KREIN S FORMULA. Two dimensional kinetic Hamiltonian The O( invariant D kinetic operator admits a one-parameter family of self-adjoint extensions (SAE which can be labelled by some energy parameter E >. The spectral decomposition of a generic element within the family reads H(E = + l= dk h k m l k k l E ψ B ψ B ( where r θ l k = expilθ} π ψ l (k r; E k ( in which specifically whereas ψ l (k r = kj l (kr l Z } (3 ψ (k r; E = A(k; E J (kr + B(k; E N (kr. (4 The coefficients A(k; E and B(k; E are such that B(k; E A(k; E = The normalizable bound state is provided by r ψ B = ψ B (κ r = Notice that the eigenstates are normalized according to ( π ln ( h k /me. (5 κ π K (κr hκ me. (6 ψ B ψ B = l k l k = δ ll δ(k k k k >. Now we aim to show that the very same conclusion is reached following the Krein s method for the resolvent. Let us begin from the resolvent of the free kinetic operator in D which is defined by (H zg (r r ; z = δ ( (r r (7 where H = p m.
The resolvent can be written in terms of the basic integral representation [ d G (r r p exp ī h ; z p (r r ] h p m z ( mw = πm h = im h H( dw w z J (q r r hq mz Imz >. h r r (8 It is worthwhile to remark that if we set z = E + iη we obtain in the limit η and E > G (r r ; z = πm ( ( mw h dw CPV J r r w E h + i π m h J ( me r r. h From the above expression it immediately follows that lim η πi G (r r; E + iη G (r r; E iη} = π Im G (r r; E = πm h ρ (D as it does owing to translation invariance where ρ (D is the density of the quantum states per unit area in two space dimensions. Notice that H ( (q r r = J (q r r + in (q r r + i π [ln(q r r ln + γ E ] r r. It follows therefrom that G (r r ; z m π h ( ( q q r r ln + ln + γ E iπ} q r r where hq is some arbitrary momentum scale. It is important to realize that the resolvent ( exhibits an ultraviolet divergence at coincident points r = r. To this concern it is convenient to introduce the renormalized (or subtracted resolvent: namely G R (r r ; z = G (r r ; z + m ( π h ln q r r. ( (9 (
The renormalized resolvent turns out to be finite although arbitrary at coincident points as we have G R ( ; z = m ( } q π h iπ ln γ E ( q up to the arbitrariness in the choice of the renormalization prescription i.e. the subtraction momentum scale hq. Now we aim to obtain the resolvent in the presence of contact interaction. To this purpose following [Alb] let us start from the Hamiltonian H = p + V (r (3 m and expand the resolvent [ ] p G(z = m + V (r z = G (z [G (zv ] n G (z. (4 If we now formally insert V (r = λδ ( (r and consider the integral kernel i.e. the Green s function we formally obtain n= G(r r ; z = G (q r r G (q r G (q r λ + G ( ; z hq mz Imz >. In order to make the denominator meaningful we have to implement the renormalization (or subtraction procedure which amounts to the replacement λ + G ( ; z + G R ( ; z = λ R + m ( } q λ R π h iπ ln γ E + constant q in which we have introduced the renormalized and scale dependent coupling parameter λ R in order to keep finite the denominator of eq. (5. A comparison with eq.s (-(6 is mandatory to fix the renormalization prescrption i.e. the arbitrary constant in eq. (6. To this aim let us consider the Bergmann-Manuel-Tarrach (BMT renormalization prescription [BMT] which is defined by + G R ( ; z λ R m ( BMT π h ln h q. (7 me In the RHS of the above expression the energy scale E is nothing but the absolute value of the bound state energy whereas the momentum scale hq is the subtraction point at which the running coupling parameter λ R is defined: namely λ R (q = π h (5 (6 m ln ( h q me. (8 3
It is worthwhile to observe that from the field theoretical point of view the behaviour of the running coupling parameter λ R (q at fixed E turns out to be somewhat peculiar: as a matter of fact the model under investigation appears to be asymptotically free and infrared stable at the same time because lim λ R(q = = lim λ R(q. q q Now taking the relationships (8(6-(8 into account we can rewrite eq. (5 in the form G(r r ; z = im h H( (q r r πm H ( (qrh( (qr ( h ln h q me (9 hq mz Imz >. Let us first check that the singularity in the denominator of the above equation just corresponds to the presence of a bound state. As a matter of fact we have lim (z + E G(r r πm(z + E H ( ; z = lim [q(zr]h( [q(zr ] [ ( ] z E z E h z ln E + = me π h K (κrk (κr = κ π K (κrk (κr hκ = me < E ( in perfect agreement with eq. (6. The continuous part of the spectrum can be read off the imaginary part of the Green s function taking the summation theorem for the Hankel s function into account [Gra] and after setting r = r cos θ r = r cos θ ϕ = θ θ : namely h q (r πm Im G r ; z = h q m = l Z } ( (q/π h + [J ( (qrj (qr ln q ln h q + π me q me ( ln h q me q π eilϕ J l (qrj l (qr ln ( h q me + π [J (qrn (qr + N (qrj (qr ]. ] + π N (qrn (qr ( It can be readily verified that if we set ψ (q r; E ( (q/π ln h q me + π [ ( h q ] ln J (qr πn (qr me = q cos[πµ(q; E ]J (qr + q sin[πµ(q; E ]N (qr ( 4
