the spectrum of the -neumann laplacian for the ball and annulus was explicitly computed by folland  . fu   determined the spectrum for the polydisc , showing that it need not be purely discrete like for the usual dirichlet laplacian . here we will exhibit the weighted case , where the weight function is (z)=|z|2 , showing that the essential spectrum is non - empty , which is equivalent to the fact that the -neumann operator ( the inverse to the -neumann laplacian ) fails to be compact  . let :cnr+ be a plurisubharmonic c2-weight function , and define the space l2(cn , e)={f : cnc:cn|f|2ed< } , where  denotes the lebesgue measure , the space l(0,q)2(cn , e ) of ( 0,q)-forms with coefficients in l2(cn , e ) , for 1qn . let ( f , g)=cnfged denote the inner product and f2=cn|f|2ed the norm in l2(cn , e ) . we consider the weighted -complex l(0,q1)2(cn , e)l(0,q)2(cn , e)l(0,q+1)2(cn , e ) , where for ( 0,q)-forms u=|j|=qujdzj with coefficients in c0(cn ) we have u=|j|=qj=1nujzjdzjdzj , and u=|k|=q1k=1nkukkdzk , where k=zkzk . there is an interesting connection between  and the theory of schrdinger operators with magnetic fields ; see , for example ,   for recent contributions exploiting this point of view . the complex laplacian on ( 0,q)-forms is defined as ,q+ , where the symbol ,q is to be understood as the maximal closure of the operator initially defined on forms with coefficients in c0 , i.e. , the space of smooth functions with compact support . ,q is a self - adjoint and positive operator , which means that ( ,qf , f)0,forfdom( ) . the associated dirichlet form is denoted by ( 1.1)q(f , g)=(f,g)+(f,g) , for f , gdom()dom( ) . the weighted -neumann operator n,q is  if it exists  the bounded inverse of ,q . we indicate that a square integrable ( 0,1)-form f=j=1nfjdzj belongs to dom( ) if and only if ej=1nzj(fje)l2(cn , e ) , where the derivative is to be taken in the sense of distributions , and that forms with coefficients in c0(cn ) are dense in dom()dom( ) in the graph norm f(f2+f2)12 ( see  ) . we consider the levi matrix m=(2zjzk)jk of  , and suppose that the sum sq of any q ( equivalently : the smallest q ) eigenvalues of m satisfies ( 1.2)lim inf|z|sq(z)>0 . we show that ( 1.2 ) implies that there exists a continuous linear operator n,q : l(0,q)2(cn , e)l(0,q)2(cn , e ) , such that ,qn,qu = u , for any ul(0,q)2(cn , e ) . if we suppose that the sum sq of any q ( equivalently : the smallest q ) eigenvalues of m satisfies ( 1.3)lim|z|sq(z)= , then the -neumann operator n,q : l(0,q)2(cn , e)l(0,q)2(cn , e ) is compact ( see   for further details ) . to find the canonical solution to f = u , where ul(0,1)2(cn , e ) is a given ( 0,1)-form such that u=0 , one can take f=n,1u , and f will also satisfy fker. for further results on the canonical solution operator to  , see  . if the weight function is (z)=|z|2 , the corresponding levi matrix m is the identity . the space a2(cn , e|z|2 ) of entire functions belonging to l2(cn , e|z|2 ) is the fock space , which plays an important role in quantum mechanics . in this case , ( 1.4),0u=u=14u+j=1nzjuzj , where udom,0l2(cn , e|z|2 ) , and ( 1.5),nu=u=14u+j=1nzjuzj+nu , where udom,nl(0,n)2(cn , e|z|2 ) . for n=1 , there is a connection to schrdinger operators with magnetic fields ( see   for properties of its spectrum ) , and to dirac and pauli operators  : the operators p+=e|z|2/2e|z|2/2,p=e|z|2/2e|z|2/2 defined on l2(c ) are the pauli operators ; p+ is also a schrdinger operator with magnetic field , and the square of the corresponding dirac operator satisfies d2=(p00p+).for n>1 and 1qn1 , the -neumann laplacian ,q acts diagonally ( see  ): for u=|j|=qujdzjdom,ql(0,q)2(cn , e|z|2 ) , we have ( 1.6),qu=(+)u=|j|=q(14uj+j=1nzjujzj+quj)dzj . in order to determine the spectrum of ,q for (z)=|z|2 , we use the following lemma ( see  [ 2 ,  lemma 1.2.2 ] ) . lemma  2.1let 
 h 
 be a symmetric operator on a hilbert space 
 h 
 with domain 
 l , and let 
 ( fk)k=1 
 be a complete orthonormal set in 
 h 
 . if each 
 fk 
 lies in 
 l 
 and there exist 
 kr 
 such that 
 hfk=kfk 
 for every 
 k , then 
 h 
 is essentially self - adjoint . moreover , the spectrum of 
 h 
 is the closure in 
 r 
 of the set of all 
 k . for the sake of simplicity , and in order to explain the general method , we start with the complex one - dimensional case . looking for the eigenvalues  of ,0 it is clear that the space a2(cn , e|z|2 ) is contained in the eigenspace of the eigenvalue =0 . let 
 h 
 be a symmetric operator on a hilbert space 
 h 
 with domain 
 l , and let 
 ( fk)k=1 
 be a complete orthonormal set in 
 h 
 . if each 
 fk 
 lies in 
 l 
 and there exist 
 kr 
 such that 
 hfk=kfk 
 for every 
 k , moreover , the spectrum of 
 h 
 is the closure in 
 r 
 of the set of all 
 k . for any positive integer k , the antiholomorphic monomial zk is an eigenfunction for the eigenvalue =k . in the following , we denote n0=n{0}. lemma  2.2let 
 n=1 
 . for 
 kn0 
 and 
 mn , the functions(2.2)uk , m(z , z)=zk+mzm+j=1m(1)j(k+m)!m!j!(k+mj)!(mj)!zk+mjzmjare eigenfunctions for the eigenvalue 
 k+m 
 of the operator 
 ,0u=uzz+zuz.for 
 kn 
 and 
 mn0 , the functions(2.3)vk , m(z , z)=zkzk+m+j=1k(1)j(k+m)!k!j!(k+mj)!(kj)!zkjzk+mjare eigenfunctions for the eigenvalue 
 k 
 of the operator 
 ,0u=uzz+zuz. let 
 n=1 
 . for 
 kn0 
 and 
 mn , the functions(2.2)uk , m(z , z)=zk+mzm+j=1m(1)j(k+m)!m!j!(k+mj)!(mj)!zk+mjzmjare eigenfunctions for the eigenvalue 
 k+m 
 of the operator 
 ,0u=uzz+zuz. for 
 kn 
 and 
 mn0 , the functions(2.3)vk , m(z , z)=zkzk+m+j=1k(1)j(k+m)!k!j!(k+mj)!(kj)!zkjzk+mjare eigenfunctions for the eigenvalue 
 k 
 of the operator 
 ,0u=uzz+zuz. proofto prove ( 2.2 ) , we set uk , m(z , z)=zk+mzm+a1zk+m1zm1+a2zk+m2zm2++am1zk+1z+amzk , and compute 2zzuk , m(z , z)=(k+m)mzk+m1zm1+a1(k+m1)(m1)zk+m2zm2++am1(k+1)zk as well as zzuk , m(z , z)=(k+m)zk+mzm+a1(k+m1)zk+m1zm1++am1(k+1)zk+1z+amkzk , which implies that the function uk , m is an eigenfunction for the eigenvalue =k+m , and from ( 2.1 ) we obtain , comparing the highest exponents of z and z , ( k+m)ma1(k+m1)=(k+m)a1 ; hence a1=(k+m)m . comparing the next lower exponents , we get a1(k+m1)(m1)a2(k+m2)=a2(k+m ) and a2=12(k+m)(k+m1)m(m1 ) . in general , we find that , for j=1,2,  ,m , aj=(1)j(k+m)!m!j!