On the two-dimensional Coulomb-like potential with a central point interaction
aa r X i v : . [ m a t h - ph ] J a n On the two-dimensional Coulomb-like potentialwith a central point interaction
P. Duclos ∗ , P. Šťovíček , M. Tušek Centre de Physique Théorique (CPT-CNRS UMR 6207) Université du Sud,Toulon-Var, BP 20132, F–83957 La Garde Cedex, France Department of Mathematics, Faculty of Nuclear Science, Czech TechnicalUniversity in Prague, Trojanova 13, 120 00 Praha, Czech Republic
Abstract
In the first part of the paper, we introduce the Hamiltonian − ∆ − Z/ p x + y , Z > , as a self-adjoint operator in L ( R ) . A general central point interac-tion combined with the two-dimensional Coulomb-like potential is constructedand properties of the resulting one-parameter family of Hamiltonians is studiedin detail. The construction is also reformulated in the momentum representa-tion and a relation between the coordinate and the momentum representationis derived. In the second part of the paper we prove that the two-dimensionalCoulomb-like Hamiltonian can be derived as a norm resolvent limit of the Hamil-tonian of a Hydrogen atom in a planar slab as the width of the slab tends to zero. PACS : 02.30.Sa, 02.30.Tb, 03.65.Db
In this paper we discuss, in the framework of nonrelativistic quantum mechanics, twosubjects related to the two-dimensional Coulomb-like potential in the plane. In thefirst part, in Section 2, we reexamine the two-dimensional Hydrogen atom. This is tosay that we consider a quantum model in the plane with the attractive potential V ( x, y ) = − Z/̺, ̺ = p x + y , (1)which we call the two-dimensional Coulomb (or hydrogenic) potential. This modelhas already been studied from various points of view in the physical literature. A ∗ Pierre Duclos passed away in January 2010. The manuscript was prepared for publication by thesecond and the third author. div E = σ where σ stands for the planar charge density, E z = 0 , and the electric fieldis supposed to be rotationally symmetric. Integration of this equation over a diskof radius ̺ together with application of Green’s theorem leads to the choice of thepotential in the form V ( x, y ) = const ln( ̺ ) . The Schrödinger equation for this potential is studied in [4].One of the goals of the present paper is to describe a central point interactioncombined with the two-dimensional Coulomb-like potential and to study its basicproperties. The construction of point interactions based on the theory of self-adjointextensions is now pretty well established. To our best knowledge, however, the two-dimensional Coulomb-like potential is not yet discussed in the literature, including thewell known monograph [2] where only the one-dimensional and the three-dimensionalcases are considered. On the other hand, there exists a general theoretical backgroundfor the construction of self-adjoint extensions with singular boundary conditions, asdescribed in paper [9], which is directly applicable to our model.Along with the construction of point interactions in the coordinate representation,and this is the standard way how to proceed, we discuss the construction also inthe momentum representation. Moreover, we derive an explicit relation between thetwo representations. This correspondence is based on the Whittaker integral trans-formation whose integral kernel is a properly normalized generalized eigenfunction ofthe two-dimensional Coulomb-like Hamiltonian depending on the spectral parameter.Remarkably, this integral transformation has been studied in the mathematical liter-ature quite recently [3, 14]. On this point we refer, first of all, to paper [11] wherethe unitarity of the eigenfunction expansion is proven for a much more general classof Schrödinger operators on a halfline.In the second part of the present paper, in Section 3, we study the Hydrogen atomin a thin planar layer of width a , called Ω a . In our model, we confine the atom to theslab by imposing the Dirichlet boundary condition on the parallel boundary planes.Our main goal in this section is to show that the resulting Hamiltonian in L (Ω a ) ,2alled H a , is well approximated in a convenient sense by the two-dimensional Coulomb-like Hamiltonian as the width of the layer approaches zero. The method we use isstrongly motivated by the paper by