PAN Wei-ye,YANG Cong-li,ZHAO Jian

(School of Mathematics and Science,Guizhou Normal University,Guiyang 550001,China)

Abstract:In this article,for α>0,we characterize several versions uncertainty principles of self-adjoint operators and linear operators for the α-fock space Fα2in the complex plane.By using the general result from functional analysis,we find two linear operators Tf=and T?=zf to construct two self-adjoint operators A and B such that[A,B]is a scalar multiple of the identity operator on,and obtain some more accurate results about the uncertainty principles for the α-fock space ,where T?is the adjoint of T,[A,B]=AB?BA is the commutator of A and B,which extends and completes the results of Qu[1]and Zhu[2].

Keywords: α-fock space;uncertainty principles;linear operators;self-adjoint operators;Gaussian measure

1 Introduction

Let C be the complex plane,for any positive parameter α,we consider

be the Gaussian measure on C,where dA(z)=dxdy is the Euclidean area measure on the complex plane.We define the α-fock spaceas follow:

where H(C)is the space of all entire functions.It is easy to show thatis a Hilbert spac

e with the following inner product inherited from L2(C,dλα):

accordingly define the norm kfk2,αby

It is well know that the Fockspace has become one of the vitally important mathematical tools of quantum physics.Thus,it is significative to study the uncertainty principle for the Fock space.In fact,it was an extensive interest in study of uncertainty principles for fock space F2.In particular,some versions uncertainty principles of self-adjoint operators for the fock space F2were obtained,for some details,see[1,2].In addition,an inequality of uncertainty principle about the average value and the covariance of self-adjoint operators for the fock space F2also was proved in[1],and this result is due to the uncertainty principle of signal analysis,see for example[3].On the other hand,The uncertainty principles of linear operators for the fock space F2also can be found in the article[4].Moreover,we invite the interested reader to see[5–7]for other perspectives in the study of uncertainty principles see[8–13].Based on these work,our goal here is to introduce a positive parameter α and extend uncertainty inequalities on two fronts.For one thing,we introduce a positive parameter α and obtain two different forms uncertainty principles of self-adjoint operators for the α-fock space,see Section 2.For another thing,we study the uncertainty principles of linear operators for α-fock space,see Section 3.

Note that all results discussed in this article are on the complex plane C,there is no explanation below.

2 Uncertainty Principles of Self-Adjoint Operators

In[14],an uncertainty principle about self-adjoint operators from functional analysis is stated as follows.

Theorem 1[14]Suppose A and B are self-adjoint operators,possibly unbounded,on Hilbert space H.Then

for all f∈Dom(AB)∩Dom(BA)and all a,b∈C,where Dom(AB)and Dom(BA)are the domains of the operators AB and BA,respectively.Here[A,B]=AB?BA is the commutator of A and B,and I is the identity operator.Furthermore,equality in(2.1)holds if and only if(A?aI)f and(B?bI)f are purely imaginary scalar multiples of one another.

Proof This result is very useful and widely known,see page 27 of[14]for a proof.

Later,a sharper inequality about(2.1)was provided in[15].

Theorem 2[15]Suppose A and B are self-adjoint operators,possibly unbounded,on Hilbert space H.Then we have

for all f∈Dom(AB)∩Dom(BA)and all a,b∈C.Here[A?aI,B?bI]+=(A?aI)(B?bI)+(B?bI)(A?aI)and I is the identity operator.

Proof This is proved.See[15]for some details.

Combining with the above theorems,we construct two natural self-adjoint operators A and B such that[A,B]is a scalar multiple of the identity operator which based on the operator of multiplication by z and constant multiple of the differentiation operator on,then an uncertainty principle arises.Next,we collect two lemmas which provided a crucial evidence in proving Theorem 3.

Proof This is proved.See[16]for some details.

It is very easy to check that

Thus we consider the following two self-adjoint operators on:

that is

It follows from[1,2]that,for a function f∈,if∈,then both Af and Bf are well defined.If both+zf and?zf are in,it’s obvious thatand zf are in.Therefore,the intersection of the domains of A and B consists of those function f such that(or zf)is stillin.It’s possible to identify the domains of AB,BA,and their intersection as well.

