'У/ ъ'//?/?:
объединенный ИНСТИТУТ ядерных исследований дубна
Е11-91-353
Yu.Yu.Lobanov, R.R.Shahbagian, E.P.Zhidkov
COMPUTATION OF GREEN FUNCTION OF THE SCHRODINGER-LIKE PARTIAL DIFFERENTIAL EQUATIONS BY THE NUMERICAL FUNCTIONAL INTEGRATION
Submitted to the International Colloquium "Differential Equations and Applications", Budapest, 21-24 August 1991.
1991 1.INTRODUCTION
Numerical functional integration is one of the perspective means of computation in many branches of contemporary science, especially in quantum and statistical physics [1]. One of the important areas of application of functional integrals [2] is the computation of various characteristics of physical systems which consist of many particles interacting with each other. The basis for the computations is the Green function Z(x,x ,t) which in Euclidean metrics _ (t=ir) is the solution of the following problem
ьг " a2z ; У — - V(x) Z = H Z at 2 ц ах2 i = i i
Z(x,x .0) = S (x-xn) (1)
Z(x,x ,t) -»- 0 when |x I -» too , k=l, 2, . . . ,n where V(x) is a given function. The Green function method is an effective means of solution of multidimensional problems in statistical mechanics, nuclear physics, quantum optics etc. The solution of (1) without any simplifying assumptions (mean field approximation, collective excitations) is a rather complicated computational problem. In the case of high dimensions (n>3) the traditional methods (finite element, finite difference) lose their efficiency because of the presence of singularities and of the necessity of solving the algebraic systems of extreme high orders. The approach based on the computation of matrix elements of the time evolution operator exp H
1 Zfx^x^t) -
TH
F[x] dp(x) (4) X some new approximation formulas exact on a class of polynomial functionals [8]. Here X is a full separable metric space, F is a real functional. In particular case of conditional Wiener measure the family of approximation formulas with the weight is derived [9]. The use of the formulas in the problems of quantum mechanics show [10] that these formulas provide the higher efficiency of computations versus other methods of evaluation of functional integrals.
3 The employment of our formulas gives the significant (by an order) economy of computer time and memory compared to the lattice Monte Carlo method in the problems wich we have considered (with the equal accuracy of results). Moreover, while solving the problem (1) by finite difference methods one needs to discretise both space and time variables, the integral formulation via lattice Monte Carlo method assumes the discretization of time only and the continuum approach based on the use of our formulas does not need any discretisation at all. The discretization is performed here only at the final step of computation of the ordinary (Rimanian) integrals which arise in the formulas. In order to solve the problem (1) by the functional integration method in the case n>l one has to evaluate the multiple functional integrals [11]. In the present paper we derive and study the approximation formulas for such integrals. We prove the theorem on convergence of approximations to the exact value of integral and estimate the speed of this convergence for some class of functionals. We illustrate the employment of our formulas by examples of computation in statistical mechanics.
2. CONSTRUCTION OF APPROXIMATION FORMULAS.
Let X™ = X-...-X be a Cartesian product of the full separable metric spaces X. The m-dimensional integral with Gaussian measure is defined as an integral built on Xго with respect to Cartesian product of the Gaussian measures ц on X.
W ... F(xi,...,xm)du(xi)...du(x|>) s F(x)dM (x) (5)
X X X One of the means of computation of integral (5) is the successive employment of some approximation formulas for the "one-dimensional" functional integrals (e.g. formulas exact
2n +1 for on a class of polynomial functionals of degree * k the variable xeX,k»l,2,...,m, which we constructed in paper 4 [8]). The more interesting is to construct the approximation formulas with the given summary degree of accuracy 2k+l, i.e. formulas which are exact for the constant functional and for the functionals m
X *•(*, J = П Fk С,) i = i l where k+k+...+k * 2k+l, F (x ) is a homogeneous 12m ' к I ^ i polynomial of degree к with respect to argument x . The example of such a formula is given by the following Theorem l.[6] let L be a linear homogeneous functional defined on a manifold of the functionals integrable with respect to the measure д. Let L satisfy the following conditions 1. L(F) = 0 for any odd functional F(x). 2. L-|<£, -XT}, •>[ = K(€,T)) for arbitrary £,T)eX'
К(£,т)) is a correlation functional of the measure д. (6) 3. Either L| |~| } * 0 and L{l)ybO,
S fo r any еХ =2 or ii П <€,''* Г ° ?1*°»?1 'Д .---»т. Let b (i=l,2,...,m) be arbitrary positive numbers.