Brummelhuis and Duclos [7]. Firstly, we applythe projection on the first transversal mode getting this way the so-called effectiveHamiltonian in L ( R ) . Then, in Subsection 3.3, we show that the norm resolventlimit of the effective Hamiltonian, as a → , exactly equals the two-dimensionalhydrogenic Hamiltonian plus the energy of the lowest transversal mode. As a nextstep we prove, in Subsection 3.4, that the full Hamiltonian H a is well approximatedby the effective Hamiltonian, again in the norm resolvent sense. Since the spectrumof H C is known explicitly one can use this approximation to derive, with the aid ofstandard perturbation methods, asymptotic formulas for the eigenvalues of H a thoughwe do not go into details at this point. Let − ∆ be the free Hamiltonian in L ( R , d x d y ) . It is known that the Coulomb-likepotential ( x + y ) − / in the plane is ( − ∆) form bounded with relative bound zero.This is a consequence of the Kato inequality; the proof is given in [6] but see also [12]where even a more general case is treated. In more detail, the inequality claims that p x + y ≤ Γ(1 / π √− ∆ . (2)Suppose Z > . By the KLMN theorem [17, Theorem X.17], the operator H C = − ∆ − Z p x + y (the form sum) (3)is self-adjoint. The form domain of the Hamiltonian H C coincides with that of thefree Hamiltonian, i.e. with the first Sobolev space. In particular, Dom( H C ) ⊂H ( R ) . Note that the same conclusions can also be deduced from the results in[19, Chp. XIII.11]. The operator H C has been studied quite intensively in the physicalliterature (see, for example, [13, 21, 16]).To introduce a central point interaction let us consider the densely defined sym-metric operator, ˙ H = − ∆ − Z p x + y , Dom( ˙ H ) = C ∞ ( R \ { } ) . Denote by H min the closure of ˙ H . Then H C is exactly the Friedrichs extension of H min .As usual, the Hilbert space naturally decomposes in the polar coordinates ( ̺, ϕ ) , L ( R , d x d y ) = ∞ M m = −∞ L ( R + , ̺ d ̺ ) ⊗ C e imϕ , H min decomposes correspondingly, H min = ∞ M m = −∞ H min ,m ⊗ , where H min ,m is the closure of the operator ˙ H m = − ∂ ∂̺ − ̺ ∂∂̺ + m ̺ − Z̺ ,
Dom( ˙ H m ) = C ∞ ( R + ) . Put also H max ,m = H † min ,m . For the maximal operator one has [20, Chapter 8] Dom( H max ,m ) = (cid:8) f ∈ L ( R + , ̺ d ̺ ); f, f ′ ∈ AC loc ( R + ) , L m f ∈ L ( R + , ̺ d ̺ ) (cid:9) , with L m = − ∂ ∂̺ − ̺ ∂∂̺ + m ̺ − Z̺ . If f ∈ Dom( H max ,m ) then H max ,m f = L m f .For z ∈ C \ R and m ∈ Z consider the equation ( L m − z ) f = 0 . Two independentsolutions are expressible in terms of the Whittaker functions, namely ̺ − / M Z/ (2 √− z ) , | m | (cid:0) √− z ̺ (cid:1) and ̺ − / W Z/ (2 √− z ) , | m | (cid:0) √− z ̺ (cid:1) , (4)with Re √− z > . From the asymptotic expansions it follows (see, for instance,[1]) that the former function in (4) is square integrable at zero but is not squareintegrable at infinity while the latter one is square integrable at infinity but is notsquare integrable at zero, except of the case m = 0 . Thus for m = 0 , the operators H min ,m = H max ,m = H m are self-adjoint while for m = 0 , H min , has deficiency indices (1 , . For a wide class of Schrödinger operators, including our case as well, an explicitconstruction of all self-adjoint extensions defined by boundary conditions can be foundin [9]. Proposition 1.
All self-adjoint extensions of H min , are H ( κ ) ⊂ H max , , κ ∈ R ∪{∞} ,with the domains Dom ( H ( κ )) = { f ∈ Dom( H max , ); f = κf } , where the boundary values f , f are defined by f = lim ̺ → ( − ln ̺ ) − f ( ̺ ) , f = lim ̺ → ( f ( ̺ ) + f ln ̺ ) . (5) The self-adjoint extension H ( ∞ ) determined by the boundary condition f = 0 coin-cides with the Friedrichs extension of H min , . All self-adjoint extensions H ( κ ) of H min are again labeled by κ ∈ R ∪ {∞} and areequal to H ( κ ) = − M m = −∞ H m ⊕ H ( κ ) ⊕ ∞ M m =1 H m . In particular, H ( ∞ ) coincides with H C . 