Eigenvalue bounds for polynomial central potentials in d dimensions
aa r X i v : . [ m a t h - ph ] S e p CUQM-122
Eigenvalue bounds for polynomial central potentialsin d dimensions Qutaibeh D. Katatbeh
Department of Mathematics and Statistics, Faculty of Science and Arts, JordanUniversity of Science and Technology, Irbid, Jordan 22110E-mail: [email protected]
Richard L. Hall
Department of Mathematics and Statistics, Concordia University, 1455 deMaisonneuve Boulevard West, Montr´eal, Qu´ebec, Canada H3G 1M8E-mail: [email protected]
Nasser Saad
Department of Mathematics and Statistics, University of Prince Edward Island,550 University Avenue, Charlottetown, PEI, Canada C1A 4P3.E-mail: [email protected]
Abstract.
If a single particle obeys non-relativistic QM in R d and has theHamiltonian H = − ∆ + f ( r ) , where f ( r ) = P ki =1 a i r q i , ≤ q i < q i +1 , a i ≥ E = E ( d ) nℓ ( λ ) are given approximately by the semi-classicalexpression E = min r> n r + P ki =1 a i ( P i r ) q i o . It is proved that this formula yieldsa lower bound if P i = P ( d ) nℓ ( q ), an upper bound if P i = P ( d ) nℓ ( q k ) and a generalapproximation formula if P i = P ( d ) nℓ ( q i ). For the quantum anharmonic oscillator f ( r ) = r + λr m , m = 2 , , . . . in d dimension, for example, E = E ( d ) nℓ ( λ ) isdetermined by the algebraic expression λ = β (cid:0) α ( m − mE − δ (cid:1) m (cid:0) α ( mE − δ ) − E ( m − (cid:1) where δ = p E m − α ( m −
1) and α, β are constants. An improved lowerbound to the lowest eigenvalue in each angular-momentum subspace is alsoprovided. A comparison with the recent results of Bhattacharya et al (Phys.Lett. A, 244 (1998) 9) and Dasgupta et al (J. Phys. A: Math. Theor., 40 (2007)773) is discussed.PACS numbers: 03.65.Ge
Keywords : Polynomial potentials, Envelope method, Kinetic Potentials, Quantumanharmonic oscillators. igenvalue bounds for polynomial central potentials in d dimensions
1. Introduction and main results
The purpose of the present work is to establish a global bound formula for the discretespectrum { E ( d ) nℓ } , n = 1 , , . . . , l = 0 , , , . . . of the d -dimension Schr¨odinger equationwith polynomial potentials given by Hψ = − ∆ + k X i =1 a i r q i ! ψ = Eψ, ≤ q i < q i +1 , (1)where ∆ is the d -dimensional Laplacian operator, r = k r k , r ∈ R d , and the coefficients a i ≥ , are not all zero. The key motivation for our present study lies in the well-knownfact that the majority of quantitative predictions of Schr¨odinger’s equation with apolynomial potential (1) in nuclear, atomic, molecular, and condensed matter physicsmust usually rely on numerical estimates [1]-[5]. Thus, a simple global eigenvalueformula can serve as a basis for exploration and also for checking different approximatemethods in quantum mechanics [6]. Another important motivation for the presentwork is a recent contribution by Dasgupta et al [7] regarding a general simple schemefor evaluating the ground state as well the excited-state energies for λr m quantumanharmonic oscillators in one dimension, see also [6]. We provide in the present worka more general scheme sufficient to generate all energy levels in arbitrary dimension,not only of r m quantum anharmonic oscillators, but also for any polynomial potentialof the form P ki =1 a i r q i a sufficient degree of accuracy to be interesting. The purposeis not merely to obtain accurate energy eigenvalues for different polynomial potentialsfor which a large number of methods exist in the literature. Rather, we proposea simple approach which provides energy bounds as well as an approximate energyformula with a reasonable accuracy and with a minimum amount of effort. Consider,as an example, the celebrated quantum anharmonic oscillator [8]-[23] Hamiltonian − ∆ + r + λr m , m = 2 , , . . . in d dimensions: we show that for any state n = 1 , , . . . ,the eigenenergy E = E ( d ) nℓ ( λ ) is determined approximately by the expression λ = 1 β (cid:18) α ( m − mE − δ (cid:19) m (cid:18) α ( mE − δ ) − E ( m − (cid:19) (2)where δ = p E m − α ( m −
