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New exact approach to singular potential problems: Tree graph method and genetics series solutions
Volume 104, number 8
PHYSICS LETTERS
17 September 1984
NEW EXACT APPROACH TO SINGULAR POTENTIAL PROBLEMS: TREE GRAPH METHOD AND GENETICS SERIES SOLUTIONS Ming.de DONG Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Bei]ing, People's Republic of China Received 26 April 1984 Revised manuscript received 9 June 1984
A new theoretical framework is formulated to overcome the fundamental difficulty of singular potential problems: the impossibility of obtaining exact irregular wavefunctions by classical methods. A unifying procedure for studying various kinds of singularity in the Schr~dinger, the Dirac, the Bethe-Salpeter equation, etc. is established. The intrinsic structure of irregular wavefunctions is discovered by the tree graph method and represented by an essentially new type of series (genetics functions), distinguished by their tree structure with inexhaustible proliferation - a natural generalization of recursion series for regular wavefunctions. Some remarkable features of the irregular wavefunctions both in mathematical and physical aspects are discussed.
1. Motivation for an exact approach. Despite extensive investigations concerning singular potential problems (SPPs), a field of paramount importance in quantum mechanics and quantum field theory, exact solutions have been obtained exclusively for a single case: the Schr/bdinger equation with V(r) = r -4 (k 4= 0, l :/: 0) [1-3]. This corresponds to the interaction of a charge with an induced dipole and then the Schr6dinger equation is reducible to a well.studied form of the Mathieu equation. Since SPPs have resisted all endeavours to yield exact analytic solutions by classical methods, up to this time one has to be satisfied by studying them by various approximate treatments, such as variation procedure, perturbation expansion, WKB technique, numerical analysis, etc. The primary purpose of this remark is to propose a new method to discover the intrinsic structure of the irregular wavefunetions for certain classes of SPPs. Moreover, due to the general character of the method invented a complete classification of the exactly solvable classes of SPPs within the present framework is given, involving the SchrOdinger equation, the Dirac equation, the Bethe-Salpeter equation, etc., together with the potential functions possessing all known types of non.fuchsian singularities. It is well recognized that the fundamental difficulty of SPPs is due to the existence of non-fuchsian singularities of the governing equation. As a result, by applying the classical method of formal solutions, one will be led inevitably to an infinite system of equations for the undetermined coefficients, namely to the infinite determinant established by Hill. Then, the transcendental dependence of the required solutions upon the parameters of the given equation can be studied only by numerical analysis. In other words, within the framework of the classical theory it is absolutely impossible to obtain exact explicit solutions for equations possessing non-fuchsian singularities. In a more general mathematical formulation this is known as the problem of irregular integrals, which has remained unsolved since the time of Poincar6 (1880). To overcome the insurmountable difficulty stemming from the presence of the non-fuchsian singularities, it is necessary to invent a new method beyond the classical treatment by deriving intrinsic solutions instead of assum. ing formal ones. For this purpose one must first derive the required solutions in a natural way by the residue theorem of function theory. The key point, however, is to spell out the intrinsic structure of the residue series solutions obtained by using the tree graph method. Consequently, it can be shown that irregular integrals (or wavefunctions) are analytic functions of a new type,
0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
405
Volume 104, number 8
PHYSICS LETTERS
17 September 1984
called genetics functions, possessing intrinsically a tree structure. The irregular integrals are the natural extension of the conventional regular ones in the sense that the recursion relation in the latter is generalized to the tree structure in the former. 2. Canonical equation ofSPPs. A reasonable classification is always indispensible for the establishment of a new theory to exhibit the exact solutions, it is also essential for a unifying treatment to signify the scope of application for a new method in relation to the SPPs concerned. It seems to be a highly difficult task to give a complete classification of SPPs, since the wave equations may be quite different (Schr6dinger, Dirac, Bethe-Salpeter equation, etc.), whereas the potential functions are known to be too numerous and diverse (for example, Lennard-Jones potential, Buckingham potential, super-Yukawa potential, etc.). Consider the wave equation in the general form of spherical symmetry: 22(r, r -1 ; rd/dr) ¢(r) = 0 .