Then the approximation formula
FWdM'-'fx) « (l-^b^d)) F(0,0 0) + x" (7) m
+ XV^I F(0,0,...,xi/v^',0,...,0) | is exact for all polynomial functionals of the third summary degree on x". Remark. The designation L (F) means that the functional L is applied to F as to the functional of argument x e X only. Formulas like (7) give a good approximation to the exact value of integral when F[x] is close to the polynomial
S functional of the third summary degree on X™. More precise approximations can be achieved for the large class of functionals if one uses the method of construction of the so-called "composite approximation formulas" which we derived in [8,9] for the 1-dimensional functional integrals. The advantages of the composite approximation formulas over the "elementary" ones have been determined in [9]. Analogously to the case of 1-dimensional functional integrals the construction of the composite approximation formulas for integral (5) is based on the use of the relation called "mixed integration formula" [6]. Applying this formula to integral (5) with respect to each component x we obtain the mixed integration formula for the multiple functional integrals
JF(W"(x) - [ехр{-Ц<«1,,,,и1»)}х
Xя RH
xl F(X -S (X )+U (U
n n, ,,, , ,, a H-^ i» U,,)«R"1 < (« .« )-i (u; ) n are arbitrary positive numbers. w: is an orthonormal basis in the Hilbert space H which is generated by the measure ц and is dense almost everywhere in X [6]. This basis is formed by eigenfunctions of correlation functional K(£,i))- Substituting the integral over X* in the right-hand •id* of (8) by the approximation formula (7), we obtain the 6 composite approximation formula of the third summary degree of accuracy for integral (5). Thus, the following theorem appears to be proved Theorem 2. Under conditions (6) and (9) the approximation formula
[F(W>(X) = (2*)-"4exp{4£(u"\um)}x Xю RN
in x Г( l-TbLd) ) F(S,(x- 0,u(n) £ (x-c 0,u I л tj 1 ' 11 mm m + Y bL {F(S (x=0,u(,)),..., (10) 1=1 iv <1, <, ) £i(xi/v^,u ),...,Zn(Xm=0,u " ))}] du + RN(F) is exact for all polynomial functionals of the third summary degree on X*. Here m dHx in the space X = J С [0,1]; x(0)=x(l)=0 1 =• С the .composite approximation formula of the third summary degree of accuracy for the multiple conditional Wiener integrals is written as follows г |F(X) dMx - (2л)-' jexp{-i £<«"',„<»>} x C" RN l F ( 6 (1, m * 5 I f n (u )».--.Z1(/~FT p(v,-),u ) (li) I 7 -t sign v, t s |v| P(v,t) (1-t) sign v, t > Ivl l(p(v,t),u(i)) = p(v,t) - S (p(v,t)) + U (u <0 1 • = •. > Sn (p(v,t),u ) = 2 £ -i sin(JTit) sign(v) cos(jrcv) 5 (u,l)) = /"IT £ uj" £ sin(JHt). J = i Proof. Since in the space С [0,1] the following relation holds = | x(t) d?(t) о we can write l l KHZ,V)( = J j" S (t,s) d?