1) and α and β are constants. Further, we show thatupper or lower bounds for the energy eigenvalues (2) for a given state are expressed interms of a single constant for any value of λ . The dependence of α and β on m and d will be discussed in a subsequent section. We obtain our global eigenvalue formula for(1) by using the so-called P -representation [24] for the Schr¨odinger spectra generatedby the pure power-law potential ( q > ǫ is written as the minimum of a function of one variable r and a parameter P : thisinduces a one-one relation between ǫ and P. More specifically, we write( − ∆+ r q ) ψ ( d ) nℓ = ǫ ( d ) nℓ ( q ) ψ ( d ) nℓ ⇒ ǫ ( d ) nℓ ( q ) = min r> (cid:20) r + (cid:16) P ( d ) nℓ ( q ) r (cid:17) q (cid:21) , (3)where P ( d ) nℓ ( q ) = h ǫ ( d ) nℓ ( q ) i (2+ q ) / q (cid:20)
22 + q (cid:21) /q (cid:20) q q (cid:21) / . (4)This may seem at first sight rather inconvenient since the computation of P requiresknowledge of ǫ . An important advantage of (4), however, is that the computation of P is independent of the potential parameters. In other words, the computation of igenvalue bounds for polynomial central potentials in d dimensions P for H = − ∆ + r q is sufficient to yield the discrete spectrum of the Hamiltonian H v = − ∆ + vr q with eigenvalues given by E ( d ) nℓ ( q ) = min r> (cid:20) r + v (cid:16) P ( d ) nℓ ( q ) r (cid:17) q (cid:21) (5)for arbitrary v >
0. This may seem unnecessary for a Hamiltonian of the form H v because a simple scaling argument shows E ( d ) nℓ ( q ; v ) = v / ( q +2) E ( d ) nℓ ( q ); but forpolynomial potentials, as in (1), where a i > , i = 1 , , . . . , k , equations (3)-(4) playan important role in establishing some of the general energy formulae [25] through thedecomposition of the Hamiltonian (1) by means of H = − ∆ + k X i =1 a i r q i = k X i =1 ω i H ( i ) (6)where H ( i ) = − ∆ + a i ω i r q i . (7)and { ω i } ki =1 is an arbitrary set of positive weights with sum equal to 1. Further, itworth mentioning that this dependence can be resolved for certain special values of q ,for example if q = 2, we know that [26] P dnℓ (2) ⇒ P dnℓ (2) = (2 n + l + d −
2) if d ≥ P n (2) = ( n − ) if d = 1 (8)The main results of the present work may be summarized by the following twotheorems: Theorem A:
Eigenvalue bounds for the spectrum { E ( d ) nℓ } of the Hamiltonian (1) aregiven by E ≡ min r> " r + k X i =1 a i ( P i r ) q i . (9) wherei) if P i = P ( d ) nℓ ( q ) , then E ≤ E ( d ) nℓ ii) if P i = P ( d ) nℓ ( q k ) then E ≥ E ( d ) nℓ Here the numbers P ( d ) nℓ ( q i ) are given by (4). Theorem B:
The eigenvalues of the Hamiltonian (1) are given approximately by thesemiclassical formula
E ≡ min r> " r + k X i =1 a i ( P i r ) q i . (10) where, for the lowest eigenvalue in each angular-momentum subspace, and d ≥ , wehavei) For n = 1 and P i = P ( d )1 ℓ ( q i ) , i = 1 . . . k , then E ≤ E ( d )1 ℓ for all d ≥ and l = 0 , , , . . . .ii) For n ≥ and P i = P ( d ) nℓ ( q i ) , i = 1 . . . k , E ≈ E ( d ) nℓ .Further, for the lowest eigenvalue in d ≥ , we haveigenvalue bounds for polynomial central potentials in d dimensions iii) E ≤ E ( d )10 if the numbers P i are replaced by the explicit lower approximations for P ( d )10 ( q i ) given by P i = (cid:18) de (cid:19) (cid:18) dq i e (cid:19) qi " Γ[1 + d )Γ(1 + dq i ) d , e = exp(1) (11) iv) E ≥ E ( d )10 if the numbers P i are replaced by the explicit upper approximations to P ( d )10 ( q i ) given by P i = (cid:18) d (cid:19) " Γ( d + q i )Γ( d ) qi . (12)The difference between the two parts of Theorem B is that, in the first part (i)-(ii), the P -numbers are to be computed from the pure-power energies by use of (4), whereas, inthe second part (iii)-(iv), the P -numbers are given explicitly in terms of the Gammafunction. We use the term “semiclassical” in the following sense: once the componentkinetic potentials have been fixed by the P -numbers, what remains is a minimizationover a real function; in the approximation, this expresses the trade off between thekinetic and potential energies; the final picture is semiclassical since the kinetic energyis reduced to 1 /r and a wave equation is no longer involved.In the next section, we discuss the proof of these two theorems. In section (3), theapplication of these two theorems to the quantum anharmonic oscillator Hamiltonianis presented. The conclusion is given in section 4.