eq. ( can be recast into the form h q (r πm Im G r ; z = h q m = q π J l (qrj l (qr cos(lϕ + ψ (q r; E ψ (q r ; E. l= (3 After direct inspection one can easily realize that eq. (3 does agree with the spectral decomposition of eq. ( whilst eq. ( does coincide with eq. (4 leading to the identification B(q; E A(q; E = ( π ln ( h q /me = tan[πµ(q; E ]. (4 To sum up we have explicitely verified that the method of boundary conditions or von Neumann method of deficiency indices to obtain all the SAE of the free D kinetic operator is equivalent to the Krein s formula for the resolvent of the hamiltonian operator involving contact interaction. The full correspondence between the two methods is achieved provided some renormalization prescription is adopted which defines the running coupling parameter of contact interaction. References [Alb] S. Albeverio F. Gesztesy R. Hoegh-Krohn and H. Holden Solvable Models in Quantum Mechanics Springer-Verlag New York (988 9-357-358. [BMT] O. Bergmann Phys. Rev. D 46 5474 (99; C. Manuel and R. Tarrach Phys. Lett. B 68 (99. [Gra] I. S. Gradshteyn and I. M. Ryzhik Table of Integrals Series and Products Academic Press San Diego (994 see 8.53. pg 993. 5
. Two dimensional uniform field Here we apply the Krein s formula to the case of a two dimensional point particle in the presence of a uniform (i.e. constant and homogeneous field such as gravity and contact interaction. The first step is to find the resolvent of the Hamiltonian in the absence of contact interaction. The resolvent is defined by where (H zg (r r ; z = δ ( (r r ( H = p m + mgx ( in which we have set r = (x x and p = (p p. Here the absence of contact interaction corresponds to assume regularity of the eigenfunctions on the whole plane that means in turn [H p ] =. It is worthwhile to notice that the presence of a uniform field just allows to introduce natural quantum gravitational wavelength and energy: namely ( h λ g κ = m g E g mg κ = h κ m. /3 (3 It is convenient to introduce dimensionless coordinates and quantum numbers in such a way that the eigenvalues equation for the Hamiltonian ( becomes ( x + y y + ɛ ψ ɛ (x y = (4 where x κx y κx ɛ E E g. (5 After setting we eventually get ψ ɛ (x y = e ipx ψ ɛp (y p p hκ (6 ψ ɛp ( y + p ɛ ψ ɛp = ɛ p R. (7 As gravity is attractive in the half plane y > the only allowed eigenfunctions which are square summable in the half plane y > are provided by ψ ɛp (x y = (π / e ipx Ai ( y + p ɛ x y ɛ p ɛ p R (8 where Ai(z is the Airy s function [Abr]. The above dimensionless improper wave-functions are normalized according to ɛ p ɛ p = δ(ɛ ɛ δ(p p. (9 6
Going back to dimensionfull quantities the eigenfunctions read Ψ Ep (x x x x E p } ( κ i = exp π he g h p x Ai κx + p h κ E E g ( which turn out to be normalized according to E p E p = δ(e E δ(p p. ( It is interesting to obtain the weak field limit g of the eigenfunctions (. To this aim if we set we easily find [Abr] κ i Ψ Ep (x x = exp π he g h p x κ i π he exp g h p x } ( κ i E = π he exp g h p x u E E g p h κ κx ζ 3 u3/ ( } Ai( u } ( u /4 sin ζ + π 4 /4 ( + 4 κx E ( g E +... sin ζ + π 4 E g g u > E E p m. (3 Now taking into account that E g = g /3 ( m h /3 ; ζ 3 ( E E g 3/ ( 3 κx E g E +... g (4 we eventually obtain Ψ Ep (x x ( m } [ /4 ī exp π h E h p π x sin 4 + ( ] 3/ E 3 E g h p x g E > p m + mgx p me. (5 Notice that the asymptotic form (5 of each vawe-functions just corresponds to the sum of two opposite progressive plane waves in the x direction up to a phase factor [Lan] whilst the eigenfunctions exponentially vanish when u < as it does in the case of negative energies. 7