Lemma 2 For the operators A and B defined above,we have[A,B]=,where I is the identity operator onand i is the imaginary unit.

Proof From(2.3)and(2.4),we have

This proves the desired result.

We now derive the first version of the uncertainty principle about self-adjoint operators on.

for all a,b∈C.Here f00is the second derivative of f.

Proof From(2.4),we get

This implies that

From the above,the inequality in(2.5)follows from(2.2).

This completes the proof of the theorem.

In order to prove Corollary 1,a discussion about the minimization is also needed.

Lemma 3 If fix some function f∈,for any a,b∈C and the operators T and T?defined above,we have

and the minimum is attained when

Similarly,we have

Also the minimum is attained when

Proof The conclusion is obviously established.We omit the details.

Corollary 1 If f is a unit vector in,f0,f00∈,then we have

Proof Since f is a unit vector,from Theorem 3 and its proof,we get

Also

which easily implies that

Then it follows from combining this with the minimization argument of Lemma 3 and

This completes the proof of corollary.

Proof This follows directly from Theorem 3 by setting a=b=0.

In fact,we can improve the argument above to obtain a more interesting result of uncertainty principle.

Proof From Corollary 2,we have the following estimates

which proves the desired result.

Extraordinarily,we now consider several versions of the uncertainty principle which are based on the geometric notions of angle and distance.

here f0,f00∈.

Proof In fact,we have

The same arguement shows that

Applying Corollary 1 with

and

Then we can obtain the desired result.

Corollary 5 Suppose f is a unit vector in,θ±are the angles between f andin,and f0,f00∈.Then for any δ>0,we have

Proof This desired result is clear by using the proofs of Corollaries 3 and 4.

Corollary 6 Suppose f a unit vector in,θ±are the angles between f and±zf in,and f0,f00∈,then we have

Proof This follows directly from Corollary 5 by δ=1.

Motivated by Corollary 4,here we get the following results.

Corollary 7 Suppose f is any function in,not identically zero,and f0,f00∈,then we have

where[f]=Cf is the one-dimensional subspace ofspanned by f and d(g,X)denotes the distance infrom g to X.

Proof This is an equivalent state of Corollary 4,because

and

Hence the result is clear from Corollary 4.

Now we can do a significant extension.Actually,all conclusions above that we have done for the α-fock spaceremains valid for any operator T and its adjoint operator T?which satisfies[T,T?]=mI,here m is a positive constant.

Corollary 8 If f is any function in,Suppose T is any operator onsuch that[T,T?]=mI,then we have

here m is a positive constant.

Proof This follows from the proofs of Lemma 2 and Theorem 3.

It is worth paying attention to the case that when the function f0(or equivalently,the function zf)also belongs to the α-fock space,which is not always the case,the interested reader could see[1]for some details.When the function f0is not in,each of the left-hand sides of the inequalities above is in finite,Hence the inequality always becomes valid.

Next we will obtain a different version uncertainty principle of self-adjoint operators,for this purpose,we first give the following definition which also be found in[1].

and

The following lemma plays an important role in proving Theorem 4.

Proof By(2.6)and(2.7),we conclude that

The same procedure may be easily adapted to obtain that

This proves the desired estimate.

Carefully examining the proof of Lemma 4,we obtain the following characterization.

for all f∈Dom(AB)∩Dom(BA).Here z=x+iy.

Proof Recall that

Hence we divide the proof into two steps.

For one thing,we consider the inequality that

Actually,we have

For another thing,we consider the another inequality that

This completes the proof of the theorem.

3 Uncertainty Principles of Linear Operators

In this section,we turn out our attention to the uncertainty principles of linear operators on.To achieve that end,we let A?and B?be the adjoint of the operators A and B respectively.Throughout the article,we shall use the notation

Definition 2[17]Suppose A is linear operator with domain and range in the same complex Hilbert space H,for any nonzero f∈Dom(A),we defined

which is equal to

More interestingly,as a generalization of Lemma 3,we have the following lemma.