2. Proof of theorems A and B
The proof of theorem A depends on the application of envelope theory and Kineticpotentials technique developed earlier by Hall [27]-[29] and used successfully sincethen. We shall outline here a brief summary of the theory to provide us with sufficientdetails to prove the theorem, and we refer the interested reader to Ref. [27]-[29] formore details. For simplicity, we present this brief summary for the case of d = 3spatial dimensions: for arbitrary d , the extension is straightforward. Consider theSchr¨odinger operators of the form H = − ∆ + vf ( r ) , (13)where f is the shape of a central potential in R , and v > ψ, Hψ ) / ( ψ, ψ ) is performed in two stages. The first stage, with h ψ, − ∆ ψ i = s fixed, involves only the shape of the potential f and leads to a family { f nℓ } of kineticpotentials f nℓ ( s ). Here s is a positive constraint variable: it only becomes the meankinetic energy when the minimization of the sum of the kinetic and potential energieshas been effected. We have E nℓ = min s> { s + vf nℓ ( s ) } (14)in which the critical value of s = h ψ, − ∆ ψ i > igenvalue bounds for polynomial central potentials in d dimensions f for fixed s , are given as a Legendre transformation { s = E ( v ) − vE ′ ( v ) , f ( s ) = E ′ ( v ) } of the function E ( v ), which describes how the eigenvalue depends on thecoupling v . They may also be defined by the following general formula f nℓ ( s ) = inf D nℓ sup ψ ∈D nℓ k ψ k =1 Z ψ ( r ) f ([( ψ, − ∆ ψ ) /s ] / r ) ψ ( r ) d r, (15)where D nℓ is the span of a set of n linearly independent functions. It is interestingto notice that the kinetic potential f nℓ can be replaced by the potential f ( r ) itselfthrough the parameterization of f nℓ ( s ) in term of the variable r (used here as a newparameter to replace s ), that is to say f nℓ ( s ) = f ( r ). We now invert this monotonefunction to give the K functions [29] s = ( f − nℓ ◦ f )( r ) = K ( f ) nℓ ( r ) . (16)It is easy to show that the K functions obey the scaling property Af (cid:16) rb (cid:17) + B → (cid:18) b (cid:19) K (cid:16) rb (cid:17) , (17)and in general they are independent of coupling and potential shifts [29]. Theeigenvalues are recovered from the K functions by the expression E nℓ = F nℓ ( v ) = min r> n K ( f ) nℓ ( r ) + vf ( r ) o (18)For the power-law potentials f ( r ) = r q , it is known by mean of simple scaling argumentthat the spectrum of the pure-power Hamiltonian satisfies − ∆ + vr q → F ( q ) nℓ ( v ) = E ( q ) nℓ (1) v / ( q +2) . (19)In order to compute the kinetic potentials f nℓ ( s ), we notice from the minimizationprocess of (14) that f ′ nℓ ( s ) = − v − and consequently we have s = F ( q ) nℓ ( v ) − vf nℓ ( s ) ⇒ f nℓ ( s ) = ddv F ( q ) nℓ ( v ) (20)which implies using (19) that f nℓ ( s ) = 2 q + 2 v − qq +2 E ( q ) nℓ (1) . (21)On the other hand, we have from the l.h.s. of (20) that v − F ( q ) nℓ ( v ) = f nℓ ( s ) − sf ′ nℓ ( s ) (22)which implies using (21) that f nℓ ( s ) = 2 q qE ( q ) nℓ q + 2 ! ( q +2) / s − q/ . (23)The K functions are then computed by means of (16) and (20)-(23) to yields K ( f ) nℓ ( r ) = (cid:18) q (cid:19) /q qE ( q ) nℓ q + 2 ! ( q +2) /q r = (cid:18) P nℓ ( q ) r (cid:19) (24)where we have defined P nℓ ( q ) = (cid:16) E ( q ) nℓ (cid:17) (2+ q ) / q (cid:20)