(t) dT)(s) where % (t,s) = J x(t) x(s) dHx. с In the case of conditional Wiener measure we have [6] К (t,s) = min(t,s) - ts ; ek = <ЛГ^ Фк (t) where X and Ф (t) are the eigenvalues and the eigenfunctions of the kernel X (t,s) Ak = 1/kV ,• Фц (t) = УГТ* sin(kirt). Taking the functional L{F) in the form L{F) - F[ p(v) ]dv(v), R v is the symmetric probabiprobabilistil c measure on R, satisfying df(v) - ; dv, v с [-1,1] 8 and substituting the last relations into (10) with Ь = l/m , k=l,2,...,m, we obtain formula (11). In many cases the use of approximation formulas with weight is preferable. For the multiple conditional Wiener integrals P(x) F(x) d x (12) и x = (x 1 ,...,x m) , dx = dx dx12 • • • d mx WWW U with the weight •n . P(x) = exp ^ [ Pi(t)x^(t) + q^tjx^t) j dt (13) P,(t),q (t) s С [0,1] for all i=l,2,...,m we obtained the following approximation formula Theorem 3. Let В (s) be the solution of differential equation H (1-s) B^s) -(l-s) B^s) - 3 B((s) = 2 p^s), s e [0,1] B((l) = -2/3 Pi(l) (14) and let the following definitions hold W^t) = exp (1-s) B^s) ds a,(t) = L (s) ds — B((s) W^s) x I v ' н (t) Г L((u) du 1 ds (15) -,(t) - [ B((s) W^s) H((s) - q((s) ] ds + c. H,(s) = L; (u) du = 0 Then the approximation formula l I * exp (1-s) В (s) ds exp (t)dt v 1 = 1 J (16) XS^ F *t(v,-) = f^v,-) - ajv,-), m 1 n(v,О f|(v.t) = sign v - 1 + B^s) Wt(s) ds sign v, ts|v| ff,(v,t) 0 , t>|v| . Proof. We employ the linear transformation x(t)—> y(t), given by the relation у = x + Ax, x € C, i=l,2,...,m where AjXl(t) = (1-t) B^s) x^s) ds , B((s) б С [0,1] The transformation A. = 1 + A. maps the space С onto itself in one-to-one correspondence [12]. Using this transformation analogous to the result which we derived in [13] for the case of integration in one dimension, we obtain now | F(x)dHx- n^o, [ F[ V, vjx 10 exp { "I f ( s [ at (V.)]2 + *, it = П D, [ * [x, xj 1 = 1 ••» C" 1 2 expA Y" ^(l-s)BJ(s)-(l-s) B^(s)-3Bi(s)Jx^(s)dsJ- dHX, 0 where Ф [Xt XJ - F[ Vl V.] < D is the Fredholm determinant D = exp , (1-s) B,(s) ds \ . 1 2 Therefore, if В (t) is the solution of the problem (14) , we have « [x,...,x] exp I Y^ p,(t) x"(t)dt} d„x = 1 m 1=1-' 0 m _ = П D"l Ф Г A"'x , . . . ,AMx 1 dx where l A; xl(t) = x|(t) -^ З^э) w^s) x((s) ds, W (s) corresponds to (15). Performing one more change of variables y((t) - z((t) + L^s) ds, where L (s) satisfy (15) after some transformations we obtain II for integral (12) with the weight (13) P(x) F(x) d x = m г exp -\Y (l-s) В (s) ds\ exp (t)dt 1-1 J (17) F A"Jx +a , . . . ,AMx +a d x |_ 11 1 ra m mj H where a correspond to (15). For the integral over C1" in the ric and side of (17) we apply the approximation formula (7). 'i„« assertion of Theorem л 3 follows now from Theorem 1 due to continuity of A~ and oc л and to the linearity of A"1. Remark. Equation (14) is in fact a Riccati equation. Its solution for p (t) = p = const < л /2 is Bl(S) = £ [ /2p( ctg ( •2pi (l-s))- ±- ] . If we set also q (t) = q = const, then a (t) can be expressed explicitly as 4. «,(t) = sin(v^7? t) sin(/p~77 (l-t)) p^osv^/a' i=1,2,...,m and the approximation formula (16) acquires the form exp — [tg/p^T? - v^7l]lL 1=1[ ^ sin/2p ' I <2P.) (18) l =1 h(•).