4 roposition 2. For the essential spectrum one has σ ess ( H C ) = [ 0 , ∞ ) and, moregenerally, σ ess ( H ( κ )) = [ 0 , ∞ ) for all κ ∈ R ∪ {∞} .Proof. Let us introduce (temporarily) the functions U ( x, y ) = − Z p x + y + 1 , U ( x, y ) = − Z p x + y + Z p x + y + 1 , and denote by U and U the corresponding multiplication operators. Put A = − ∆+ U .Since U ( x, y ) is bounded and tends to zero at infinity one knows that σ ess ( A ) =[ 0 , ∞ ) (see, for instance, [5, Theorem 4.1]). Note that, by the closed graph theorem, ( A + k ) − / ( − ∆ + 1) / is bounded for k > sufficiently large. Moreover, U is arelatively form bounded perturbation of A and H C equals the form sum A + U . It isshown below, in the proof of Lemma 5, that the operator ( − ∆ + 1) − / U ( − ∆ + 1) − / is Hilbert-Schmidt and hence compact. Consequently, U is a relatively form-compactperturbation of A and, by the results in [19, Chp. XIII.4] related to the Weyl theorem, σ ess ( H C ) = σ ess ( A ) . To extend the equality to all H ( κ ) it suffices to observe that, bythe Krein formula, the resolvent of H ( κ ) is a rank-one perturbation of the resolventof H ( ∞ ) .By the general theory of Sturm-Liouville operators, the resolvent kernels G m ( z ; ̺, ̺ ′ ) of the partial Hamiltonians H ( ∞ ) , if m = 0 , and H m = H min ,m = H max ,m , if m = 0 ,are equal to G m ( z ; ̺, ̺ ′ ) = 12(2 | m | )! √− z √ ̺̺ ′ Γ (cid:18)
12 + | m | − Z √− z (cid:19) × M Z/ (2 √− z ) , | m | (cid:0) √− z̺ < (cid:1) W Z/ (2 √− z ) , | m | (cid:0) √− z̺ > (cid:1) Here ̺ < , ̺ > denote the smaller and the greater out of ̺ , ̺ ′ , respectively.The Green function G κ ( z ; ̺, ̺ ′ ) for the Hamiltonian H ( κ ) , κ ∈ R , can be con-structed using the Krein resolvent formula that guarantees existence of a function φ κ ( z ) such that G κ ( z ; ̺, ̺ ′ ) = G ( z ; ̺, ̺ ′ ) + φ κ ( z ) √ ̺̺ ′ W Z/ (2 √− z ) , (cid:0) √− z̺ (cid:1) W Z/ (2 √− z ) , (cid:0) √− z̺ ′ (cid:1) , (6)with z ∈ C \ R . Since the integral kernel must satisfy the same boundary conditionas that defining Dom H ( κ ) we have φ κ ( z ) = 12 √− z Γ (cid:18) − Z √− z (cid:19) (cid:18) γ + ln(2 √− z ) + Ψ (cid:18) − Z √− z (cid:19) + κ (cid:19) − (7)where Ψ( z ) = Γ ′ ( z ) / Γ( z ) is the polygamma function and γ = − Γ ′ (1) is the Eulerconstant. The Green function of H ( κ ) , κ ∈ R ∪ {∞} , expressed in polar coordinates,equals G κ ( z ; ̺, ϕ, ̺ ′ , ϕ ′ ) = 12 π ∞ X m = −∞ G m ( z ; ̺, ̺ ′ ) e im ( ϕ − ϕ ′ ) + φ κ ( z )2 π √ ̺̺ ′ W Z/ (2 √− z ) , (cid:0) √− z̺ (cid:1) W Z/ (2 √− z ) , (cid:0) √− z̺ ′ (cid:1) . (8)5he point spectrum of H C equals the union of the point spectra of H m , m ∈ Z (with H ≡ H ( ∞ ) ). The eigenvalues of H C jointly with eigenfunctions are computedin [21] and correspond to the poles of the respective Green functions. Thus we recallthat all eigenvalues of H m , m ∈ Z , are simple and are equal to λ m,n = − Z (2 | m | + 2 n + 1) , n ∈ Z + , (9)(here Z + = { n ∈ Z ; n ≥ } ). Denote by N = | m | + n + 1 the principal quantumnumber and put λ N = λ m,n for | m | + n = N − , i.e. λ N = − Z / (2 N − , N ∈ N .Then the multiplicity of λ N in the spectrum of H C is N − . The correspondingnormalized eigenfunctions are ψ m,n ( ̺, ϕ ) = (cid:18) n !2 π ( n + 2 | m | )! (cid:19) / Z (2 | m | + 2 n + 1) / (cid:18) Z̺ | m | + 2 n + 1 (cid:19) | m | × L (2 | m | ) n (cid:18) Z̺ | m | + 2 n + 1 (cid:19) e − Z̺/ (2 | m | +2 n +1) e imϕ , where L (2 | m | ) n stands for the associated Laguerre polynomial.Using a similar reasoning as in [8] one concludes that the point spectrum of H ( κ ) , κ ∈ R , contains the eigenvalues λ N with multiplicities N − (hence λ is missing).In fact, the point spectrum of H m is simple and is formed by the eigenvalues λ N , N ≥| m | + 1 . If a point interaction is switched on then the spectrum of the component H is deformed while the point spectra of the components H m , m = 0 , remain untouched.On the other hand, if κ ∈ R then additional eigenvalues emerge in the spectrum of H ( κ ) , the so called point levels. They are simple and negative. Let us denote themin ascending order by ǫ j ( Z ; κ ) , j = Z + .From the general theory concerned with Friedrichs extensions [17, Theorem X.23]and location of discrete spectra of self-adjoint extensions [20, Chp. 8.3] one deducesthat the points levels are located as follows ǫ ( Z ; κ ) < λ < ǫ ( Z ; κ ) < λ < ǫ ( Z ; κ ) < λ < . . . < . Using the substitution ǫ j ( Z ; κ ) = − Z k j ( κ ) , κ = κ + ln Z, one finds from (6) and (7) that the equation on point levels takes the form γ + ln(2 k ) + Ψ (cid:18) − k (cid:19) + κ = 0 , (10)with the unknown k = k j ( κ ) > . This implies the scaling property ǫ j ( Z ; κ ) = Z ǫ j (1; κ + ln Z ) , j = 0 , , , . . . .