22 + q (cid:21) /q (cid:20) q q (cid:21) / (25) igenvalue bounds for polynomial central potentials in d dimensions E nℓ = min r> { (cid:18) P nℓ ( q ) r (cid:19) + vr q } (26)In ordered to obtained a definite bound, Hall [27] used interesting geometricinterpretation in terms of envelopes. If the potential shape f ( r ) = g ( h ( r )) is a smoothtransformation g of a soluble potential h , then the kinetic potentials associated with f ( r ) are given by f ( r ) = g ( h ( r )) → f nℓ ( s ) ≈ g ( h nℓ ( s )) (27)and the corresponding K functions satisfies K ( f ) nℓ = ( g ◦ h nℓ ) − ◦ ( g ◦ h ) = h − nℓ ◦ h = K ( h ) nℓ (28)Therefore f = g ( h ) → K ( f ) ∼ = K ( h ) (29)and the eigenvalue approximations are given by E nℓ ≈ min r> { K ( h ) nℓ ( r ) + vf ( r ) } (30)in which g no longer appears. This expression yields upper or lower bounds depending,respectively, whether g is concave or convex [25]-[26]. For f ( r ) = g ( r q ) = P ki =1 a i r q i :since q i < q i +1 , clearly g is convex if q = q (lower bound) and concave if q = q k (upper bound). We therefore have, by using (30) with h ( r ) = r q , E nℓ = min r> ((cid:18) P nℓ ( q ) r (cid:19) + v k X i =1 a i r q i ) . (31)Or, equivalently, and by a change in the minimization variable, E nℓ = min r> ( r + v k X i =1 a i ( P nℓ ( q ) r ) q i ) . (32)With v = 1 , this equation yields (9) and we obtain a lower bound if q = q and anupper bound if q = q k . This complete the proof of the theorem.
The first part of theorem B was introduced [25] to improve the lower bounds for theground state energy obtained in theorem A. The second part is based on the Barneset al’s [30] general lower-bound formula for the lowest eigenvalue of the Schr¨odingeroperator H = − ∆ + V ( r ) in d ≥ igenvalue bounds for polynomial central potentials in d dimensions
3. Fractional anharmonic oscillator
Before we study specific problems in quantum mechanics, we first consider theapplication of theorem A and Theorem B to the class of arbitrary fractionallyanharmonic oscillator Hamiltonians [32]: H = − ∆ + X δ ∈ Z g δ r δ (33)where Z is an arbitrary finite set of the integer or rational numbers and the coupling g δ , δ ∈ Z are chosen so that the Hamiltonian supports the existence of a discretespectrum. It is known [32]-[35] that this class of Hamiltonian possesses elementarysolutions for certain particular cases of the coupling g δ . For consistency, we assume δ ≥
2, although the conclusion of theorems A and B are perfectly applicable forall δ ≥ −
1, where, for example, the P number in the case δ = − P d ≥ nℓ ( −
1) =( n + l + d/ − / H = − ∆ + Cr α + Dr β , β > α > q = min δ ∈ Z { δ } and Q = max δ ∈ Z { δ } . By using theorem A, we immediately find analytic expressions forlower bounds ǫ dnℓ and upper bounds E dnℓ for the eigenvalues of the Hamiltonian (33):these can be written explicitly as ǫ dnℓ ( δ ) ≈ min r> r + X δ ∈ Zq = min δ ∈ Z { δ } g δ ( P ( q ) r ) δ , (35)and E dnℓ ( δ ) ≈ min r> r + X δ ∈ ZQ =max δ ∈ Z { δ } g δ ( P ( Q ) r ) δ . (36)Here the numbers P ( q ) and P ( Q ) are computed numerically by means of Eq.(3) forrational q, Q = − , − ∆ + r q ) ψ = E q ψ and ( − ∆ + r Q ) ψ = E Q ψ . An interestingimprovement for the eigenvalues ǫ d ℓ ( δ ) and E d ℓ ( δ ) can be obtained through theapplication of theorem B. The cost, however, is that the exact eigenvalues of therational power-law potentials V ( r ) = r δ for each δ ∈ Z must be computed numerically.Less accurate bounds can be obtain directly using the explicit P numbers (11) and(12). An important class [32] of the fractional anharmonic oscillator Hamiltonians(33) that have found many applications [32] in quantum field theory [33] is given by H = − ∆ + V ( r ) = − ∆ + q +1 X j =1 g j r j , g q +1 = a > . (37)This class of Hamiltonians has found many applications not only in quantummechanics (where V ( r ) represents [32] an arbitrary potential in the limit q → ∞ ) igenvalue bounds for polynomial central potentials in d dimensions ǫ dnℓ ( δ ) ≈ min r> r + q +1 X j =1 g j ( P dnℓ (2) r ) j . (38)and E dnℓ ( δ ) ≈ min r> r + q +1 X j =1 g j ( P dnℓ (2 q + 1) r ) j . (39)where P dnℓ (2) is given by Eq.(8) and P dnℓ (2 q + 1) is given by (3), respectively.
