|∇W |2 ≤ (γ1 (|V | + |B| + 1) − Re(λ) − | Im(λ)| − γ2 )+
Then
e
1−ε
W
3
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 ∈ C0∞ (Ω), we have
1
k(−i∇ + A)eW f k2
2
Z
1
(Im V )2 + 12d
|B|2
+
|eW f |2
+ Re V
|V | + |B| + 1
Ω
Re `(f, e2W f ) + Im `(f, Φe2W f ) ≥
− 9 |∇Φ|2 + |∇Ψ|2 + |∇W |2
dx
Summary of results
Main assumption
3
|∇V (x)| + |∇B(x)| = o (|V (x)| + |B(x)|) 2 + 1 ,
(Re V (x))− = o |V (x)| + |B(x)| + 1 ,
|x| → ∞
Summary of results
Main assumption
3
|∇V (x)| + |∇B(x)| = o (|V (x)| + |B(x)|) 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 ) 6= ∅
Summary of results
Main assumption
3
|∇V (x)| + |∇B(x)| = o (|V (x)| + |B(x)|) 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 ) 6= ∅
• separation property of Dom(L )
Dom(L ) = Dom((−i∇ + A)2 ) ∩ Dom(V )
Summary of results
Main assumption
3
|∇V (x)| + |∇B(x)| = o (|V (x)| + |B(x)|) 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 ) 6= ∅
• 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
3
|∇V (x)| + |∇B(x)| = o (|V (x)| + |B(x)|) 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 ) 6= ∅
• 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
Invited speakers:
5 – 9 June 2017
Marseille, France
Wolfgang Arendt (Ulm)
Anne Sophie Bonnet-BenDhia (Paris)
Lyonell Boulton (Edinburgh)
Nicolas Burq (Orsay)
Cristina Câmara (Lisbon)
A.F.M. ter Elst (Auckland)
Luca Fanelli (Rome)
Eduard Feireisl (Prague)
Didier Felbacq (Montpellier)
Eva A. Gallardo Gutiérrez (Madrid)
Ilya Goldsheid (London)
Bernard Helffer (Orsay)
Patrick Joly (Paris)
Martin Kolb (Paderborn)
Marseille, colonie grecque 1869 by Pierre Puvis de Chavannes
Musée des beaux-arts de Marseille
Vadim Kostrykin (Mainz)
Stanislas Kupin (Bordeaux)
Yehuda Pinchover (Haifa)
http://www.ujf.cas.cz/NSAatCIRM
Zdeněk Strakoš (Prague)
Christiane Tretter (Bern)
Organisers:
David Krejčiřı́k (Prague)
Petr Siegl (Bern)
Advisory board:
Guy Bouchitté (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 Mathématiques
Corollaries of results
Domain truncations [Bögli-S-Tretter’15]12
• L = −∆ + V in L2 (R), |V (x)| → ∞ as |x| → ∞ (⇒ compact resolvent of L )
12
13
S. Bögli, P Siegl, and C. Tretter. arXiv: 1512.01826. 2015.
Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466.
Corollaries of results
Domain truncations [Bögli-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.
12
13
S. Bögli, P Siegl, and C. Tretter. arXiv: 1512.01826. 2015.
Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466.
Corollaries of results
Domain truncations [Bögli-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 ).
12
13
S. Bögli, P Siegl, and C. Tretter. arXiv: 1512.01826. 2015.
Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466.
Corollaries of results
Domain truncations [Bögli-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 | ≤
12
13
ψ Rd \ Bn (0)
kψk
S. Bögli, P Siegl, and C. Tretter. arXiv: 1512.01826. 2015.
Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466.
Corollaries of results
Domain truncations [Bögli-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)
kψk
Corrolaries of [KRRS16]
• exponential convergence rate of |λ − λn |
12
13
S. Bögli, P Siegl, and C. Tretter. arXiv: 1512.01826. 2015.
Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466.
Corollaries of results
Domain truncations [Bögli-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)
kψk
Corrolaries of [KRRS16]
• exponential convergence rate of |λ − λn |
• extension of results of Almog-Helffer on the completeness of eigensystem13
12
13
S. Bögli, P Siegl, and C. Tretter. arXiv: 1512.01826. 2015.
Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466.
Examples
L = −∂x2 + ix3
• all eigenvalues are real14
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