6y an elementary analysis of equation (10) one can show that ǫ j ( Z ; κ ) are strictlyincreasing functions in the parameter κ ∈ R , and one has the asymptotic formulas ǫ j ( Z ; κ ) = − Z (2 j + 1) − Z (2 j + 1) κ + O ( κ − ) as κ → + ∞ , for all j ≥ , and ǫ ( Z ; κ ) = − e − γ − κ + O ( e − κ ) as κ → −∞ ,ǫ j ( Z ; κ ) = − Z (2 j − − Z (2 j − κ + O ( κ − ) as κ → −∞ , j ≥ . Figure 1 depicts several first point levels as functions of κ for Z = 1 .Finally note that from the form of the Green function one can also derive normal-ized eigenfunctions corresponding to the point levels, namely η j ( κ ; ̺, ϕ ) = s Z π̺ k j ( κ ) (cid:18) k j ( κ ) + 12 Ψ ′ (cid:18) − k j ( κ ) (cid:19)(cid:19) − / Γ (cid:18) − k j ( κ ) (cid:19) × W / (2 k j ( κ )) , (cid:0) k j ( κ ) Z̺ (cid:1) , where k j ( κ ) = ( − ǫ j ( Z ; κ )) / /Z , j = 0 , , , . . . . The normalized generalized eigenfunctions for H m , m ∈ Z , (with H ≡ H ( ∞ ) ) areknown including the correct normalization [21]. One has, with k > , ψ m ( k, ̺ ) = 1(2 | m | )! (cid:18)
21 + e − πZ/k (cid:19) / | m |− Y s =0 (cid:18) s + 12 (cid:19) + Z k ! / i m √ ik̺ M Z/ (2 ik ) , | m | (2 ik̺ ) . In a comparatively recent paper [11] a large class of Schrödinger operators on a halflinewith strongly singular potentials is studied, with the results being directly applicableto operators H m , m ∈ Z . In that article, a measure on the dual space is constructedwith the aid of the associated Titchmarsh-Weyl m -function, and unitarity of the eigen-function expansion, involving both proper and generalized eigenfunctions, is proven(one can also consult paper [14] which appeared later on and covers a less generalclass of potentials but with our example still being included). As a consequence onededuces that the integral transform T m : L ( R + , ̺ d ̺ ) → L ( R + , k d k ) , T m [ f ]( k ) = ˆ ∞ ψ m ( k, ̺ ) f ( ̺ ) ̺ d ̺ is a well defined bounded operator. Denote by H m, pp the closure of the subspacein L ( R + , ̺ d ̺ ) spanned by the eigenfunctions ψ m,n ( ̺ ) , n ∈ Z + , and by H m, ac itsorthogonal complement. Then the kernel of T m equals H m, pp , and the restriction T m, ac := T m (cid:12)(cid:12) H m, ac : H m, ac → L ( R + , k d k )
7s a unitary mapping. Moreover, T m, ac transforms H m (cid:12)(cid:12) H m, ac into the multiplicationoperator by the function k acting in L ( R + , k d k ) . It follows that the essential spec-trum of H m is in fact absolutely continuous. The same is also true for all H ( κ ) , κ ∈ R . Proposition 3.
For all m ∈ Z one has σ ess ( H m ) = σ ac ( H m ) = [ 0 , ∞ ) and σ pp ( H m ) = { λ m,n ; n ∈ Z + } (with the only accumulation point being just ). Similarly, for all κ ∈ R one has σ ess ( H ( κ )) = σ ac ( H ( κ )) = [ 0 , ∞ ) and σ pp ( H ( κ )) = { ǫ j ( κ ); j ∈ Z + } .Moreover, the spectra of H m , m ∈ Z , and H ( κ ) , κ ∈ R , are simple. In particular, thesingular continuous spectra of H m and H ( κ ) are empty. The transformation inverse to T m, ac is T − m, ac : L ( R + , k d k ) → H m, ac , T − m, ac [ g ]( ̺ ) = ˆ ∞ ψ m ( k, ̺ ) g ( k ) k d k. Thus one concludes that H m in L ( R + , ̺ d ̺ ) is unitarily equivalent to ˆ H m in ˆ H m = ℓ ( Z + ) ⊕ L ( R + , k d k ) . Dom( ˆ H m ) is formed by those ˆ f = { ˆ f n } ∞ n =0 + ˆ f ( k ) ∈ ˆ H m for which k ˆ f ( k ) ∈ L ( R + , k d k ) .If ˆ f ∈ Dom( ˆ H m ) then ˆ H m ˆ f = { λ m,n ˆ f n } ∞ n =0 + k ˆ f ( k ) . The unitary mapping ˆ H m → L ( R + , ̺ d ̺ ) : ˆ f f is given by f ( ̺ ) = ∞ X n =0 ˆ f n ψ m,n ( ̺ ) + ˆ ∞ ψ m ( k, ̺ ) ˆ f ( k ) k d k. Conversely, ˆ f n = h ψ m,n , f i , ˆ f ( k ) = T m [ f ]( k ) . One has k ˆ f k = ∞ X n =0 | ˆ f n | + ˆ ∞ | ˆ f ( k ) | k d k = ˆ ∞ | f ( ̺ ) | ̺ d ̺ = k f k . One can use the momentum representation for an alternative and equivalent con-struction of point interactions. It again turns out that an nontrivial result can bederived only in the sector m = 0 to which we confine our attention. A symmetricrestriction A of ˆ H is obtained by requiring that f (0) = 0 if ˆ f ∈ Dom A ⊂ Dom ˆ H .More details follow. From now on we omit, in the notation, the hat over elements f ∈ ˆ H .Let us denote the normalization factor of generalized eigenfunctions as N ( k ) = (cid:18)