4. Quantum anharmonic oscillator
In this section, we consider the Schr¨odinger equation (cid:0) − ω ∆ + ar + br m (cid:1) ψ = E ( ω, a, b ) ψ, m = 2 , , , . . . (40)where ω, a and b are positive parameters and the potential in (33) is a single-wellpotential which describes for m = 2 , , , , . . . the quartic, sextic, octic, and decadicoscillators, and so on. It is easy to check that for the energy in (40) the followingscaling relation holds E ( ω, a, b ) = ( aω ) / E (cid:18) , , bω ( m − / a ( m +1) / (cid:19) ,ψ ( r ; ω, a, b ) = ψ (cid:18)(cid:16) aω (cid:17) r ; 1 , , bω ( m − / a ( m +1) / (cid:19) . (41)Thus the original problem (40) is essentially a single-parameter problem which we nowwrite as H ( m ) ψ = (cid:0) − ∆ + r + λr m (cid:1) ψ = E ( λ ) ψ, m = 2 , , , . . . (42)where E ( λ ) = E (1 , , λ ) and λ = bω ( m − / a ( m +1) / . The Schr¨odinger equation with thequantum anharmonic oscillators (42) are among the most widely studied models inquantum mechanics. In spite of their simplicity, they give rise to interesting problems,both computationally and conceptually [23]. A rigorous analysis of the mathematicalproperties of the anharmonic oscillator Hamiltonians H (2) was made by Simon [8] andby the seminal work of Bender and Wu [9]. The aim in the discussion we presentbelow is to derive simple upper- and lower-bound formulae based on Theorems A andB. For the anharmonic oscillator potentials f ( r ) = r + λr m , m = 2 , , . . . (43)Theorem A implies that E ( λ ) ≈ min r> (cid:20) r + αr + λβr m (cid:21) . (44)where • E ( λ ) ≤ E ( λ ) is an lower bound, if α = ( P nℓ (2)) and β = ( P nℓ (2)) m . • E ( λ ) ≥ E ( λ ) is an upper bound, if α = ( P nℓ (2 m )) and β = ( P nℓ (2 m )) m . igenvalue bounds for polynomial central potentials in d dimensions α = ( P nℓ (2)) and β = ( P nℓ (2 m )) m , then E ( λ ) ≤ E ( λ ) for n = 1, and E ( λ ) ≈ E ( λ ) for all n ≥
2. Let x = r , we note that theminimization of (44) occurs at − x + α + mλβx m − = 0 . Multiplying through by x and solving for λβx m , we can easily show that the minimization of (44) occurs at r = mE − p E m − α ( m − α ( m −
1) (45)and consequently we have λ = 1 β (cid:18) α ( m − mE − δ (cid:19) m (cid:18) α ( mE − δ ) − E ( m − (cid:19) (46)where δ = p E m − α ( m − H ( m ) in (42) one has to solve Eq.(46) for the given λ . It is clear that at λ = 0, equation (46) implies E = 2 √ α , with α = ( P ( q ) nℓ (2)) =(2 n + l + d − , as given by (8). Consequently, E = 4 n + 2 l + d −
4, the result forthe d -dimensional harmonic oscillator [39]. When equation (46) is used to determinethe lower or the upper bounds to the exact eigenvalues of the λr m oscillator, it isclear that the formula is expressed in terms of a single constant for any value of λ .This follows from the fact that, for lower or upper bound, α m = β and equation (46)reduces to λ = 2 m ( m − ( m − ( m + 1) ( − E + p m E − α ( m − mE − p m E − α ( m − m . (47)This is a remarkable simple formula that gives a global lower and upper bound to theexact eigenvalues for a given λ for all n = 1 , , . . . and l = 0 , , , . . . in d dimensions,accordingly as α = (2 n + l + d − and α = ( P ( d ) nℓ (2 m )) , respectively. In particular,a global formula that gives a lower bound for all n = 1 , , . . . , m = 2 , , . . . is λ = 2 m ( m − ( m − ( m + 1) ( − E + q m E − n + l + d − ( m − mE − q m E − n + l + d − ( m − m . (48)Note in the case of d = 1, we should set either l = − l = 0 to obtain a lowerbound to even or odd (exact) eigenvalues, respectively. Despite the generality of(47), we should like to make two immediate remarks concerning the application oftheorem A: (i) Formula (47), in general, gives a loose bound; (ii) the upper bound α =( P ( d ) nℓ (2 m )) , m = 2 , , . . . which is obtained by means of Eq.