The spectral shift function for Dirac operators
with electrostatic -shell interactions
Jussi Behrndt (TU Graz)
with Fritz Gesztesy (Waco) and Shu Nakamura (Tokyo)
Linear and Nonlinear Dirac Equation, Como, 2017
Jussi Behrndt
Spectral shift function for Dirac operators with -potential
Contents
I. Spectral shift function
II. Representation of the SSF via Weyl function
III. Dirac operators with electrostatic -shell interactions
Jussi Behrndt
Spectral shift function for Dirac operators with -potential
PART I
Spectral shift function
Jussi Behrndt
Spectral shift function for Dirac operators with -potential
A very brief history of the spectral shift function
This introduction is based on:
BirmanYafaev’92: The spectral shift function. The papers of
M. G. Krein and their further development
BirmanPushnitski’98: Spectral shift function, amazing and
multifaceted
Yafaev’92 and ’10: Mathematical Scattering Theory
General assumption
H Hilbert space, A, B selfadjoint (unbounded) operators in H
I. M. Lifshitz, 1952
B A finite rank operator. Then exists ⇠ : R ! R such that
formally
Z
tr '(B)
'(A) =
Jussi Behrndt
R
'0 (t)⇠(t) dt
Spectral shift function for Dirac operators with -potential
Krein’s spectral shift function (1953 and 1962)
Theorem
Assume B A trace class operator, i.e. B
exists real-valued ⇠ 2 L1 (R) such that
Z
1
1
tr (B
)
(A
)
=
R
and
R
R ⇠(t) dt
= tr (B
A 2 S1 . Then
⇠(t)
(t
)2
dt
A).
R
tr '(B) '(A) = R '0 (t)⇠(t) dt for '(t)R= (t
) 1
Extends to Wiener class W1 (R): '0 (t) = e itµ d (µ)
Corollary
If
= (a, b) and \
⇠(b )
ess (A)
= ; then
⇠(a+) = dim ran EB ( )
dim ran EA ( )
Spectral shift function for U, V unitary, V
Jussi Behrndt
U 2 S1
Spectral shift function for Dirac operators with -potential
Krein’s spectral shift function (1953 and 1962)
Theorem
Assume
1
1
2 S1 ,
2 ⇢(A) \ ⇢(B).
(1)
R
Then exists ⇠ 2 L1loc (R) such that R |⇠(t)|(1 + t 2 ) 1 dt < 1 and
(B
)
tr (B
)
(A
)
1
(A
)
1
=
Z
⇠(t)
R
(t
)2
dt.
The function ⇠ is unique up to a real constant.
Trace formula for '(t) = (t
) 1 and '(t) = (t
) k
Large class of ' in trace formula in Peller’85
Birman-Krein formula
Assume (1). The scattering matrix {S( )} of {A, B} satisfies
det S( ) = e
2⇡i⇠( )
Jussi Behrndt
for a.e.
2R
Spectral shift function for Dirac operators with -potential
Krein’s spectral shift function: Generalizations
L.S. Koplienko 1971
Assume ⇢(A) \ ⇢(B) \ R 6= ; and for some m 2 N:
(B
)
m
(A
Then exists ⇠ 2 L1loc (R) such that
tr (B
)
m
(A
)
m
)
R
m
R |⇠(t)|(1
=
Z
R
(2)
2 S1 .
(t
+ |t|)
(m+1) dt
<1
m
⇠(t) dt.
)m+1
D.R. Yafaev 2005
Assume
m 2 N odd. Then exists ⇠ 2 L1loc (R) such
R (2) for some (m+1)
that R |⇠(t)|(1 + |t|)
dt < 1
Z
m
m
m
tr (B
)
(A
)
=
⇠(t) dt.