21 + e − πZ/k (cid:19) / , k > . For g ∈ ˆ H such that g ( k ) ∈ L ( R + , k d k ) put S ( g ) = ∞ X n =0 Z (2 n + 1) / g n + ˆ ∞ N ( k ) g ( k ) k d k. ξ ∈ C and f ∈ ˆ H such that f ( k ) − ξN ( k ) / ( k + Z ) ∈ L ( R + , k d k ) put S ( ξ, f ) = ∞ X n =0 Z (2 n + 1) / f n + ˆ ∞ N ( k ) (cid:18) f ( k ) − ξN ( k ) k + Z (cid:19) k d k. Clearly, if ξ exists then it is unambiguously determined by f , and S ( g ) ≡ S (0 , g ) .Observe that ∀ g ∈ Dom( ˆ H ) , g ( k ) ∈ L ( R + , k d k ) , and one has ˇ g (0) = S ( g ) where ˇ g ( ̺ ) = ∞ X n =0 g n ψ ,n ( ̺ ) + ˆ ∞ ψ ( k, ̺ ) g ( k ) k d k. (11)One defines A ⊂ ˆ H by Dom( A ) = { g ∈ Dom( ˆ H ); S ( g ) = 0 } . It is not difficult to check that f ∈ Dom( A ∗ ) if and only if there exist (necessarilyunique) ξ ∈ C and η ∈ ˆ H such that for all n ∈ Z + and almost all k > , λ ,n f n = η n + 2 Zξ (2 n + 1) / , k f ( k ) = η ( k ) + ξN ( k ) . In that case, A ∗ f = η . Note that if f ∈ Dom( A ∗ ) and ξ , η are as above then f ( k ) − ξN ( k ) k + Z = Z f ( k ) k + Z + η ( k ) k + Z ∈ L ( R + , k d k ) , Let us now discuss self-adjoint extensions of A . The deficiency indices of A are (1 , . For z ∈ C \ R , Ker( A ∗ − z ) = C f z where ∀ n ∈ Z + , ( f z ) n = 2 Z (2 n + 1) / ( λ ,n − z ) ; ∀ k > , f z ( k ) = N ( k ) k − z . (12)For the computational convenience the spectral parameter is chosen to be z = iZ / .For e iα ∈ T let A α be the self-adjoint extension of A defined by Dom( A α ) = Dom( A ) + C (cid:0) f z + e iα f ¯ z (cid:1) ,A α (cid:0) g + t (cid:0) f z + e iα f ¯ z (cid:1)(cid:1) = Ag + t (cid:0) zf z + ¯ ze iα f ¯ z (cid:1) . If f = g + t (cid:0) f z + e iα f ¯ z (cid:1) ∈ Dom( A α ) , with g ∈ Dom( A ) , t ∈ C , (13)then there exists ξ ∈ C (necessarily unique) such that f ( k ) − ξN ( k ) k + Z ∈ L ( R + , k d k ) , namely ξ = t (1 + e iα ) . Furthermore, one has S ( ξ, f ) = t (cid:0) S (1 , f z ) + e iα S (1 , f ¯ z ) (cid:1) . S (1 , f z ) = − Re (cid:18) Ψ (cid:18) i (cid:19)(cid:19) − γ −
72 ln(2) + i + iπ iπ (cid:16) π (cid:17) . The computation is based on the following identities: for a / ∈ Z + + 1 , ∞ X n =0 n + 1)(2 n + 1 − a ) = 12 a (cid:18) Ψ (cid:18) (cid:19) − Ψ (cid:18) − a (cid:19)(cid:19) , (14)and, for a / ∈ ( −∞ , , ˆ ∞ y (1 + e πy )( y + a ) d y = −
14 ln(4 a ) + Ψ( √ a ) −
12 Ψ (cid:18) √ a (cid:19) . (15)Notice that S (1 , f ¯ z ) = S (1 , f z ) . Put ˆ κ = 11 + e iα (cid:16) S (1 , f z ) + e iα S (1 , f z ) (cid:17) = − Re (cid:18) Ψ (cid:18) i (cid:19)(cid:19) − γ −
72 ln(2) + (cid:16) π π (cid:16) π (cid:17)(cid:17) tan (cid:16) α (cid:17) . Still assuming (13) one has S ( ξ, f ) = ξ ˆ κ. Let us redenote A α = ˆ H (ˆ κ ) .One concludes that the one-parameter family of all self-adjoint extensions of A is ˆ H (ˆ κ ) , ˆ κ ∈ R ∪ {∞} . A vector f ∈ ˆ H belongs to Dom( ˆ H (ˆ κ )) iff there exists ξ ∈ C (necessarily unique) such that f ( k ) − ξN ( k ) k + Z ∈ L ( R + , k d k ) , k f ( k ) − κ S ( ξ, f ) N ( k ) ∈ L ( R + , k d k ) . (16)Then ˆ H (ˆ κ ) f = (cid:26) λ ,n f n − Zξ (2 n + 1) / (cid:27) ∞ n =0 + (cid:18) k f ( k ) − κ S ( ξ, f ) N ( k ) (cid:19) . In addition one has S ( ξ, f ) / ˆ κ = ξ . Clearly, ˆ H ( ∞ ) = ˆ H .Let us check the point spectrum of ˆ H (ˆ κ ) , ˆ κ ∈ R . Suppose = f ∈ ˆ H and λ ∈ R fulfill ˆ H (ˆ κ ) f = λf . This means that λ ,n f n − Zξ (2 n + 1) / = λf n for n ∈ Z + , k f ( k ) − ξN ( k ) = λf ( k ) for k > . (17)Clearly, λ must be negative since otherwise f ( k ) = N ( k ) / ( k − λ ) would not be L integrable. Furthermore, the point spectrum of ˆ H (ˆ κ ) is disjoint with the pointspectrum of ˆ H . In fact, suppose λ = λ ,p for some p ∈ Z + . Then from the firstequation in (17), with n = p , it follows that ξ = 0 . Moreover, (17) implies f n = 0 for n = p , and f ( k ) = 0 for k > . Necessarily, f p = 0 . Then S ( ξ, f ) = 2 Z (2 p + 1) − / f p = 0 λ < and λ = λ ,n , ∀ n , is an eigenvalue. Then there exists one indepen-dent eigenvector f corresponding to λ for which one can put ξ = 1 , f n = 2 Z (2 n + 1) / ( λ ,n − λ ) for n ∈ Z + , f ( k ) = N ( k ) k − λ for k > . (18)The eigenvalue equation reads S (1 , f ) = ˆ κ , with f given in (18), i.e. ∞ X n =0 Z (2 n + 1) ( λ ,n − λ ) + ˆ ∞