•• • .v^m + (v, .) Theorem 4. Let for almost all veR, with respect to measure f(v) the following convergence hold S (p(v)) > p(v) when n —> и , i=l,2,...,m. (19) 1 Let F(x) be a continuous on Xm functional, satisfying the condition I m |F(x)l = g( A (xi,xi),...,A (xm,xm) ) (20) where Ak(x ,x )- is a non-negative quadratic functional '(x , x ) = V у (x ,e ) - (21) v к' к' L I * к' i'I Y n\ < °° , г. г o, i=i,2. (22) g(x) is a non-decreasing function, and g ( A'(XI,X) /(pM.plvD+A't^,^) X-R (23) A (x ,x ) ) dv(v) &u (x) < o> . m m Then the remainder of the approximate formula (10) R (F) —> 0 when n-> и, i=l,2,...,m. Proof, Without any restrictions of generality we suppose that », - 7, and A (xk,xk) sA(xk,xk), k=l,2,...,m. Using (20)-(22) we obtain |F(S (x ) ,/T(p(v)-S (p(v)))+S (x ) S (x))| = n 1 n n к n m 1 к к m 1 F( I 1» ••' 1<х„'еЛ eJ ' £ (24) *( I'i I we,>sa ) g( A(x ,x ) , .. .,A(p(v),p(v))+A(x ,x ),...,A(x ,x ) ), ll к к mm for all k=l,2,...,m. Consider the functional W XJ = F(S (x),...,,Tm (p(v)-S (p(v)))+S (x ),..., n 1 n n к I 1 к к S (x ) ) dv(v) T (x) is integrable on X1" with respect to measure д. Using the mixed integration formula (8) we get г m ,m) /г T (x) dn (x) = exp ( u ,u H (2п)-" l"i L * Г (25) Л T (X-S (x )+U (um) x-S (x )+U (u<")))djic,,) INlnln mnmn xdu. J 1 1 mm One can transform the functional ww+u» 14 - I f F( s„ R (k) "5T(p(v)-s (P(V)))+S (x+s (x )+u (u )),..., П П К П К П к к к к Sn (x+S (x >«J (u'"")) ) dv(v) = (26) F( U (u'1'),..., /lT(p(v)-S (P(v)))+U (u(k>),..., I v n n n U (u(m)) ) di>(v) . m Substituting (26) into (25) and taking into account du("°(x) = 1 we obtain , (m /г T (Xt ) du '(x) = (2пУ» exp{-l£(u"\u")},< F( ., iTro (p(v)-S (p(v)))+U (u"")c f (if "•.<"'"'• "k nk X" R = (2Я)-"2 [еЦ-Ц^'Чи'»)^ RN (1) (lc) У F( U (u ) v^lT(p(v)-Sn (p(v)))+Un (u ) k=l J 1 к к R U (u(B>) ) df(v) du . Hence, the integral IS F(x)du(ra) (x) can be represented in the form ta) I *4*)*i (x) = i TM(X) dw (x) + RN(F) For almost i-all xeX™ with respect to the measure д there holds the convergence S (x ) —» x when n -»oo, i-1,2, . . . ,m . Therefore •~nT(p(v)-S (p(v))) + Sn (xj > xR к к when n —» F( Sn (Xj) /Tir(p(v)-Sn (p(v))) + Sn (xj,..., 1 к к S (X ) ) > F(X) m by the simultaneous approach of all n to the infinity. It follows from (23) and (24) that the sequence { T (x) } , i=l,2 ,m i is bounded by the integrable function. Now we can apply the Lebesgue theorem "on the passage to the limit under the integral sign" (m TN(x) du '(x) F(x)du (x) asn->w, i=l,2, X"' Xm which completes the proof of the theorem. The estimate of the remainder R (F) in dependence on N is established by the following Theorem 5. If the integrable with respect to measure