)m+1
R (t
Jussi Behrndt
Spectral shift function for Dirac operators with -potential
Some references and contributors
BuslaevFaddeev’60
BirmanSolomyak’72
KreinYavryan’81
GesztesySimon’96
Pushnitski’97
Adamyan, Albeverio, Azamov, Bony, Bruneau, Carey,
Chattopadhyay, Combes, Demirel, Dimassi, Dodds, Dykema,
Fernandez, Geisler, Hislop, Hundertmark, Ivrii, Khochman,
Killip, Klopp, Kohmoto, Koma, Kostrykin, Krüger, Langer,
Latushkin, Makarov, Malamud, Mohapatra, Motovilov, Müller,
Naboko, Nakamura, Neidhardt, Nichols, Pavlov, Peller, Petkov,
Potapov, Raikov, Rummenigge, Ruzhansky, Rybkin,
Sadovnichii, Safronov, Scharder, Sinha, Shubin, Sobolev,
Skripka, de Snoo, Stollmann, Sukochev, Teschl, Tomilov,
Veselic...
Jussi Behrndt
Spectral shift function for Dirac operators with -potential
PART II
Representation of the SSF via Weyl function
Jussi Behrndt
Spectral shift function for Dirac operators with -potential
Quasi boundary triples
S ⇢ S ⇤ closed symmetric operator in H with infinite defect
Def. [Bruk76, Kochubei75; DerkachMalamud95; B. Langer07]
{G, 0 , 1 } quasi boundary triple for S ⇤ if G Hilbert space and
T ⇢ T = S ⇤ and 0 , 1 : dom T ! G such that
(i) (Tf , g)
(ii)
:=
(iii) A = T
(f , Tg) = ( 1 f ,
0
1
Example 1: (
Sf =
S⇤f =
Tf =
0 g)
( 0f ,
1 g),
: dom T ! G ⇥ G dense range
ker
0
f , g 2 dom T
selfadjoint
+ V on domain ⌦, @⌦ of class C 2 , V 2 L1 real)
f + Vf {f 2 H 2 (⌦) : f |@⌦ = @⌫ f |@⌦ = 0}
f + Vf {f 2 L2 (⌦) : f 2 L2 (⌦)} 6⇢ H s (⌦), s > 0
f + Vf H 2 (⌦)
Here (Tf , g) (f , Tg) = (f |@⌦ , @⌫ g|@⌦ ) (@⌫ f |@⌦ , g|@⌦ ).
Choose G = L2 (@⌦), 0 f := @⌫ f |@⌦ , 1 f := f |@⌦ .
Jussi Behrndt
Spectral shift function for Dirac operators with -potential
Example 2: Dirac operators with -shell interactions
⌃ boundary of bdd. C 1 -domain ⌦+ ⇢ R3 , ⌦ := R3 \ ⌦+
Sf =
✓
◆
icrf+ + mc 2 f+
, dom S = H01 (⌦+ )
icrf + mc 2 f
H01 (⌦ )
Tf =
✓
◆
icrf+ + mc 2 f+
, dom T = H 1 (⌦+ )
icrf + mc 2 f
H 1 (⌦ )
S⇤f
=
✓
◆
icrf+ + mc 2 f+
, dom S ⇤ maximal in L2
icrf + mc 2 f
For f 2 dom T = H 1 (⌦+ )
0f
:= ic↵ · ⌫(f+ |⌃
f |⌃ )
H 1 (⌦ ) set
and
Jussi Behrndt
1f
:=
1
1
(f+ |⌃ + f |⌃ ) +
2
⌘
0f .
Spectral shift function for Dirac operators with -potential
Example 2: Dirac operators with -shell interactions
Proposition
Then T = S ⇤ and {L2 (⌃),
0f
:= ic↵ · ⌫(f+ |⌃
f |⌃ )
for f 2 dom T = H 1 (⌦+ )
With this choice of
A=T
ker
0, 1}
0
0
=
and
and
1f
1
1
1
(f+ |⌃ + f |⌃ ) +
2
⌘
0f
we have
icr + mc 2 ,
ker
:=
H 1 (⌦ ).
and
A⌘ := T
QBT, where
1
=
dom A = H 1 (R3 ),
icr + mc 2 + ⌘
⌃
A⌘ selfadjoint in L2 (R3 ) for ⌘ 6= {±2c, 0} (Holzmann lecture!)