21 + e − πZ/k (cid:18) k − λ − k + Z (cid:19) k d k = ˆ κ. One can get rid of the parameter Z using the substitution λ = − Z /x . With the aidof (14) and (15) one finds that λ = − Z /x is an eigenvalue iff x solves the equation π tan (cid:16) π x (cid:17) + ln( x ) − Ψ (cid:18) x (cid:19) − γ − κ. (19) We wish to compare the operators ˆ H (ˆ κ ) and H ( κ ) . The domain of the latter Hamil-tonian in the coordinate representation is given by a boundary condition at the origin.So we have to determine the asymptotic behavior of ˇ g ( ̺ ) as ̺ → for an arbitrary g ∈ Dom ˆ H (ˆ κ ) , with ˇ g being given in (11).As a first step we find a relation between the basis function f z of the deficiencysubspace given in (12) and ˇ f z ( ̺ ) , a basis function of the deficiency subspace in thecoordinate representation. To simplify the notation let us temporarily set Z = 1 . Weput ˇ f z ( ̺ ) = 1 √ ̺ W / (2 √− z ) , (2 √− z ̺ ) , z ∈ C \ [ 0 , + ∞ ) . This can be rewritten in terms of confluent hypergeometric functions, ˇ f z ( ̺ ) = √ − z ) / e −√− z ̺ U (cid:18) − √− z , , √− z ̺ (cid:19) . One knows that ∞ X n =0 ( f z ) n ψ ,n ( ̺ ) + ˆ ∞ f z ( k ) ψ ( k, ̺ ) k d k = C ( z ) ˇ f z ( ̺ ) where C ( z ) is a holomorphic function on C \ [ − , + ∞ ) .By unitarity, ∞ X n =0 | ( f z ) n | + ˆ ∞ | f z ( k ) | k d k = | C ( z ) | ˆ ∞ | ˇ f z ( ̺ ) | ̺ d ̺. z < − . In that case, ˆ ∞ ˇ f z ( ̺ ) ̺ d ̺ = 2 √− z + Ψ ′ (cid:16) − √− z (cid:17) − z )Γ (cid:16) − √− z (cid:17) . Furthermore, ∞ X n =0 ( f z ) n = 14( − z ) / (cid:18) Ψ ′ (cid:18) − √− z (cid:19) − Ψ ′ (cid:18)
12 + 12 √− z (cid:19)(cid:19) . Using the identity ˆ ∞ πx/ ( x + a ) d x = 1 πa Ψ ′ (cid:18) a (cid:19) one finds that ˆ ∞ f z ( k ) k d k = − z + 14( − z ) / Ψ ′ (cid:18)
12 + 12 √− z (cid:19) . Finally one arrives at the equality ∞ X n =0 ( f z ) n ψ ,n ( ̺ ) + ˆ ∞ f z ( k ) ψ ( k, ̺ ) k d k = 1( − z ) / Γ (cid:18) − √− z (cid:19) √ ̺ W √− z , (2 √− z ̺ ) . (20)Using a simple scaling one can return back to a general parameter Z > . Consid-ering the limit z → − in (20) one derives the asymptotic formula ˆ ∞ ψ ( k, ̺ ) N ( k ) k + Z k d k = − ln( Z̺ ) − γ + 3 ln(2) + O ( ̺ ln( ̺ )) as ̺ → . Suppose f ∈ Dom( ˆ H (ˆ κ )) . Then ( ξ ∈ C is introduced in the definition of Dom( ˆ H (ˆ κ )) ) ˇ f ( ̺ ) = ∞ X n =0 f n ψ ,n ( ̺ )+ ˆ ∞ ψ ( k, ̺ ) (cid:18) f ( k ) − ξN ( k ) k + Z (cid:19) k d k + ξ ˆ ∞ ψ ( k, ̺ ) N ( k ) k + Z k d k. Hence ˇ f ( ̺ ) = S ( ξ, f ) + ξ ( − ln( Z̺ ) − γ + 3 ln(2)) + o (1) as ̺ → . Recall that S ( ξ, f ) = ξ ˆ κ . One concludes that ∀ f ∈ Dom( ˆ H (ˆ κ )) , ˇ f ( ̺ ) = ξ (cid:0) − ln( Z̺ ) + ˆ κ − γ + 3 ln(2) (cid:1) + o (1) as ̺ → . (21)Since the domain of H ( κ ) is determined by the asymptotic behavior at ̺ = 0 , ˇ f ( ̺ ) = − α ln( ̺ ) + α + o (1) as ̺ → , where α = κα , (22)one finds, by comparing (21) and (22), that the operators H ( κ ) and ˆ H (ˆ κ ) are uni-tarily equivalent if κ = ˆ κ − ln( Z ) − γ + 3 ln(2) . A Hydrogen atom in a thin layer
We wish to discuss a model describing a Hydrogen atom or a Hydrogen-like ion con-fined to an infinite planar slab Ω a of width a . Thus we denote Ω a = R × (cid:16) − a , a (cid:17) ⊂ R . Our goal is to consider the limit when the width a tends to zero. Let us first introducethe notation and recall some related results.For Ω ⊂ R n , an nonempty open set with a Lipschitz continuous boundary oneach component, denote by H m (Ω) the m th Sobolev space and by H m (Ω) the closureof C ∞ (Ω) in H m (Ω) . One has a natural isometric embedding H m (Ω) ⊂ H m ( R n ) .Furthermore, D , ( R n ) denotes the completion of C ∞ ( R n ) with respect to the norm u (cid:18) ˆ R n |∇ u | d x (cid:19) / . In this case one has a continuous embedding