и m (x) functional F(x) can be expressed in the form N. F(x+xQ) = Рз(х) + r(x;xo) (27) where P (x) is a polynomial functional of the third summary я degree on X and the remainder r(x;xQ) is estimated by the expression m 2 |r(x;xQ)| * f] (А'Сх,^)) [ Ciexp { C.,A' (Xl+x°, X|+X°) } + l + c3 exp | c2A (x°,x°)J ] , (28) c( > 0 , (i=l,2,3) сгН) 0 , к = 1,2,... , i = 1,2,...,m 2 k 2 V r"'at < a ; (ek , v^ p(v)) ^ ak; ak,v e R (29) then for the remainder of approximation formula (10) there holds the estimate m oo m 03 2 + (( У >а )2) (30) RN(F) =0(fl (£\"') ) X^° Z " х - k = l i =1 к =n +1 1 Proof. Since the formula (10) is exact for all polynomial functionals of the third summary degree, it follows that it's remainder can be expressed as follows R (F) (2Л) sxp ,u N l"=Z-. Г r(x-S (x);U (u))d|i '(x) m ...0,Л p(v)-S (p(v)),0,...,0;U (u))dy(v) du= n n I К - К . 1 2 According to (28) , we get 2 1 (1 |KJ , (2Я)- [exP{-i^(u" ,u >)} ") , ,г,г 1 = 1 *— k=n +1 k=n 1 k=n +1 n dM,m,(x) du (2 )-П1/г - (U ,U )} , -П И i = 1 ^"«wv k=n +1 Xx i 00 2 .expb I у (x ,e ) + с £C0 } k=n +1 1 V^ <1) , UK 2) + с exp{ dM(x,) du' n /z f] (2n)- i |du(x,) 3XP|-- (U ,U )| x 1 ^ ( V <П / k=n +1 I I» a = П П d-aea,«')-f £ C(*..V > ,=1 k=1 J £•=•„+, X t 00 l) 2 =iexp{c2£ ^ (x|fek) }. + с du(*,) k=n +1 1 П П (1-2о."»)-л I»'. 1=1 k = l Analogously for К we have 1, 1/2 |K2| = £ П (l-2cX )- f(f т;"(Лр(У),вк)'| 1=1 k_1 J k=n +1 R i oa 2 + c_ dv(v) 2iexp|c2 V r"'(vT P(v),ek) l k=n +1 1 n ,, i/a *t*— • kп = l d-acx )- с. 1 =1 Consider the integral f x e )2 IJX) = ехр/лс.^Г ' k"( 1' k } dM(Xi),i=l,2, J k=n+l X 1 It is well known [6], that it's analytic value ,/а Il(A) = [j (1-2Лсаг^»)- . k=nui After some more transformations we obtain i; 1 1 П (1-ахо »»,)-1Я 'а-глс^ ')" 1 [i -" 00 2 2 a £ (y;") (i-2xc2^>)- . + 2 П Since , 1/a i;(D - П (i-av« )- I »r(b2V;'v L J=n( + l CO l) 1/a + 2 и (i-2c T« )" > (r;")^(i-2c,;'v«-2V;"--;")> П a I 7 and • Ш -I £ w 2 i;'(0) = + 2 > u;») J=n(+1 j=n,+l then х;»-0|П d-2oX»>- £ *;" и 2 1> 2 2 £ (г; ) (1-2сгУ; )- | + J=n,+» Г m I2 ю + 2c3;_ (,«',« and there follows 20 , к = о П J Ч-J-n +1 > •j=ni+i Using the condition (29) we get I»' - [ o, exp {c2 £ <\] + c3 ] g ,«', k=n +1 k=n +1 I i K rma k k - = Z°IL I 4-k=k=n +1 J ' 2 1=1 V k=n +1 l 1 1 =1 Thus the proof of the theorem is complete. 4.NUMERICAL CALCULATIONS. We will illustrate the use of the formulas with examples of m-dimensional quantum models, characterized by the Hamiltonian m н = IУ -^~,+ v<*> • <32) <£- ax 1 = 1 i We shall study the energy E of the ground state and wave function Ф (x) . The basis for the computation is the Green function (2). The expressions for the principal quantities of Euclidean quantum mechanics in the form of functional integrals with conditionaf(tl) =Wiene - \ rmz(T measur) e are [10] Z(T) = Tr exp {- -} • J Z(X,X,T) dX i(X,X,T) - (art)"1''2 expj-T V(v^T" x(t) + X)dtlj