(1)
We represent the singular potential by generalized Laurent series with an infinite number of positive and negative powers. V(r) = ~
(2)
Onrc°
n
([22] is the order of the operator 22, n = 0, -+1, -+2, ...). Then the complete classification of the singularities is (1) irregular pole: co(n) -- n, (n = -N1, ... 0 ..... N2), (2) essential singularity: co(n) = n, (N1, N 2 ~ ~), (3) branch point: co(n) = Pn/qn, (rational number), (4) transcendental singularity: co(n) = lrn (transcendental number), (5) logarithmic singularity (using a suitable-transformation for the variable), (6) singular line. The existence of a singular line instead of isolated singularities implies liran ~±** F(n +l)/co(n) = 0 (Hadamard condition). A unifying procedure for exactly solvable classes of SPPs by means of the tree graph method may be formulated by casting the governing equation into the following canonical form 0 = 22 tp - (L(p d/dp) - ~ pW(n)Mn(.p d/dp)] ~ , \ n /
L = I-I (p d/dp - ~o), a
Mn = I-l. an(p d[dp - "Yn,/), l
(3)
p is the dimensionless r; (a, ~3,~/) are the equation's parameters (k, l, on .... ); the order of operator [L] t> [M], o = 1,2, ..., [L],]= 1,2 ..... [Mn]. The most interesting case is co(n) = n integer, then 22 =--L - Y'n rnMn • A further simplification is obtained if M n = an, then 22 =- L - Z h an r n. From this follows Hill's equation with periodic potential by imposing a n = a - n , [L] = 2 [41. It is not difficult to consider multi-component wavefunctions by substituting the matrix equation. In the case of a nonlocal potential, such as the correlation effect, M n may be extended to an integro-differential operator. 3. Irregular wavefunctions. By the method of Green functions one obtains from the canonical equation a system of integro-differential equations Co = e¢) ¢o(~) - f G(~ - ~') ~ exp[w(n)~'] Mn(d/d~' ) ea(~") d~'= ~b0(~'),
(4)
+n
where Ca and ¢0 are respectively the basic sets of solutions of the canonical equation and its degenerate form L ¢ ° = 0. For the difference kernel and the variable coefficients in the exponential functions, the integral transform (s ~ ~ (=In p)) will give exact solutions automatically through a residue calculation:
406
Volume 104, number 8
PHYSICS LETTERS
(
@o(p) = ~ res pS 1 - L - l ( s ) ~ exp[-co(n)as] Mn(S )
17 September 1984
)-1
(s - fla) -1 .
(5)
+-n
From the Faltung theorem follows the representation theorem: there exists a map D a ( p ; ~ , ~, 7) from @0 to Sa: (6)
$o(P) = Do(o)~b°(P) •
Thus, the main task consists in obtaining an exact expression for the mapping function Da(.p ). It is observed that there are two kinds of Do(p) represented by irregular integrals find regular integrals, depending on whether the powers are contractable or not. In the residue expression this implies the contraction of creation and annihilation operators 2~± exp [-co(n)8 s ], a common characteristic shared by all irregular integrals. Otherwise, one has to do only with regular integrals with a recursion relation, even though the equations may possess non-fuchsian singularities. The most intriguing thing is to derive the irregular integrals, from which follows immediately their degenerate case, i.e. regular integrals. The repeated action of this operator produces, aside from normal terms, also contraction terms of all possible modes, which in a residue calculation lead to pole-sequences of arbitrarily high order. Since the contraction modes of consecutive terms cannot be represented by the method of direct induction, the resulting solution is a collection of apparently unrelated terms, radically differing from the recursion series hitherto used in analysis. Thus, the crucial point is to invent a method, known as tree graph method, to treat the above residue series in a sufficiently natural and general way as to spell out the intrinsic structure of the solutions obtained. To accomplish this one must employ the genetics analogy for the tree graph to sum the solutions, only then one gets a new representation called tree series or genetic functions, provided the region of convergence is established. Extending the analysis in ref. [4], the mapping function can be derived in explicit form: Da(.p) = p~O ~
dompW(m),
(m = 0, +1, +2 .... , a = 1, 2 ..... [L]),
(7)
m
5o =
2
"(~')~o ,
(~)~ o = (~')5(s)ls=O o ,
^k,-~2 ~r (X)dom = O(unl ~n2 "'" ~nr)'
dom = ~ (X)dom , 2
'(~)dom = (~)dm(S)ls=oo ,
)~ = )~1 + ~2 + "" + ~r
The successive sequence is characterized by simultaneous approximation both in the irregular exponents and in the expansion coefficients. This is connected intimately with the tree structure of the parametric series 5o(a, [], 7) and dora(co , ~, 7), as the limit of corresponding sequences: n ( s - ¢%)M_n(S (2)6(s) = ~_l M -~($'~L-~_--'~n ) ) ,
(3)(~(s)= * ~ ~ M n ' ( s - ¢°nl)Mn2(S - con` - Wn2)M-nx-n~(S) n, n2 L ' ( s ) L ( s - ¢On,)L(s- Wnl- con2) '
(4)(~ ~ _ j Mnx(s-- wn')Mn2(s- c°n' - con2)( M-n,(S + C°n,)M-n2(S) =
M n~(S - ~ n , ) M n l ( S ) +
~ +z_,
L ( s - COn, - c o n ~ ) L ( s
- COn, )
0 s"*l~o
ns
Mns(S - COn, - ¢On2 - cona)M_n,_n2_ns(S) \ . . . . . 1
- con1 - ¢°,,2 - ¢ ° n s ) I '
"=
d,n(S) = gm(O + 23 ~n,(OHm_n~(S + con,) + 2 3 2 3 ~n~(s)~,,,(~ ÷ con,)~m-,u-nJS + con, ÷ CO,Q ÷ .... n,
Hm(S) = Mm(s)/L(s + COrn).