Jussi Behrndt
Spectral shift function for Dirac operators with -potential
-field and Weyl function
S ⇢ T ⇢ T = S ⇤ , {G,
Definition
Let f 2 ker (T
),
0, 1}
quasi boundary triple (QBT).
2 C\R. -field and Weyl function:
( ) : G ! H,
M( ) : G ! G,
0f
0f
7! f ,
7!
1f
2 ⇢(A)
2 ⇢(A)
,
( ) solves boundary value problem in PDE
M( ) Dirichlet-to-Neumann (Neumann-to-Dir. map) in PDE
+ V , QBT {L2 (@⌦), @⌫ f |@⌦ , f |@⌦ }
Example 1:
) = {f 2 H 2 (⌦) :
Here ker (T
( ) : L2 (@⌦)
where (
f + Vf = f } and
H 1/2 (@⌦) ! L2 (⌦),
+ V )f = f and @⌫ f |@⌦ = ', and
M( ) : L2 (@⌦)
H 1/2 (@⌦) ! L2 (@⌦),
Jussi Behrndt
' 7! f ,
' = @⌫ f |@⌦ 7! f |@⌦ .
Spectral shift function for Dirac operators with -potential
Example 2: Dirac operators with -shell interactions
Tf =
✓
◆
icrf+ + mc 2 f+
, dom T = H 1 (⌦+ )
icrf + mc 2 f
and QBT {L2 (⌃),
0f
0 , 1 },
= ic↵ · ⌫(f+ |⌃
and ((A
)
1 f )(x)
where
f |⌃ ),
=
R
H 1 (⌦ )
R3
1f
G (x
=
1
1
(f+ |⌃ + f |⌃ ) +
2
⌘
0f ,
y )f (y )dy .
-field and Weyl function defined on ran 0 = H 1/2 (⌃) are
Z
2
2
3
( ) : L (⌃) ! L (R ), ' 7!
G (x y )'(y )d (y ),
⌃
M( ) : L2 (⌃) ! L2 (⌃), ' 7! lim
Z
G (x
"!0
|x y |>"
Jussi Behrndt
1
y )'(y )d (y ) + '
⌘
Spectral shift function for Dirac operators with -potential
Quasi boundary triples and selfadjoint extensions
Perturbation problems for selfadjoint operators in QBT scheme:
Lemma
Assume A, B selfadjoint and S = A \ B,
Sf := Af = Bf ,
dom S = f 2 dom A \ dom B : Af = Bf
densely defined. Then exists T ⇢ T = S ⇤ and QBT {G,
such that
A=T
and
(B
where
)
1
ker
(A
and B = T
0
)
1
=
ker
( )M( )
1
1
( ¯ )⇤ ,
and M are the -field and Weyl function of {G,
Jussi Behrndt
0, 1}
0 , 1 }.
Spectral shift function for Dirac operators with -potential
Main abstract result: First order case
Theorem
A, B selfadjoint, S = A \ B densely defined and {G,
A=T
Assume
(A
and (
(B
µ)
0)
1
(B
2 S2 , M(
)
1
(A
ker
µ)
1)
and B = T
0
1
1, M(
)
1
ker
0, 1}
QBT
1
for some µ 2 ⇢(A) \ ⇢(B) \ R
2)
=
Im log(M( )) 2 S1 (G) for all
bounded for
( )M( )
1
2 C \ R and
1
tr Im log(M(t + i"))
"!+0 ⇡
⇠(t) = lim
0 , 1 , 2 Then:
¯
( )⇤ 2 S1
for a.e. t 2 R
is a spectral shift function for {A, B}, in particular,
Z
1
1
1
tr (B
)
(A
)
=
⇠(t) dt.
