HOMOGENEOUS SCHR ODINGER OPERATORS ON HALFLINE JAN DEREZI NSKI - PowerPoint PPT Presentation

HOMOGENEOUS SCHR ¨ ODINGER OPERATORS ON HALFLINE JAN DEREZI ´ NSKI Dep. Math. Meth. in Phys., Faculty of Physics, University of Warsaw 1. HOMOGENEOUS SCHR ¨ ODINGER OPERATORS (in collaboration with LAURENT BRUNEAU and VLADIMIR GEORGESCU) 2. ALMOST HOMOGENEOUS SCHR ¨ ODINGER OPERATORS (in collaboration with SERGE RICHARD) 3. HOMOGENEOUS RANK ONE PERTURBATIONS

HOMOGENEOUS SCHR ¨ ODINGER OPERATORS (in collaboration with LAURENT BRUNEAU and VLADIMIR GEORGESCU) Consider a formal differential expression � 1 − 1 L α = − ∂ 2 � x + 4 + α x 2 . We would like to interpret it as a well-defined (unbounded) operator. To do this we need to specify its domain.

We will obtain operators with surprisingly rich mathe- matical phenomenology, which should be close to physi- cists’ hearts: the “running coupling constant” flows under the action of the “renormalization group”, there are two ”phase transitions”, attractive and repulsive fixed points, limit cycles, breakdown of conformal symmetry, etc. I will discuss both the self-adjoint and non-self-adjoint cases. The latter have quite curious properties and I am looking for their physical applications.

Let U τ be the group of dilations on L 2 [0 , ∞ [ , that is ( U τ f )( x ) = e τ/ 2 f ( e τ x ) . We say that B is homogeneous of degree ν if = e ντ B. U τ BU − 1 τ Clearly, L α is homogeneous of degree − 2 .

Here are two natural questions: 1. If α ∈ R , how to interpret L α as a self-adjoint operator on L 2 [0 , ∞ [ ? When is it homogeneous of degree − 2 ? 2. If α ∈ C , how to interpret L α as a closed operator on L 2 [0 , ∞ [ ? When is it homogeneous of degree − 2 ?

L α , and closely related operators H m , which we intro- duce shortly, are interesting for many reasons. • They appear as the radial part of the Laplacian in all dimensions, in the decomposition of the Aharonov- Bohm Hamiltonian, in the membranes with conical sin- gularities, in the theory of many body systems with contact interactions and in the Efimov effect. • They have rather subtle and rich properties illustrat- ing various concepts of the operator theory in Hilbert spaces (eg. the Friedrichs and Krein extensions, holo- morphic families of closed operators).

• Essentially all basic objects related to H m , such as their resolvents, spectral projections, wave and scat- tering operators, can be explicitly computed. • A number of nontrivial identities involving special func- tions, especially from the Bessel family, find an ap- pealing operator-theoretical interpretation in terms of the operators H m . Eg. the Barnes identity leads to the formula for wave operators.

Two naive interpretations of L α : 1. The minimal operator L min α : We start from L α on C ∞ c ]0 , ∞ [ , and then we take its closure. 2. The maximal operator L max : We consider the domain α consisting of all f ∈ L 2 [0 , ∞ [ such that L α f ∈ L 2 [0 , ∞ [ . Clearly, Dom( L min ) ⊂ Dom( L max ) and � L max � Dom( L min ) = L min . � In other words L min ⊂ L max .

We will see that it is often natural to write α = m 2 Theorem 1 . 1. For 1 ≤ Re m , L min m 2 = L max m 2 . 2. For − 1 < Re m < 1 , L min m 2 � L max m 2 , and the codimension of their domains is 2 . α ) ∗ = L max 3. ( L min . Hence, for α ∈ R , L min is Hermitian. α α 4. L min and L max are homogeneous of degree − 2 . α α

Notice that 1 2 ± m = 0 . Lx Let ξ be a compactly supported cutoff equal 1 around 0 . 1 2 + m ξ belongs to Dom L max Let − 1 < Re m . Note that x m 2 . This suggests to define the operator H m to be the re- striction of L max m 2 to 1 Dom L min 2 + m ξ. m 2 + C x

Theorem 2 . 1. For 1 ≤ Re m , L min m 2 = H m = L max m 2 . 2. For − 1 < Re m < 1 , L min m 2 � H m � L max m 2 and the codi- mension of the domains is 1 . 3. H ∗ m = H m . Hence, for m ∈ ] − 1 , ∞ [ , H m is self-adjoint. 4. H m is homogeneous of degree − 2 . 5. sp H m = [0 , ∞ [ . 6. { Re m > − 1 } ∋ m �→ H m is a holomorphic family of closed operators.

Theorem 3 . 1. For α ≥ 1 , L min = H √ α is essentially self-adjoint on α C ∞ c ]0 , ∞ [ . 2. For α < 1 , L min is not essentially self-adjoint on C ∞ c ]0 , ∞ [ . α 3. For 0 ≤ α < 1 , the operator H √ α is the Friedrichs extension and H −√ α is the Krein extension of L min α . 4. H 1 2 is the Dirichlet Laplacian and H − 1 2 is the Neumann Laplacian on halfline. 5. For α < 0 , L min has no homogeneous selfadjoint ex- α tensions.