(7a) 407
Volume 104, number 8
PHYSICS LETTERS
17 September 1984
As is well known, the P-symbol was indispensible for Riemann in developing his theory of hypergeometric functions, but it loses completely its significance for equations having non-fuchsian singularities. To study the structure and transformation property of irregular integrals (wave/unctions), it is convenient to introduce a new symbol, or function: I
/9
C'/) 6o(+n), a+n
/31,/32 .... /3[L]
/
~'+n,1..... T+n,[Mn] ~ ,
(8)
/
(m(-n), a_ n T-n,1 ..... T-n,[Mnl ] (+n: 1, 2 ..... N l ; - n : -1, - 2 ..... -N2) , where the parameter sets (co(n), % ) (/3a) and (Tn,/) are situated respectively in the third, first and fourth quadrant. As for their mathematical and physical meaning, the a denote coupling constants,/3 Green poles or regular exponents, T cut-off parameters. By the representation theorem the basic set of independent solutions (ffa~)} can be expressed by ~ka(p) = Da(p ) ~bO~) =
d =_
~/9#a
-w(m)+6a a,mP ]
/9 =c/)a co(n), a n
/
I
/31..... 0IL] Tn,1 ..... Tn,[Mnl (±n)
}
(9)
In this sense the ~ a function is not only a symbol but represents a de/mite class of analytic functions, all parameters being written down according to the tree structure, given by (7a). From the ~ symbol by the simple substitution p' = 1//9 one readily gets the solutions at infinity. It is not difficult to prove that ~ functions are invariant under a bilinear transformation. Thus, the most significant property of genetics functions is that they are higher automorphic functions (Poincar6).
4. Some remarkable features of SPPs. (1) Problem o/secular terms. Comparison with perturbative solutions. By direct verification, ~ functions satisfy the original wave equation identically. A most remarkable fact is that the contraction terms are eventually equal to the secular terms, being killed unnaturally in perturbation theory. It has been demonstrated that the secular terms are not only mathematically summable but also physically meaningful [4]. This is the crucial point to settle conclusively Poincar6's problem of irregular integrals, or wave/unctions in SPPs. (2) Analyticity proof. For SPPs with potentials represented by Laurent series, the genetics series solutions are uniformly convergent within the given annular region, def'med by the coefficients of the equation. The parametric series of irregular exponents (/3a + 50) and expansion coefficient dam are both analytic in (a) (T) and rational in
(/3). The tree series solutions thus obtained satisfy the original wave equation generation after generation, instead of a sequence of perturbation equations as in perturbation technique. (3) Classification ofwavefunetions. Irregular wave/unctions can be classified according to the distribution of Green poles (/3o). For the Schr6dinger and the Dirac equation there exist 3 types of wave/unctions in the normal case: elliptic:/31,2 real, waves with radius-dependent amplitude; parabolic:/31 =/32 real and equal, ~2 is unphysieal due to a logarithmic singularity; hyperbolic:/31 = ~2 complex conjugate, waves oscillating rapidly near r = 0. For the B-S equation of fourth order there are four Green poles and one gets a combination of these three types of functions. (4) Boundary conditions. From the exact solutions follow naturally the boundary conditions of physical waves at the origin and at infinity, depending on the character of the potential, repulsive or attractive. (5) Monodromy theorem. For potentials of Laurent type the basic set of solutions possesses the monodromy group, the indices of which are irregular exponents, represented explicitly by parametric series. This implies also 408
Volume 104, number 8
PHYSICS LETTERS
17 September 1984
that Hill's int'mite determinants can be solved explicitly. Thus, the irregular indices (Floquet exponents) can be defined by an explicit analytic expression with tree structure, the dynamical behavior of which is higher than almost periodic motion (H. Bohr).