H ( R n ) ⊂ D , ( R n ) . Recall also that theDirichlet Laplacian − ∆ D is the unique self-adjoint operator associated with the closedpositive form q ( f, g ) = h∇ f, ∇ g i defined on H (Ω) (the scalar product is taken in L (Ω , d x ) ). The form representation theorem implies that Dom ∆ D = H (Ω) ∩ H (Ω) .Below we employ the Hardy inequality in R which states that for any u ∈ D , ( R ) , ˆ R | u ( x ) | | x | d x ≤ ˆ R |∇ u ( x ) | d x . (23)The Hardy inequality is extended to domains with boundaries in [22] where one canfind additional references. In the case of the Dirichlet boundary condition, however,one can simply make use of the chain of embeddings H (Ω) ⊂ H ( R ) ⊂ D , ( R ) .Hence inequality (23) holds for any u ∈ H (Ω) where Ω ⊂ R is still supposed to havethe above stated properties.In our model we introduce the Hamiltonian H a = − ∆ D − Zr , (24)with r = p x + y + z and Z > , in the Hilbert space L (Ω a ) . To see that a self-adjoint operator is well defined by relation (24) it suffices to show that the potential /r is ( − ∆ D ) bounded with a relative bound less than one (or even zero) and to referto the Kato-Rellich theorem. In fact, by the Hardy inequality (23), the estimate k r − ψ k ≤ k∇ ψ k = 4 h ψ, − ∆ D ψ i ≤ (cid:18) ǫ k − ∆ D ψ k + 1 ǫ k ψ k (cid:19) holds for all ψ ∈ Dom ∆ D and ǫ > . Thus one has Dom H a = Dom( − ∆ D ) = H (Ω a ) ∩ H (Ω a ) , Q ( H a ) = H (Ω a ) Q ( A ) stands for the form domain of A ).Using the scaling x → Z x one can readily see that H aZ (the Hamiltonian H a for agiven constant Z ) is unitarily equivalent to Z H aZ =1 . This is why we can set, withoutloss of generality, Z = 1 , and this is what we do in the remainder of the paper. The operator − ∆ D can be decomposed with respect to the basis in L (( − a/ , a/ , d z ) formed by the transversal modes, − ∆ D = ∞ M n =1 ( − ∆ x,y + E an ) ⊗ h χ an , · i χ an , with E an = n π a , χ an ( z ) = r a ( cos( nπz/a ) if n is odd sin( nπz/a ) if n is even , n ∈ N . Here − ∆ x,y is the free Hamiltonian in L ( R , d x d y ) . Put P an = 1 ⊗ h χ an , · i χ an , n ∈ N . Using the projection on the lowest transversal mode we define the effective Hamilto-nian, H a eff = P a H a P a . This Hamiltonian may be regarded as an operator on L ( R ) , H a eff = − ∆ x,y + E a − V a eff ( ̺ ) (25)where the effective potential is defined by V a eff ( ̺ ) = 2 a ˆ a/ − a/ cos ( πz/a ) p ̺ + z d z (26)and ̺ = p x + y . Note that < V a eff ( ̺ ) < /̺ for all ̺ > . Hence, if the Coulomb-like potential is ( − ∆) form bounded with relative bound zero than the same is truefor V a eff . Thus the RHS in (25) makes sense as a form sum and Q ( H a eff ) = H ( R ) .Moreover, − E a ≤ H C + E a ≤ H a eff . (27)with H C being defined in (3) (also denoted by H ( ∞ ) in the previous section). Let usalso note that σ ess ( H a eff ) = [ E a , ∞ ) .It is even true that V a eff is ( − ∆) bounded with relative bound zero. In fact, recallthat for any α > there is β such that ∀ f ∈ H ( R ) , k f k ∞ ≤ α k ∆ f k + β k f k , (28)14ith k · k being the L norm, see [17, Theorem IX.28]. Moreover, one observes that V a eff ( ̺ ) = − a ln( ̺ ) + O (1) as ̺ → , and V a eff ( ̺ ) decays like /̺ at infinity. Hence V a eff , regarded as a function on R , issquare integrable at the origin and tends to zero at infinity. It follows that for every ǫ > there exists a decomposition V a eff = V + V , with V ∈ L ∞ ( R ) , V ∈ L ( R ) , (29)such that k V k ∞ < ǫ . Combining (28) and (29) one finds that, for all f ∈ H ( R ) , k V a eff f k ≤ k V k ∞ k f k + k V kk f k ∞ ≤ α k V kk ∆ f k + ( β k V k + k V k ∞ ) k f k . This shows the relative boundedness and thus one can apply the Kato-Rellich theorem.In particular,
Dom H a eff coincides with Dom( − ∆) = H ( R ) . Moreover, the existenceof decomposition (29) implies that σ ess ( H a eff ) = [ E a , ∞ ) , see [19, Theorem XIII.15]. a Here we show that the Hamiltonian H a eff − E a converges to the two-dimensional hy-drogenic Hamiltonian H C in the norm resolvent sense as a → . Lemma 4.