Let us come to the evaluation of the Green s function wich is defined to be G (r r ; z r (H z r + + dw = dp Ψ wp (x x Ψ wp w z (x x = κ + } i dp exp π he g h p (x x (6 + dw Ai (y weg Ai (y weg. w + p m z The above expression can be rewritten in terms of the following integral representation: namely G (r r im ; z = lim ε π h ε + it exp (x x } 4(ε + it exp it mz h i t3 3 iyy + i } (7 t 4t (y + y t Imz >. Notice that the limit has been introduced in order to carefully perform the integration over the degeneracy variable p in eq. (6 and to treat correctly the case of coincident points. As a matter of fact it is not difficult to realize that the limits r r and ε do not commute. After some straightforward algebra we obtain G (r r ; z = m [ ] mz π h t exp it h κ3 (x + x i t3 κ 6 + i } 4t (r r Imz > r r. (8 Now some comments are in order. First we remark that one should expect that translation invariance holds true as we are in the presence of a uniform field and in the absence of contact interaction. Translation invariance becomes manifest if we turn to the new complex energy variable ξ z h κ 3 4m (x + x Imξ > (9 in such a way that we can employ the manifestly translation invariant form of the Green s function i.e. G (r r ; ξ = m π h t exp it mξ h i t3 κ 6 + i } 4t (r r ( Imξ > r r. The second remark concerns the limit of vanishing uniform field. It can be readily checked that lim G (r r ; ξ = m g π h t exp it mz h + i } 4t (r r ( = im ( mz h H( r r Imz > h 8
in agreement with eq. (8 of section. Thanks to translation invariance and taking eq. (7 properly into account we can also extract the density of the quantum states per unit volume: namely ρ (D g (E lim η πi G ( ; z = E + iη G ( ; z = E iη} π Im G ( ; z = E m ( me = lim ε h t + ε ε cos h t κ6 = m π ( } h + me t sin h t κ6 t3 t3 ( } me + t sin h t κ6 t3 E R. ( As a check we observe that lim ρ (D g g (E = ϑ(e mπ h = ρ (D (3 where ϑ is the Heaviside s step distribution. Now we are ready to obtain the renormalized Green s function G R (r r ; ξ. To this aim it is convenient to rewrite eq. ( in the form G (r r ; ξ = im h H( + m π h ( mξ h t exp Imξ > r r. r r it mξ h + i } ( } 4t (r r exp it 3 κ6 As a consequence it is clear that the ultraviolet divergence at coincident points is subtracted by the very same term as in the absence of gravity: namely (4 G R (r r ; ξ = G (r r ; ξ + m ( π h ln q r r. (5 according to eq. ( of section where hq is some arbitrary momentum scale. Again the renormalized Green s function turns out to be finite although arbitrary at coincident points as we have G R ( ; z = m ( mz π h iπ ln h q } γ E + I (z g (6 where I(z g ( t eitz exp img h t 3 } 4 9. (7
In order to introduce contact interaction and obtain the corresponding Krein s formula for the Green s function - see eq. (5 of section - we have to define the renormalized coupling of the contact interaction. This can be done in close analogy with eq. (6 of section : namely λ + G ( ; z + G R ( ; z = λ R + m ( } mz λ R π h iπ ln h γ q E + I(z g + constant in which again we have introduced the renormalized and scale dependent coupling parameter λ R in order to keep finite the denominator of the Krein s formula. Keeping as well the BMT renormalization prescription eq. (8 becomes equivalent to the following pair of equations + G R ( ; z λ R m } BMT π h ln ( ze I(z g (9a λ R (q = π h (8 m ln ( h q me E >. (9b In the RHS of the above expression the energy scale E is nothing but the absolute value of the bound state energy in the absence of gravity whereas the momentum scale hq is the subtraction point at which the running coupling parameter λ R is defined. It is apparent from eq. (9a that there are no poles on the real axis of the energy variable z as long as g. As a matter of fact the basic quantity I(z g always contains an imaginary part when Imz = g. On the other hand the bound state arises in the limit g = as I(z g = = - see eq. (7 - according to eq. ( of section. This means that in the presence of gravity and contact interaction the spectrum is purely continuous and coincident with the whole real energy axis. Consequently once gravity is switched on no matter how weak it is the unperturbed bound state due to pure contact interaction becomes a metastable state whose decay wih will be evaluated below. Now we are ready to obtain the Krein s formula in the presence of gravity. From the expression G (r; z = m π h t exp itz it mg x i } 4 mg h t 3 + i mr h t (3 Imz > r we obtain the Krein s formula for the Green s function in the presence of gravity and contact interaction: namely G(r r ; z = G (r r ; z + π h m Imz >. G (r; zg (r ; z ln ( ze I(z g (3
The above exact expression for the Green s function exhibits the loss of translation invariance along the x -direction. This means that the translation operator p does not commute with the Hamiltonian. As a consequence we can no longer use the eigenvalues of p to label the degeneracy of the eigenstates of the self-adjoint Hamiltonians H(E whose Green s functions are provided by eq. (3. References [Abr] M. Abramowitz and I. A. Stegun Handbook of Mathematical Functions Dover New York (97 446-45. [Lan] L. D. Landau and E. M. Lifsits Meccanica quantistica - Teoria non relativistica Editori Riuniti Roma (976 p..