(k+mj)!(mj ) ! , which proves ( 2.2).in order to show ( 2.3 ) , we set vk , m(z , z)=zkzk+m+b1zk1zk+m1+b2zk2zk+m2++bk1zzm+1+bkzm , and compute 2zzvk , m(z , z)=k(k+m)zk1zk+m1+b1(k1)(k+m1)zk2zk+m2++bk1zzm+1 as well as zzvk , m(z , z)=kzkzk+m+b1(k1)zk1zk+m1++bk1zzm+1 which implies that the function vk , m is an eigenfunction for the eigenvalue =k , for each mn , and from ( 2.1 ) we obtain , comparing the highest exponents of z and z , k(k+m)b1(k1)=kb1 ; hence b1=(k+m)k . comparing the next lower exponents , we get b1(k1)(k+m1)b2(k2)=b2k and b2=12(k+m)(k+m1)k(k1 ) . in general , we find that , for j=1,2,  ,k , bj=(1)j(k+m)!k!j!(k+mj)!(kj ) ! , which proves ( 2.3 ) . now we are able to prove the following theorem . to prove ( 2.2 ) , we set uk , m(z , z)=zk+mzm+a1zk+m1zm1+a2zk+m2zm2++am1zk+1z+amzk , and compute 2zzuk , m(z , z)=(k+m)mzk+m1zm1+a1(k+m1)(m1)zk+m2zm2++am1(k+1)zk as well as zzuk , m(z , z)=(k+m)zk+mzm+a1(k+m1)zk+m1zm1++am1(k+1)zk+1z+amkzk , which implies that the function uk , m is an eigenfunction for the eigenvalue =k+m , and from ( 2.1 ) we obtain , comparing the highest exponents of z and z , ( k+m)ma1(k+m1)=(k+m)a1 ; hence a1=(k+m)m . comparing the next lower exponents , we get a1(k+m1)(m1)a2(k+m2)=a2(k+m ) and a2=12(k+m)(k+m1)m(m1 ) . in general , we find that , for j=1,2,  ,m , aj=(1)j(k+m)!m!j!(k+mj)!(mj ) ! , which proves ( 2.2 ) . in order to show ( 2.3 ) , we set vk , m(z , z)=zkzk+m+b1zk1zk+m1+b2zk2zk+m2++bk1zzm+1+bkzm , and compute 2zzvk , m(z , z)=k(k+m)zk1zk+m1+b1(k1)(k+m1)zk2zk+m2++bk1zzm+1 as well as zzvk , m(z , z)=kzkzk+m+b1(k1)zk1zk+m1++bk1zzm+1 which implies that the function vk , m is an eigenfunction for the eigenvalue =k , for each mn , and from ( 2.1 ) we obtain , comparing the highest exponents of z and z , k(k+m)b1(k1)=kb1 ; hence b1=(k+m)k . comparing the next lower exponents , we get b1(k1)(k+m1)b2(k2)=b2k and b2=12(k+m)(k+m1)k(k1 ) . in general , we find that , for j=1,2,  ,k , bj=(1)j(k+m)!k!j!(k+mj)!(kj ) ! , which proves ( 2.3 ) . the spectrum of 
 ,0 
 consists of all non - negative integers 
 { 0,1,2 ,  } , each of which is of infinite multiplicity , so  0  is the bottom of the essential spectrum . the spectrum of 
 ,1 
 consists of all positive integers 
 { 1,2,3 ,  } , each of which is of infinite multiplicity . the spectrum of 
 ,0 
 consists of all non - negative integers 
 { 0,1,2 ,  } , each of which is of infinite multiplicity , so  0  is the bottom of the essential spectrum . the spectrum of 
 ,1 
 consists of all positive integers 
 { 1,2,3 ,  } , each of which is of infinite multiplicity . proofwe already know that the whole bergman space a2(c , e|z|2 ) is contained in the eigenspace of the eigenvalue 0 of the operator ,0 and , for any positive integer k , the antiholomorphic monomial zk is an eigenfunction for the eigenvalue =k . in addition , all functions of the form zz with ,n0 can be expressed as a linear combination of functions of the form ( 2.2 ) and ( 2.3 ) . for a fixed kn , the functions of ( 2.3 ) have infinite multiplicity , as the parameter mn0 is free . all the eigenfunctions considered so far yield