(4), requires knowledgeof the exact eigenvalues of Schr¨odinger equation ( − ∆ d + r m ) ψ = ǫ ( d ) nℓ (2 m ) ψ . Inthis paper we have found the values of P ( d ) nℓ (2 m ) by the numerical integration ofthe Schr¨odinger equation just mentioned, and then we used Eq.(4) to find thecorresponding P -numbers. For immediate use of equations (47) and (48), we reportin Table 1 the values of P (1)10 (2 m ) and β = ( P (1) nℓ (2 m )) m for different values of m .Table 1: Values of P (1)10 (2 m ) and β = ( P (1) nℓ (2 m )) m for different m . igenvalue bounds for polynomial central potentials in d dimensions m P (1)10 (2 m ) β − d dr + r + 0 . r . Eq.(48)gives a lower bound 1.00248 and Eq.(47) with α = ( P (1)10 (2 m )) gives an upper boundof 1.32038. The exact eigenvalue in this case reads 1.00737. In order to improve thesebounds, we can make use of Theorem B. For the ground state eigenvalues in d ≥ P -numbers, thanks to the explicit approximate values of P ( d )10 (2 m ) given by (11) and(12). In either case, equation (46) gives a simple general formula for the energy boundof E = E ( λ ) with reasonable accuracy12 m ( m − ( m − ( m + 1) ( − E + p m ( E − d ) + d )( mE − p m ( E − d ) + d ) m = βλ, (49)for m = 2 , , . . . where now if β = ( P ( d )10 (2 m )) m as given by Table (1), Eq.(48) gives alower bound for given λ . The results of this formula is illustrated in the last columnsof Tables 2 and 3. On other hand, if β is a fixed number given by (11) and (12), thenEq.(49) gives lower and upper bound, respectively. Note that Theorem B, still allowsus to conclude that Eq.(49) yields a reasonable approximation to the excited-stateenergies for n ≥
2. However, in this case β = ( P ( d ) n (2 m )) m is strictly given by (4).In the case of d = 1, Eq.(48) reads ( m = 2 , , . . . )12 m ( m − ( m − ( m + 1) ( − E + p m ( E −
1) + 1)( mE − p m ( E −
1) + 1) m = βλ, (50)where β = ( P (1)10 (2 m )) m is given by (11) and (12) for a lower and an upper boundrespectively. Equation (50) can be compared with the recent formula introduced byBhattacharya et al [6] for the approximate ground state energy of the Hamiltonian − ∆ + r + λr m in one dimension, namely( E ( m ) ) ( m +1) − ( E ( m ) ) ( m − / ( m +2+ λ )) = ( K ( m )0 ) ( m +1) λ (51)where K (2)0 = 1 . K (3)0 = 1 . K (4)0 = 1 . E L obtained using (49), where β -valueswere given by table 1, with the approximate eigenvalues of the ground state energy E b computed by Bhattacharya et al [6] using formula (50). Recently, Dasgupta et al [7]have extend the work of Bhattacharya et al [6] to evaluate the excited state energies,still in the one-dimensional case. As in the case of the ground state (50), they foundthat the excited-state energies for the λr m oscillator defined by the one-dimensional igenvalue bounds for polynomial central potentials in d dimensions H = − d /dr + r + λr m are also a polynomial equation of thesame degree and are given by (cid:18) E ( m,n ) n + 1 (cid:19) ( m +1) − (cid:18) E ( m,n ) n + 1 (cid:19) ( m − = ( K ( m,n )0 ) ( m +1) λ (52)where E ( m,n ) is the n th excited state energy of the λr m oscillator and K ( m,n )0 areconstants [7]. Our formulas (46) and (47) are more general and seems to yield moreaccurate results, even for large values of the coupling parameter λ .