5. Discussion. A general mathematical framework is established for solving exactly certain classes of singular potential problems. Within this framework a complete classification of singularities is given. The irregular integrals are represented by genetics functions with tree structure, which extend in a natural way the recursion series of regular integrals. The introduction of the symbolic c~a-function plays a central role in the study of SPPs concerning the classification, the intrinsic structure and the group properties of the wavefunctions. The exact analysis permits one also to gain a deeper insight into the correspondence between singular potential theory and nonrenormalizable field theories: an infinite number of counter-terms required in the latter is connected to the inexhaustible proliferation of the tree structure inherent to the former. For the singular interactions in quantum field theories it is evident that there exists also a mapping relation ~ba = D a ~0, but the analytic representation of the mapping function requires a more sophisticated calculation. In conclusion, the above serves only as a brief survey of some main results concerning exactly solvable classes of SPPs within the framework of the tree graph method. All mathematical details as well as physical application will be published elsewhere [5]. It is a pleasure to thank Professor M.R.C. McDowell for his helpful interest and valuable suggestions.
References [1] W.M. Frank, D.J. Land and R.M. Specter, Rev. Mod. Phys. 41 (1971) 36. [2] H.H. Aly, W. Guttinger and H.J.W. Mt~iler,in: Lectures on particles and fields, ed. H.H. Aly (Gordon and Breach, New York, 1970). [3] R.G. Newton, Scattering theory of waves and particles (McGraw-Hill,New York, 1966). [4] M. Dong, Phys. Lett. 97A (1983) 275; 98A (1983) 156; M. Dong and J.-H. Chu, Phys. Lett. 101A (1984) 473. [5 ] M. Dong, Exactly solvable classes of singular potential problems (I) ... (VI), to be published.
409
PHYSICS LETTERS
17 September 1984
NEW EXACT APPROACH TO SINGULAR POTENTIAL PROBLEMS: TREE GRAPH METHOD AND GENETICS SERIES SOLUTIONS Ming.de DONG Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Bei]ing, People's Republic of China Received 26 April 1984 Revised manuscript received 9 June 1984
A new theoretical framework is formulated to overcome the fundamental difficulty of singular potential problems: the impossibility of obtaining exact irregular wavefunctions by classical methods. A unifying procedure for studying various kinds of singularity in the Schr~dinger, the Dirac, the Bethe-Salpeter equation, etc. is established. The intrinsic structure of irregular wavefunctions is discovered by the tree graph method and represented by an essentially new type of series (genetics functions), distinguished by their tree structure with inexhaustible proliferation - a natural generalization of recursion series for regular wavefunctions. Some remarkable features of the irregular wavefunctions both in mathematical and physical aspects are discussed.