a complete orthogonal basis of l2(c , e|z|2 ) , since the hermite polynomials { h0(x)hk(y),h1(x)hk1(y),  ,hk(x)h0(y ) } for k=0,1,2 ,  form a complete orthogonal system in l2(r2,ex2y2 ) ( see for instance  ) , and , since x=1/2(z+z),y = i/2(zz ) , we can apply lemma  2.1 and obtain that the spectrum of ,0 is n0 . for several variables we can adopt the method from above to obtain the following result . we already know that the whole bergman space a2(c , e|z|2 ) is contained in the eigenspace of the eigenvalue 0 of the operator ,0 and , for any positive integer k , the antiholomorphic monomial zk is an eigenfunction for the eigenvalue =k . in addition , all functions of the form zz with ,n0 can be expressed as a linear combination of functions of the form ( 2.2 ) and ( 2.3 ) . for a fixed kn , the functions of ( 2.3 ) have infinite multiplicity , as the parameter mn0 is free . all the eigenfunctions considered so far yield a complete orthogonal basis of l2(c , e|z|2 ) , since the hermite polynomials { h0(x)hk(y),h1(x)hk1(y),  ,hk(x)h0(y ) } for k=0,1,2 ,  form a complete orthogonal system in l2(r2,ex2y2 ) ( see for instance  ) , and , since x=1/2(z+z),y = i/2(zz ) , we can apply lemma  2.1 and obtain that the spectrum of ,0 is n0 . the spectrum of 
 ,q 
 consists of all integers 
 { q , q+1,q+2 ,  } , each of which is of infinite multiplicity . the spectrum of 
 ,q 
 consists of all integers 
 { q , q+1,q+2 ,  } , each of which is of infinite multiplicity . the factor q in the last formula is responsible for the fact that the eigenvalues start with q , which can be seen , in each component separately , by 14uj+j=1nzjujzj=(q)uj . then the function uk1,m1(z1,z1)uk2,m2(z2,z2)ukn , mn(zn , zn ) is a component of an eigenfunction for the eigenvalue j=1n(kj+mj ) of the operator ,q , which follows from ( 1.6 ) and ( 2.2).similarly , it follows from ( 1.6 ) and ( 2.3 ) that , for k1,k2,  ,knn and m1,m2,  ,mnn0 , the function vk1,m1(z1,z1)vk2,m2(z2,z2)vkn , mn(zn , zn ) is an eigenfunction for the eigenvalue j=1nkj.all other possible n - fold products with factors ukj , mj or vkj , mj ( also mixed ) appear as eigenfunctions of ,q.from this we obtain that all expressions of the form z11z11znnznn for arbitrary j,jn0,j=1,  ,n , can be written as a linear combination of eigenfunctions of ,q , which proves that all these eigenfunctions constitute a complete basis in l(0,q)2(cn , e|z|2 ) ; see the proof of theorem  2.3 . the factor q in the last formula is responsible for the fact that the eigenvalues start with q , which can be seen , in each component separately , by 14uj+j=1nzjujzj=(q)uj . then the function uk1,m1(z1,z1)uk2,m2(z2,z2)ukn , mn(zn , zn ) is a component of an eigenfunction for the eigenvalue j=1n(kj+mj ) of the operator ,q , which follows from ( 1.6 ) and ( 2.2 ) . similarly , it follows from ( 1.6 ) and ( 2.3 ) that , for k1,k2,  ,knn and m1,m2,  ,mnn0 , the function vk1,m1(z1,z1)vk2,m2(z2,z2)vkn , mn(zn , zn ) is an eigenfunction for the eigenvalue j=1nkj . all other possible n - fold products with factors ukj , mj or vkj , mj ( also mixed ) appear as eigenfunctions of ,q . from this we obtain that all expressions of the form z11z11znnznn for arbitrary j,jn0,j=1,  ,n , can be written as a linear combination of eigenfunctions of ,q , which proves that all these eigenfunctions constitute a complete basis in l(0,q)2(cn , e|z|2 ) ; see the proof of theorem  2.3 . remark  2.5(i ) since in all cases the essential spectrum is non - empty , the corresponding -neumann operator fails to be with compact resolvent ( see for instance  ) .