5. Conclusion
The application of envelope theory and kinetic-potential techniques to polynomialpotentials has yielded fairly general and good energy bounds for arbitrary values ofthe coupling constants. As specific examples, the application of theorems A and B tothe quantum anharmonic oscillators has produced a global energy formula sufficient togenerate all energy levels in arbitrary dimension for r m anharmonic oscillators with afair degree of accuracy. The main emphasis of this paper has been on energy formulasthat are also bounds. However the energy formula of Theorem B (ii), namely E ≈ min r> " r + k X i =1 a i (cid:16) P ( d ) nℓ ( q i ) r (cid:17) q i , which does indeed yield a lower bound for the bottom of spectrum ( n = 1) in eachangular-momentum subspace, is a remarkably general and accurate approximation: itrequires the input of the pure-power P numbers, and it then predicts approximately,for all the eigenvalues in all dimensions, how the spectrum generated by the potentialsum depends on the mixing parameters { a i } ; it also has the attractive collocationproperty that it is exact whenever all but one of the potential coefficients are zero. Acknowledgments
Partial financial support of this work under Grant Nos. GP3438 and GP249507from the Natural Sciences and Engineering Research Council of Canada is gratefullyacknowledged by two of us ([RLH] and [NS]). igenvalue bounds for polynomial central potentials in d dimensions Calculated values of upper and lower bounds, using (43), to the ground stateenergy of the quartic anharmonic oscillator, along with exact values, for differentvalues of λ . The comparison between the lower bound E L given by Eq.(43) usingthe exact values of β by means of table (1), and the approximate eigenvalues E b ofBhattacharya et al [6] using (44) are also shown. λ Exact value Lower bound Upper bound E b E L using Eq.(43) using Eq.(43) Ref. [6] using Eq.(43)0.001 1.00075 1.00062 1.00075 1.00079 1.000710.01 1.00737 1.00614 1.00739 1.00783 1.006970.1 1.06529 1.05585 1.06620 1.07005 1.062750.2 1.11829 1.10288 1.12062 1.12702 1.114731.0 1.39235 1.35510 1.40332 1.41155 1.387544.0 1.90314 1.83699 1.92881 1.91489 1.8989510.0 2.44917 2.35648 2.48862 2.45005 2.4457550.0 4.00399 3.841639 4.078522 3.99621 4.00182100.0 4.99942 4.79395 5.09516 4.99161 4.997661000.0 10.63979 10.19449 10.85151 10.63521 10.638962000.0 13.38844 12.82706 13.65591 13.38474 13.38778Table 3: Calculated values of upper and lower bound to the ground state energiesof the sextic anharmonic oscillator along with exact values for different values of λ .The comparison between the lower bound E L given by Eq.(43) and the approximateeigenvalues E b of Bhattacharya et al [6] using (44) are also shown. λ Exact value Lower bound Upper bound E b E L using Eq.(43) using Eq.(43) Ref. [6] using Eq.