1. Motivation for an exact approach. Despite extensive investigations concerning singular potential problems (SPPs), a field of paramount importance in quantum mechanics and quantum field theory, exact solutions have been obtained exclusively for a single case: the Schr/bdinger equation with V(r) = r -4 (k 4= 0, l :/: 0) [1-3]. This corresponds to the interaction of a charge with an induced dipole and then the Schr6dinger equation is reducible to a well.studied form of the Mathieu equation. Since SPPs have resisted all endeavours to yield exact analytic solutions by classical methods, up to this time one has to be satisfied by studying them by various approximate treatments, such as variation procedure, perturbation expansion, WKB technique, numerical analysis, etc. The primary purpose of this remark is to propose a new method to discover the intrinsic structure of the irregular wavefunetions for certain classes of SPPs. Moreover, due to the general character of the method invented a complete classification of the exactly solvable classes of SPPs within the present framework is given, involving the SchrOdinger equation, the Dirac equation, the Bethe-Salpeter equation, etc., together with the potential functions possessing all known types of non.fuchsian singularities. It is well recognized that the fundamental difficulty of SPPs is due to the existence of non-fuchsian singularities of the governing equation. As a result, by applying the classical method of formal solutions, one will be led inevitably to an infinite system of equations for the undetermined coefficients, namely to the infinite determinant established by Hill. Then, the transcendental dependence of the required solutions upon the parameters of the given equation can be studied only by numerical analysis. In other words, within the framework of the classical theory it is absolutely impossible to obtain exact explicit solutions for equations possessing non-fuchsian singularities. In a more general mathematical formulation this is known as the problem of irregular integrals, which has remained unsolved since the time of Poincar6 (1880). To overcome the insurmountable difficulty stemming from the presence of the non-fuchsian singularities, it is necessary to invent a new method beyond the classical treatment by deriving intrinsic solutions instead of assum. ing formal ones. For this purpose one must first derive the required solutions in a natural way by the residue theorem of function theory. The key point, however, is to spell out the intrinsic structure of the residue series solutions obtained by using the tree graph method. Consequently, it can be shown that irregular integrals (or wavefunctions) are analytic functions of a new type,
0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
405
Volume 104, number 8
PHYSICS LETTERS
17 September 1984
called genetics functions, possessing intrinsically a tree structure. The irregular integrals are the natural extension of the conventional regular ones in the sense that the recursion relation in the latter is generalized to the tree structure in the former. 2. Canonical equation ofSPPs. A reasonable classification is always indispensible for the establishment of a new theory to exhibit the exact solutions, it is also essential for a unifying treatment to signify the scope of application for a new method in relation to the SPPs concerned. It seems to be a highly difficult task to give a complete classification of SPPs, since the wave equations may be quite different (Schr6dinger, Dirac, Bethe-Salpeter equation, etc.), whereas the potential functions are known to be too numerous and diverse (for example, Lennard-Jones potential, Buckingham potential, super-Yukawa potential, etc.). Consider the wave equation in the general form of spherical symmetry: 22(r, r -1 ; rd/dr) ¢(r) = 0 .
(1)
We represent the singular potential by generalized Laurent series with an infinite number of positive and negative powers. V(r) = ~
(2)
Onrc°
n
([22] is the order of the operator 22, n = 0, -+1, -+2, ...). Then the complete classification of the singularities is (1) irregular pole: co(n) -- n, (n = -N1, ... 0 ..... N2), (2) essential singularity: co(n) = n, (N1, N 2 ~ ~), (3) branch point: co(n) = Pn/qn, (rational number), (4) transcendental singularity: co(n) = lrn (transcendental number), (5) logarithmic singularity (using a suitable-transformation for the variable), (6) singular line. The existence of a singular line instead of isolated singularities implies liran ~±** F(n +l)/co(n) = 0 (Hadamard condition). A unifying procedure for exactly solvable classes of SPPs by means of the tree graph method may be formulated by casting the governing equation into the following canonical form 0 = 22 tp - (L(p d/dp) - ~ pW(n)Mn(.p d/dp)] ~ , \ n /
L = I-I (p d/dp - ~o), a
Mn = I-l. an(p d[dp - "Yn,/), l
(3)
p is the dimensionless r; (a, ~3,~/) are the equation's parameters (k, l, on .... ); the order of operator [L] t> [M], o = 1,2, ..., [L],]= 1,2 ..... [Mn]. The most interesting case is co(n) = n integer, then 22 =--L - Y'n rnMn • A further simplification is obtained if M n = an, then 22 =- L - Z h an r n. From this follows Hill's equation with periodic potential by imposing a n = a - n , [L] = 2 [41. It is not difficult to consider multi-component wavefunctions by substituting the matrix equation. In the case of a nonlocal potential, such as the correlation effect, M n may be extended to an integro-differential operator. 3. Irregular wavefunctions. By the method of Green functions one obtains from the canonical equation a system of integro-differential equations Co = e¢) ¢o(~) - f G(~ - ~') ~ exp[w(n)~'] Mn(d/d~' ) ea(~") d~'= ~b0(~'),
(4)
+n
where Ca and ¢0 are respectively the basic sets of solutions of the canonical equation and its degenerate form L ¢ ° = 0. For the difference kernel and the variable coefficients in the exponential functions, the integral transform (s ~ ~ (=In p)) will give exact solutions automatically through a residue calculation:
406
Volume 104, number 8
PHYSICS LETTERS
(
@o(p) = ~ res pS 1 - L - l ( s ) ~ exp[-co(n)as] Mn(S )
17 September 1984
)-1
(s - fla) -1 .