It is easy to see that � 1 − 1 x − 1 1 � � − ∂ 2 � x + 4 + α x 2 ± 1 x 2 2 � 1 x − 1 − 1 = − ∂ 2 � x∂ x + 4 + α x 2 ± 1 , which is the (modified) Bessel operator. Therefore, it is not surprising that various objects related to H m can be computed with help of functions from the Bessel family.

Theorem 4 . If R m ( λ ; x, y ) is the integral kernel of the operator ( λ − H m ) − 1 , then for Re k > 0 we have � √ xyI m ( kx ) K m ( ky ) if x < y, R m ( − k 2 ; x, y ) = √ xyI m ( ky ) K m ( kx ) if x > y, where I m is the modified Bessel function and K m is the MacDonald function.

Proposition 5 . For 0 < a < b < ∞ , the integral kernel of 1 l [ a,b ] ( H m ) is � √ b √ xyJ m ( kx ) J m ( ky ) k d k, 1 l [ a,b ] ( H m )( x, y ) = √ a where J m is the Bessel function.

Let F m be the operator on L 2 (0 , ∞ ) given by � ∞ √ F m : f ( x ) �→ J m ( kx ) kxf ( x )d x 0 F m is the so-called Hankel transformation. Define also the operator Xf ( x ) := xf ( x ) . Theorem 6 . F m is a bounded invertible involution on L 2 [0 , ∞ [ diagonalizing H m and anticommuting with the self-adjoint generator o dilations A = 1 2i ( x∂ x + ∂ x x ) : F 2 = 1 l , F m H m F − 1 m = X 2 , F m A = − A F m .

Theorem 7 Set Ξ m ( t ) = e i ln(2) t Γ( m +1+i t ) I f ( x ) = x − 1 f ( x − 1 ) , 2 . Γ( m +1 − i t ) 2 Then F m = Ξ m ( A ) I . Therefore, we have the identity H m := Ξ − 1 m ( A ) X − 2 Ξ m ( A ) (Result obtained independently by Bruneau, Georgescu, D, and by Richard and Pankrashkin).

Theorem 8 . The wave operators associated to the pair H m , H k exist and Ω ± t →±∞ e i tH m e − i tH k m,k := lim = e ± i( m − k ) π/ 2 F m F k = e ± i( m − k ) π/ 2 Ξ k ( A ) Ξ m ( A ) .

The formula H m := Ξ − 1 m ( A ) X − 2 Ξ m ( A ) (1) valid for Re m > − 1 can be used as an alternative defini- tion of the family H m also beyond this domain. It defines a family of closed operators for the parameter m that be- longs to { m | Re m � = − 1 , − 2 , . . . } ∪ R . (2) Its spectrum is always equal to [0 , ∞ [ and it is analytic in the interior of (2).

In fact, Ξ m ( A ) is a unitary operator for all real values of m . Therefore, for m ∈ R , (1) is well-defined and self- adjoint. Ξ m ( A ) is bounded and invertible also for all m such that Re m � = − 1 , − 2 , . . . . Therefore, the formula (1) defines an operator for all such m .

One can then pose various questions: • What happens with this operator along the lines Re m = − 1 , − 2 , . . . ? • What is the meaning of the operator to the left of Re = − 1? (It is not a differential operator!)

The definition (or actually a number of equivalent defi- nitions) of a holomorphic family of bounded operators is quite obvious and does not need to be recalled. In the case of unbounded operators the situation is more sub- tle.

Suppose that Θ is an open subset of C , H is a Banach space, and Θ ∋ z �→ H ( z ) is a function whose values are closed operators on H . We say that this is a holo- morphic family of closed operators if for each z 0 ∈ Θ there exists a neighborhood Θ 0 of z 0 , a Banach space K and a holomorphic family of injective bounded operators Θ 0 ∋ z �→ B ( z ) ∈ B ( K , H ) such that Ran B ( z ) = D ( H ( z )) and Θ 0 ∋ z �→ H ( z ) B ( z ) ∈ B ( K , H ) is a holomorphic family of bounded operators.

We have the following practical criterion: Theorem 9 . Suppose that { H ( z ) } z ∈ Θ is a function whose values are closed operators on H . Suppose in addition that for any z ∈ Θ the resolvent set of H ( z ) is nonempty. Then z �→ H ( z ) is a holomorphic family of closed opera- tors if and only if for any z 0 ∈ Θ there exists λ ∈ C and a neighborhood Θ 0 of z 0 such that λ belongs to the resol- vent set of H ( z ) for z ∈ Θ 0 and z �→ ( H ( z ) − λ ) − 1 ∈ B ( H ) is holomorphic on Θ 0 .

The above theorem indicates that it is more difficult to study holomorphic families of closed operators that for some values of the complex parameter have an empty resolvent set. Conjecture 10 . It is impossible to extend { Re m > − 1 } ∋ m �→ H m to a holomorphic family of closed operators on a larger connected open subset of C .

ALMOST HOMOGENEOUS SCHR ¨ ODINGER OPERATORS (in collaboration with SERGE RICHARD) For any κ ∈ C ∪ {∞} let H m,κ be the restriction of L max m 2 to the domain f ∈ Dom( L max � Dom( H m,κ ) = m 2 ) | for some c ∈ C , x 1 / 2 − m + κx 1 / 2+ m � ∈ Dom( L min � f ( x ) − c m 2 ) � around 0 , κ � = ∞ ; f ∈ Dom( L max � Dom( H m, ∞ ) = m 2 ) | for some c ∈ C , f ( x ) − cx 1 / 2+ m ∈ Dom( L min � m 2 ) around 0 .