One has k ( − ∆ + 2) / ( H C + 2) − / k ≤ C I where C I = 18 π Γ (cid:18) (cid:19) + s Γ (cid:18) (cid:19) + 64 π . (30) Proof.
Put L = ( − ∆ + 2) / ( H C + 2) − / . Then L is bounded by the closed graph the-orem but one can derive an upper bound explicitly with the aid of the Kato inequality(2). Since h ψ, ( − ∆ + 2) − / ̺ − ( − ∆ + 2) − / ψ i ≤ Γ (cid:0) (cid:1) π k ( − ∆) / ( − ∆ + 2) − / ψ k ≤ Γ (cid:0) (cid:1) π k ψ k one has L † L = 1+( H C +2) − / ̺ ( H C +2) − / ≤
1+ Γ (cid:0) (cid:1) π ( H C +2) − / ( − ∆+2) / ( H C +2) − / . It follows that k Lψ k ≤ k ψ k + Γ (cid:0) (cid:1) π k ( H C + 2) − / k k ψ k k Lψ k . For k ( H C + 2) − / k ≤ we get k L k ≤ (cid:0) (cid:1) π k L k . Consequently, k L k ≤ C I . 15 emma 5. Suppose W ∈ L ( R + , d ̺ ) and ≤ W ≤ . Put V a ( ̺ ) = 1 ̺ (cid:16) − W (cid:16) ̺a (cid:17)(cid:17) , a > . (31) Then for any a , < a < / , one has (cid:13)(cid:13) ( − ∆ + 2) − / (cid:0) ̺ − − V a (cid:1) ( − ∆ + 2) − / (cid:13)(cid:13) ≤ a ln ( a ) (cid:18) ˆ R + W ( ̺ ) d ̺ (cid:19) + 32 a ˆ R + W ( ̺ ) d ̺. (32) Proof.
Put T a = ( ̺ − − V a ) / ( − ∆ + 1) − / . (33)Then T † a T a = ( − ∆ + 1) − / ( ̺ − − V a ) ( − ∆ + 1) − / and k T † a T a k = k T a T † a k . Let us estimate the Hilbert-Schmidt norm of T a T † a . Theintegral kernel of T a T † a is K ( x , x ) = 12 π r ̺ W (cid:16) ̺ a (cid:17) K ( | x − x | ) r ̺ W (cid:16) ̺ a (cid:17) where x i = ̺ i (cos ϕ i , sin ϕ i ) . Since the modified Bessel function K is positive andstrictly decreasing on R + we get k T a T † a k ≤ I ( R + × R + ) where the symbol I ( M ) , M ⊂ R + × R + measurable, is defined by I ( M ) = ˆ M W (cid:16) ̺ a (cid:17) K ( | ̺ − ̺ | ) W (cid:16) ̺ a (cid:17) d ̺ d ̺ . For < a < / and | ̺ − ̺ | > a one has [1] K ( | ̺ − ̺ | ) < K ( a ) < (cid:18) ln (cid:18) a (cid:19) − γ (cid:19) I ( a ) + 12 I ( √ a ) − < (cid:18) ln (cid:18) a (cid:19) − γ (cid:19) I (cid:18) (cid:19) + 12 I (cid:18) √ (cid:19) − < − a ) . Consequently, I ( {| ̺ − ̺ | > a } ) ≤ a ln ( a ) (cid:18) ˆ R + W ( ̺ ) d ̺ (cid:19) . If | ̺ − ̺ | < a < / then K ( a | ̺ − ̺ | ) < − | ̺ − ̺ | . Moreover, for ≤ W ≤ , ˆ | ̺ − ̺ | < W ( ̺ ) ln ( | ̺ − ̺ | ) d ̺ ≤ ˆ | v | < ln | v | d v = 4 .