(5)
+-n
From the Faltung theorem follows the representation theorem: there exists a map D a ( p ; ~ , ~, 7) from @0 to Sa: (6)
$o(P) = Do(o)~b°(P) •
Thus, the main task consists in obtaining an exact expression for the mapping function Da(.p ). It is observed that there are two kinds of Do(p) represented by irregular integrals find regular integrals, depending on whether the powers are contractable or not. In the residue expression this implies the contraction of creation and annihilation operators 2~± exp [-co(n)8 s ], a common characteristic shared by all irregular integrals. Otherwise, one has to do only with regular integrals with a recursion relation, even though the equations may possess non-fuchsian singularities. The most intriguing thing is to derive the irregular integrals, from which follows immediately their degenerate case, i.e. regular integrals. The repeated action of this operator produces, aside from normal terms, also contraction terms of all possible modes, which in a residue calculation lead to pole-sequences of arbitrarily high order. Since the contraction modes of consecutive terms cannot be represented by the method of direct induction, the resulting solution is a collection of apparently unrelated terms, radically differing from the recursion series hitherto used in analysis. Thus, the crucial point is to invent a method, known as tree graph method, to treat the above residue series in a sufficiently natural and general way as to spell out the intrinsic structure of the solutions obtained. To accomplish this one must employ the genetics analogy for the tree graph to sum the solutions, only then one gets a new representation called tree series or genetic functions, provided the region of convergence is established. Extending the analysis in ref. [4], the mapping function can be derived in explicit form: Da(.p) = p~O ~
dompW(m),
(m = 0, +1, +2 .... , a = 1, 2 ..... [L]),
(7)
m
5o =
2
"(~')~o ,
(~)~ o = (~')5(s)ls=O o ,
^k,-~2 ~r (X)dom = O(unl ~n2 "'" ~nr)'
dom = ~ (X)dom , 2
'(~)dom = (~)dm(S)ls=oo ,
)~ = )~1 + ~2 + "" + ~r
The successive sequence is characterized by simultaneous approximation both in the irregular exponents and in the expansion coefficients. This is connected intimately with the tree structure of the parametric series 5o(a, [], 7) and dora(co , ~, 7), as the limit of corresponding sequences: n ( s - ¢%)M_n(S (2)6(s) = ~_l M -~($'~L-~_--'~n ) ) ,
(3)(~(s)= * ~ ~ M n ' ( s - ¢°nl)Mn2(S - con` - Wn2)M-nx-n~(S) n, n2 L ' ( s ) L ( s - ¢On,)L(s- Wnl- con2) '
(4)(~ ~ _ j Mnx(s-- wn')Mn2(s- c°n' - con2)( M-n,(S + C°n,)M-n2(S) =
M n~(S - ~ n , ) M n l ( S ) +
~ +z_,
L ( s - COn, - c o n ~ ) L ( s
- COn, )
0 s"*l~o
ns
Mns(S - COn, - ¢On2 - cona)M_n,_n2_ns(S) \ . . . . . 1
- con1 - ¢°,,2 - ¢ ° n s ) I '
"=
d,n(S) = gm(O + 23 ~n,(OHm_n~(S + con,) + 2 3 2 3 ~n~(s)~,,,(~ ÷ con,)~m-,u-nJS + con, ÷ CO,Q ÷ .... n,
Hm(S) = Mm(s)/L(s + COrn).
(7a) 407
Volume 104, number 8
PHYSICS LETTERS
17 September 1984
As is well known, the P-symbol was indispensible for Riemann in developing his theory of hypergeometric functions, but it loses completely its significance for equations having non-fuchsian singularities. To study the structure and transformation property of irregular integrals (wave/unctions), it is convenient to introduce a new symbol, or function: I
/9
C'/) 6o(+n), a+n
/31,/32 .... /3[L]
/
~'+n,1..... T+n,[Mn] ~ ,
(8)
/
(m(-n), a_ n T-n,1 ..... T-n,[Mnl ] (+n: 1, 2 ..... N l ; - n : -1, - 2 ..... -N2) , where the parameter sets (co(n), % ) (/3a) and (Tn,/) are situated respectively in the third, first and fourth quadrant. As for their mathematical and physical meaning, the a denote coupling constants,/3 Green poles or regular exponents, T cut-off parameters. By the representation theorem the basic set of independent solutions (ffa~)} can be expressed by ~ka(p) = Da(p ) ~bO~) =
d =_
~/9#a
-w(m)+6a a,mP ]
/9 =c/)a co(n), a n
/
I
/31..... 0IL] Tn,1 ..... Tn,[Mnl (±n)
}
(9)
In this sense the ~ a function is not only a symbol but represents a de/mite class of analytic functions, all parameters being written down according to the tree structure, given by (7a). From the ~ symbol by the simple substitution p' = 1//9 one readily gets the solutions at infinity. It is not difficult to prove that ~ functions are invariant under a bilinear transformation. Thus, the most significant property of genetics functions is that they are higher automorphic functions (Poincar6).