(43)0.001 1.00185 1.000932 1.001859 1.00143 1.001440.01 1.01674 1.008994 1.017387 1.01374 1.013660.1 1.10908 1.070681 1.119935 1.10565 1.099200.2 1.17389 1.119782 1.192805 1.17513 1.162611.0 1.43653 1.334560 1.484050 1.44870 1.424004.0 1.83044 1.675050 1.916177 1.83193 1.8205810.0 2.20572 2.004582 2.322916 2.19235 2.1973450.0 3.15902 2.850163 3.348809 3.13471 3.15304100.0 3.71698 3.347427 3.946987 3.69348 3.711871000.0 6.49235 5.828630 6.914382 6.47694 6.489412000.0 7.70174 6.911387 8.205757 7.68861 7.69925 igenvalue bounds for polynomial central potentials in d dimensions References [1] M. Znojil, D. Yanovich and V. P. Gerdt, J. Phys. A: Math. Gen. 36 (2003) 6531.[2] A. S. de Castro and A. de Souza Dutra, Phys. Lett. A 269 (2000) 281.[3] A de Souza Dutra, A. S. de Castro, E. A. da Silva and L. C. O. Castilho, J. Phys. A: Math.Gen. 36 (2003) 1711.[4] A. De Freitas, P. Mart´ın, E. Castro, and J.L. Paz, Phys. Lett. A 362 (2007) 371.[5] G. Campoy and A. Palma, Int. J. Quant. Chem. 30 (2004) 33.[6] R. Bhattacharya, D. Roy and S. Bhowmick, Phys. Lett. A, 244 (1998) 9.[7] A. Dasgupta, D. Roy and R. Bhattacharyaa, J. Phys. A: Math. Theor., 40 (2007) 773.[8] B. Simon. Ann. Phys. NY 58 (1970) 76.[9] C. M. Bender and T. T. Wu. Phys. Rev. 184 (1969) 1231.C. M. Bender and T. T. Wu., Phys. Rev. Lett. 27 (1971) 461.C. M. Bender and T. T. Wu. Phys. Rev. D 7 (1973) 1620.[10] L.C. Kwek, Y. Liu, C. H. Oh, and X. B. Wang, Phys. Rev. A, 62 (2000) 052107.[11] Jing-Ling Chen, L. C. Kwek, C. H. Oh and Yong Liu, J. Phys. A: Math. Gen., 34 (2001) 8889.[12] J. Killingbeck, J. Phys. A: Math. Gen., 13 (1980) 49.[13] S. N. Biswas, K. Datta, R. P. Saxena, P. K. Srivastava, and V. S. Varma, J. Math. Phys., 14(1973) 1190.[14] K. Banerjee, Proc. R. Soc. A, 364 (1978) 265.[15] L. Sk´ala, J. ˇCiˇzek and J. Zamastil, J. Phys. A: Math. Gen.32 (1999) 5715[16] Marco N´u˜nez, Physical Review E, 68 (2003) 016703.[17] R. L. Hall, Phys. Rev. D, 23 (1981) 1421, formula (7.8).[18] V. Singh, N. S. Biswas and K. Datta, Phys. Rev. D 18 (1978) 1901.[19] T. Barakat, Int. J. Mod. Phys. A, 22 (2007) 203.[20] T. Barakat, Phys. Lett. A 344 (2005) 411.[21] F. J. G´omez and J. Sesma, J. Phys. A: Math. Gen. 38 (2005) 3193.[22] F. T. Hioe and E.W. Montroll. J. Math. Phys. 16 (1975) 1945.F. T. Hioe, Don MacMillen, and E. W. Montroll J. Math. Phys. 17 (1976) 1320.[23] E. J. Weniger. Ann. Phys. 246 (1996) 133.E. J. Weniger, J. ˇCiˇzek and F. Vinette, Phys. Lett. A 156 (1991) 169.[24] R. L. Hall, Phys. Rev. A, 39 (1989) 5500.[25] R. L. Hall, J. Math. Phys, 33 (1992) 1710.[26] R. L. Hall and Q. D. Katatbeh, J. Phys. A: Math. Gen., 35 (2002) 8727.[27] R. L. Hall, J. Math. Phys, 24 (1983) 324.[28] R. L. Hall, J. Math. Phys, 25 (1984) 2708.[29] R. L. Hall, J. Math. Phys, 34 (1993) 2779.[30] J. F. Barnes, H. J. Brascamp, and E. H. Leib,
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