Lemma 5 If fix some function f∈,suppose A and B are linear operators on.Then for any a,b∈C,we have

Furthermore,the minimum of(3.3)and(3.4)are attained when.Similarly,we have

Also the minimum of(3.5)and(3.6)are attained when

Proof A direct calculation shows that

which easily implies that

or equivalently by(3.1)and(3.2),we haveand the minimum is attained when

This finishes the proof of the lemma.

The following theorem on the commutator is another generalization of the Heisenberg uncertainty principle.

Theorem 5[17]Let A and B be linear operators with domain and range in the same complex Hilbert space H,for any nonzero f∈Dom(A|B),there holds

Proof It is very easy to verify that for any nonzero f∈Dom(A|B),

This implies that

For any a,b∈C,we replace A and B above by A?aI and B?bI,respectively to obtain that

According to Lemma 5,we conclude that

from which(3.8)follows.and the minimum value is attained uniquely atand.This completes the desired result.

From Theorem 5,we know that if we find out two linear operators and their adjoint on,then an uncertainty principle arises.To this end,we still consider the operator T and its adjoint T?which defined from Lemma 1.To simplify notation,let A=T,and B=T?,namely A=and B=zf.It is obvious that A and B are linear operators on,and A?f=Bf,B?f=Af.Then we characterize the first uncertainty principle of linear operators onas follows.

Theorem 6 Let f,f0∈for any a∈C,then we have

Proof From Lemma 1,we have[A,B]f=.Following the method used in the proof of Theorem 5,we have

More specifically

This proves the desired result.

Note that when f0is not in,the left-hand sides of the inequality of(3.9)is in finite,so the inequality becomes trivial.

Corollary 9 Suppose f is a unit vector inand f0∈,then we have

Proof Since f is a unit vector,by Lemma 5,we get

and

Then the desired corollary can be proved by Theorem 5.

The results in Theorem 6 and Corollary 9 rely upon Theorem 5,on the other hand,the proof of Theorem 5 rely upon(3.8).While in[13],the author try to find two operators U and V to reduce the upper bound in(3.8)and require that f∈ Dom(A|B)∩Dom(A|U)∩Dom(B|V)∩Dom(V|U)and such that

then we need find two linear operators onwhich satisfies(2.17),this is a formidable task.For any a,b0,b1∈C,we might as well suppose V to be a multiple of the identity,namely V=aI,and let another operator U=b0I+b1A.Then,it is very easy to check that

Also

[V,U]f=(aU?aU)f=0,which shows that U and V satisfies(3.10).Hence we can obtain the following theorem.

Theorem 7 Let f,f0∈,for any a,b0,b1∈C,then we have

for all f∈Dom(A|B)∩Dom(A|U)∩Dom(B|V)∩Dom(V|U),and a,b0,b1∈C.

Proof By(3.10),we get

This implies that

It is very easy to verify that

Then we conclude that

From Lemma 5,we have and the minimum is attained whenThen combining this with the Cauchy-Schwarz inequality to the left hand side of(3.11)to obtain the inequality that

This completes the proof of the desired theorem.

When f is a unit vector,we have the following corollary.

Corollary 10 Let f is a unit vector in,f0∈,then we have

Proof When f is a unit vector on,and by minimization argument,we have

Then we may reach the conclusion just like the computation we performed in the proof of Theorem 7.

For the general case,we can modify the argument above to obtain something more interesting.Before this,we take the the operators U and V to be of the form

where n is a positive integer and b0,b1,···,bn,a ∈ C.Let f be an element in

Note that D1(A|B)=D(A|B).

Theorem 8 Let f is k-th derived function onand f(k)∈U and V are defined by(3.12),then we have

for all f∈Domn(A|B),and b0,b1,···,bn,a∈.

Proof By Theorem 7 and its proof,we have

Calculate directly that

Consequently,combining with the minimization argument of Lemma 5 and Cauchy-Schwarz inequality,we obtain

The proof of the theorem is completed.

In addition,it is also possible to obtain an inequality of the case that f is a unit vector.

Corollary 11 Let f is n-th derived function onand f(n)∈,kfk2,α=1,suppose U and V are defined by(2.19),then we have

for all f∈Domn(A|B),and b0,b1,···,bn,a∈.

Proof This follow directly from Corollary 10 and Theorem 8 and their proofs.