)2
R (t
Jussi Behrndt
Spectral shift function for Dirac operators with -potential
Main abstract result: Higher order case
Theorem
A, B selfadjoint, S = A \ B densely defined and {G,
A=T
ker
0
and B = T
ker
0, 1}
QBT
1
Assume
(A
M(
1)
µ)
1, M(
1
2)
(B
µ)
1
bounded for
for some µ 2 ⇢(A) \ ⇢(B) \ R
1,
2
and for some k 2 N:
dp
dq
(
)
M( ) 1 ( ¯ )⇤ 2 S1 , p + q = 2k ,
d p
d q
p
dq
1 ¯ ⇤ d
M(
)
(
)
( ) 2 S1 , p + q = 2k ,
d q
d p
dj
M( ) 2 S 2k +1 ,
j = 1, . . . , 2k + 1.
j
d j
Jussi Behrndt
Spectral shift function for Dirac operators with -potential
Main abstract result: Higher order case
Theorem
A, B selfadjoint, S = A \ B densely defined and {G,
A=T
Assume
(A
M(
1)
(B
µ)
1, M(
)
1
ker
(B
µ)
2 ) bounded
(2k +1)
(A
and B = T
0
1
ker
0, 1}
QBT
1
for some µ 2 ⇢(A) \ ⇢(B) \ R
for
)
1 , 2 and Sp -conditions. Then:
(2k +1) 2 S
1
For any ONB ('j ) in G and a.e. t 2 R
⇠(t) =
X
j
lim
"!+0
1
Im log(M(t + i"))'j , 'j
⇡
is a spectral shift function for {A, B}, in particular,
Z
2k + 1
(2k +1)
(2k +1)
tr (B )
(A )
=
⇠(t) dt
)2k +2
R (t
Jussi Behrndt
Spectral shift function for Dirac operators with -potential
Remarks
If A, B semibounded then
(A
µ)
1
(B
µ)
1
()
AB
in accordance with ⇠(t) = ⇡1 tr (Im log(M(t + i0)))
0
Representation of SSF via M-function:
Rank 1, k = 0 [LangerSnooYavrian’01]
Rank n < 1, k = 0 [B. MalamudNeidhardt’08]
Other representation via modified perturbation determinant
for M for k = 0 [MalamudNeidhardt’15]
Representation of scattering matrix {S( )} 2R of {A, B} for the
trace class case (k = 0):
q
⇣
⌘ 1q
S( ) = I 2i Im M( + i0) M( + i0)
Im M( + i0)
[AdamyanPavlov’86], [B. MalamudNeidhardt’08 and ’15],
[MantilePosilicanoSini’15]
Jussi Behrndt
Spectral shift function for Dirac operators with -potential
PART III
Dirac operators with electrostatic -shell
interactions
Jussi Behrndt
Spectral shift function for Dirac operators with -potential
A QBT for Dirac operators
⌃ boundary of bdd. C 1 -domain ⌦+ ⇢ R3 , ⌦ := R3 \ ⌦+
Sf =
✓
◆
icrf+ + mc 2 f+
, dom S = H01 (⌦+ )
icrf + mc 2 f
H01 (⌦ )
Tf =
✓
◆
icrf+ + mc 2 f+
, dom T = H 1 (⌦+ )
icrf + mc 2 f
H 1 (⌦ )
Then {L2 (⌃),
0f
0, 1}
QBT, where
:= ic↵ · ⌫(f+ |⌃
and A = T
ker
0
f |⌃ )
=
A⌘ = B = T
and
1f
:=
1
1
(f+ |⌃ + f |⌃ ) +
2
⌘
0f
icr + mc 2 free Dirac operator,
ker
1
Jussi Behrndt
=
icr + mc 2 + ⌘
⌃
Spectral shift function for Dirac operators with -potential
-field and Weyl function, strength ⌘
2
2
3
( ) : L (⌃) ! L (R ), ' 7!