(ii ) if one considers the weight function (z)=(|z1|2+|z2|2++|zn|2)for>1 , the situation is completely different : the operators ,q are with compact resolvent ( see  ) , so the essential spectrum must be empty . ( i ) since in all cases the essential spectrum is non - empty , the corresponding -neumann operator fails to be with compact resolvent ( see for instance  ) . ( ii ) if one considers the weight function (z)=(|z1|2+|z2|2++|zn|2)for>1 , the situation is completely different : the operators ,q are with compact resolvent ( see  ) , so the essential spectrum must be empty . we can use the results from above to settle the corresponding questions for the so - called witten laplacian which is defined on l2(cn ) . for this purpose , we set zk=zk+12zk and zk=zk+12zk , and we consider ( 0,q)-forms h=|j|=qhjdzj , where  means that we sum up only increasing multiindices j=(j1,  ,jq ) and where dzj = dzj1  dzjq . we define ( 2.4)dq+1h=k=1n|j|=qzk(hj)dzkdzj and ( 2.5)dqh=k=1n|j|=qzk(hj)dzkdzj , where dzkdzj denotes the contraction , or interior multiplication by dzk , i.e.  we have ,dzkdzj=dzk,dzj for each ( 0,q1)-form . the complex witten laplacian on ( 0,q)-forms is then given by ( 2.6)(0,q)=dqdq+dq+1dq+1 , for q=1,  ,n1 . in particular , we get , for a function vl2(cn ) , ( 2.10)(0,0)v = d1d1v=j=1nzjzj(v ) , and , for a ( 0,1)-form g==1ngdzl(0,1)2(cn ) , we obtain ( 2.11)(0,1)g=(d1d1+d2d2)g=((0,0)i)g+mg , where we set mg=j=1n(k=1n2zkzjgk)dzj and ( (0,0)i)g=k=1n(0,0)gkdzk . in general , we have that ( 2.12)(0,q)=e/2,qe/2 , for q=0,1  ,n . for more details , see  . in our case (z)=|z1|2++|zn|2 , we get ( 2.13)(0,q)h=|j|=q(14hj+12j=1n(zjhjzjzjhjzj)+14|z|2hj+(qn2)hj)dzj , for h=|j|=qhjdzjdom(0,q)l(0,q)2(cn ) . the spectrum of (0,0 ) , in an even more general form , was calculated by ma and marinescu ; see   and . using ( 2.12 ) and lemma  2.1 , we get that (0,q ) and ,q have the same spectrum . hence , by theorem  2.4 , we obtain the following theorem . theorem  2.6let 
 (z)=|z1|2++|zn|2 
 and 
 0qn 
 . the spectrum of the witten laplacian 
 (0,q ) 
 consists of all integers 
 { q , q+1,q+2 ,  } , each of which is of infinite multiplicity . the spectrum of the witten laplacian 
 (0,q ) 
 consists of all integers 
 { q , q+1,q+2 ,  } , each of which is of infinite multiplicity .
the spectrum of the -neumann laplacian on the fock space l2(cn , e|z|2 ) is explicitly computed . 
 it turns out that it consists of positive integer eigenvalues , each of which is of infinite multiplicity . 
 spectral analysis of the -neumann laplacian on the fock space is closely related to schrdinger operators with magnetic fields and to the complex witten laplacian .