4. Some remarkable features of SPPs. (1) Problem o/secular terms. Comparison with perturbative solutions. By direct verification, ~ functions satisfy the original wave equation identically. A most remarkable fact is that the contraction terms are eventually equal to the secular terms, being killed unnaturally in perturbation theory. It has been demonstrated that the secular terms are not only mathematically summable but also physically meaningful [4]. This is the crucial point to settle conclusively Poincar6's problem of irregular integrals, or wave/unctions in SPPs. (2) Analyticity proof. For SPPs with potentials represented by Laurent series, the genetics series solutions are uniformly convergent within the given annular region, def'med by the coefficients of the equation. The parametric series of irregular exponents (/3a + 50) and expansion coefficient dam are both analytic in (a) (T) and rational in
(/3). The tree series solutions thus obtained satisfy the original wave equation generation after generation, instead of a sequence of perturbation equations as in perturbation technique. (3) Classification ofwavefunetions. Irregular wave/unctions can be classified according to the distribution of Green poles (/3o). For the Schr6dinger and the Dirac equation there exist 3 types of wave/unctions in the normal case: elliptic:/31,2 real, waves with radius-dependent amplitude; parabolic:/31 =/32 real and equal, ~2 is unphysieal due to a logarithmic singularity; hyperbolic:/31 = ~2 complex conjugate, waves oscillating rapidly near r = 0. For the B-S equation of fourth order there are four Green poles and one gets a combination of these three types of functions. (4) Boundary conditions. From the exact solutions follow naturally the boundary conditions of physical waves at the origin and at infinity, depending on the character of the potential, repulsive or attractive. (5) Monodromy theorem. For potentials of Laurent type the basic set of solutions possesses the monodromy group, the indices of which are irregular exponents, represented explicitly by parametric series. This implies also 408
Volume 104, number 8
PHYSICS LETTERS
17 September 1984
that Hill's int'mite determinants can be solved explicitly. Thus, the irregular indices (Floquet exponents) can be defined by an explicit analytic expression with tree structure, the dynamical behavior of which is higher than almost periodic motion (H. Bohr).
5. Discussion. A general mathematical framework is established for solving exactly certain classes of singular potential problems. Within this framework a complete classification of singularities is given. The irregular integrals are represented by genetics functions with tree structure, which extend in a natural way the recursion series of regular integrals. The introduction of the symbolic c~a-function plays a central role in the study of SPPs concerning the classification, the intrinsic structure and the group properties of the wavefunctions. The exact analysis permits one also to gain a deeper insight into the correspondence between singular potential theory and nonrenormalizable field theories: an infinite number of counter-terms required in the latter is connected to the inexhaustible proliferation of the tree structure inherent to the former. For the singular interactions in quantum field theories it is evident that there exists also a mapping relation ~ba = D a ~0, but the analytic representation of the mapping function requires a more sophisticated calculation. In conclusion, the above serves only as a brief survey of some main results concerning exactly solvable classes of SPPs within the framework of the tree graph method. All mathematical details as well as physical application will be published elsewhere [5]. It is a pleasure to thank Professor M.R.C. McDowell for his helpful interest and valuable suggestions.
References [1] W.M. Frank, D.J. Land and R.M. Specter, Rev. Mod. Phys. 41 (1971) 36. [2] H.H. Aly, W. Guttinger and H.J.W. Mt~iler,in: Lectures on particles and fields, ed. H.H. Aly (Gordon and Breach, New York, 1970). [3] R.G. Newton, Scattering theory of waves and particles (McGraw-Hill,New York, 1966). [4] M. Dong, Phys. Lett. 97A (1983) 275; 98A (1983) 156; M. Dong and J.-H. Chu, Phys. Lett. 101A (1984) 473. [5 ] M. Dong, Exactly solvable classes of singular potential problems (I) ... (VI), to be published.
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