2
2
M( ) : L (⌃) ! L (⌃), ' 7! lim
Z
Z
G (x
y )'(y )d (y ),
⌃
G (x
"!0
|x y |>"
1
y )'(y )d (y ) + '
⌘
are continuous and admit closures defined on L2 (⌃). Recall
from [ArrizabalagaMasVega’15] that
sup
2[
mc 2 ,mc 2 ]
kM( )
⌘
1
k = M0 < 1.
Assumptions on ⌘
|⌘| 
1
M0
and ⌘ 6= {±2c, 0}.
In this case gap preserved: (A⌘ ) \ ( mc 2 , mc 2 ) = ;
Jussi Behrndt
Spectral shift function for Dirac operators with -potential
Spectral shift function for the pair {A, A⌘ }, ⌘ 6= ±2c
Theorem
(A⌘
)
3
(A
)
3
1
2 S1 (L2 (R3 )) for
For ⌘ 2 (0, M0 ), any ONB ('j ) in
⇠+ (t) =
X
lim
"!+0
L2 (⌃)
2 ⇢(A0 ) \ ⇢(A⌘ )
and a.e. t 2 R
1
Im log(M(t + i"))'j , 'j
⇡
is a spectral shift function for {A, A⌘ }
For ⌘ 2 ( M0 1 , 0), any ONB ('j ) in L2 (⌃) and a.e. t 2 R
⇠ (t) =
X
lim
"!+0
1
Im log( M(t + i")
⇡
1 )'
j , 'j
is a spectral shift function for {A⌘ , A}
The following trace formula holds for 2 ⇢(A0 ) \ ⇢(A⌘ ):
Z
3
3
3
tr (A⌘
)
(A
)
=⌥
⇠± (t) dt
)4
R (t
Jussi Behrndt
Spectral shift function for Dirac operators with -potential
Sketch of the proof
⌘ 2 (0, M0 1 ) and
(A⌘
)
1
2 ( mc 2 , mc 2 )
= (A
)
1
 (A
)
1
dk
d k
( ¯ )⇤ = k ! 1 (A
dk
d k
( )2S
dk
M(
d k
4
2k +1
)
( ) ⌘
k 1
2S
4
2k +1
1
+ M( )
1
( )⇤
L2 (R3 ), L2 (⌃)
L2 (⌃), L2 (R3 )
) = k ! 1 (A
)
k
( ) 2 S2/k (L2 (⌃))
Use S1/x · S1/y = S1/(x+y ) and conclude
dp
dq
1 ( ¯ )⇤ 2 S ,
p + q = 2,
1
d p ( ) d q M( )
p
dq
d
1
⇤
(¯)
p + q = 2,
q M( )
p ( ) 2 S1 ,
d
dj
M(
d j
d
) 2 S3/j ,
j = 1, 2, 3.
Apply Main Theorem with k = 1.
Jussi Behrndt
Spectral shift function for Dirac operators with -potential
References
N. Arrizabalaga, A. Mas, L. Vega
Shell interactions for Dirac operators
J. Math. Pures Appl. 102 (2014)
N. Arrizabalaga, A. Mas, L. Vega
Shell interactions for Dirac operators: on the point
spectrum and the confinement
SIAM J. Math. Anal. 47 (2015)
N. Arrizabalaga, A. Mas, L. Vega
An Isoperimetric-type inequality for electrostatic shell
interactions for Dirac operators
Comm. Math. Phys. 344 (2016)
J. Behrndt, P. Exner, M. Holzmann, V. Lotoreichik
On the spectral properties of Dirac operators with
electrostatic -shell interactions
J. Behrndt, F. Gesztesy, S. Nakamura
Spectral shift functions and Dirichlet-to-Neumann maps
Jussi Behrndt
Spectral shift function for Dirac operators with -potential
Thank you for your attention
Jussi Behrndt
Spectral shift function for Dirac operators with -potential