dx (33) 21 EQ = liltl f (T) (34) T-*oo G(T) = <х(0)х(т)> ш <™>'"г expj-T V(/TT x(t) + X)dt| -ю С x[ v/~f"x(T/t) + X ]d x X dx ЛЕ = Ea - Eo = - lim g^ In G(T) I* (x)l2 = lira [ exp I BT } Z(x,x,T) ] . (35) T-»co \ 1 Consider the harmonic oscillator with n V(x) =;£х* . (36) The theoretical values are E* = (m+l/2)n (37) n l**(x)|2 = (l/n)n/2 exp { - £x2 } . (38) i =i Using the formula with weight, one can evaluate the integral Z as follows n ,/ 2 2 Z(yt yn,T) = П 1/(271 shT) exp |-th(T/2)y J. (39) Consequently, the analytic values for finite T are Em - - cth(T/2) —-> E* « 2 *••• 22 Т) 2 т ! |^ (У)| = expj Е< 'т} 2(у,Т)т—* |Ф*(у)| The values of E_ and E for various T and N obtained on the CDC-6500 computer, usinоg the composite formula (10) whith various n and n are given in Tables 1 and 3 for n = 2 1 2 ап((з. The convergence of the numerical results of energy E to E^T) (T=3) is represented in Tables 2 and 4. Table 1 E(TI T E 0 n /n o V 2 4 1.007 1.037 1 5 1.004 1.013 2 6 1.002 1.004 3 Table 2 n./n2 1 2 3 4 Eo 1.024 1.009 1.005 1.004 Table 3 T E n /n o -Г' V 2 4 1.507 1.555 1 5 1.503 1.520 2 6 1.500 1.500 3 1.5 Table 4 n/na 1 2 3 4 E 1.511 o 1.515 1.509 1.507 23 The CPU time of computation of E has been ca 2 0 sec. It follows from Tables 1 and 3 that the good approximations E of the theoretical value are achieved at relatively small values of T and N. Table 5 T, 2 X2 l*0 CO I" K c*>l -2.8 .618E-04 .584E-04 -2.4 .805E-03 .777E-03 -1.6 .198E-01 .195E-01 -0.5 .173E+00 .173E+00 0.5 .173E+00 .173E+00 1.6 .198E-01 .195E-01 2.4 .805E-03 .777E-03 2.8 .618E-04 .584E-04 The numerical results of calculation 1Фо (x)l and (T) 2 1Ф (x)| obtained using the composite formula (10) with T=6, n =n =1, n=2, are given in Table 5 in the form of dependence on the parameter x with x=0.5. We compute all integrals using the Gaussian quadrature with the relative accuracy 0.1 % . The CPU time of computation of |Ф (х)|а has been ca 2 sec on the CDC-6500 computer. Consider the Calogero model which is characterized by the Hamiltonian n a2 H - - I -r + lu2I REFERENCES 1. Bogoliubov N.N. and Shirkov D.V. Introduction to the Theory of Quantized Fields. - Moscow : Nauka, 1984 (in Russian). 2. Mazmanishvili A'.S. Functional Integration as a Method of Solving the Physical Problems.- Kiev: Naukova Dumka, 1987 (in Russian). 3. Milstein A.I. Phys. Lett. A, 1987, vl22, Nol, pl-5. 4. Carlson J. Phys. Rev. C, 1987, v36, No5, p2026-2033. 23 5. Negele J.W. J. Stat. Phys., 1986, v43, No5/6, p991-1015. 6. Janovich L.A. Approximate Evaluation of Functional Integrals with Gaussian Measures.-Minsk: Nauka i technika, 1976 (in Russian). 7. Lobanov Yu.Yu. et al. In: Algorithms and Programs for solution of Some Problems in Physics, v6, KFKI-1989-62/M, Budapest, 1989, pl-28. 8. Gregus M.,Jr.,Lobanov Yu.Yu.,Sidorova O.V.,Zhidkov E.P. J. Comput. Appl. Math., 1987, V20, p247-256. 9. Lobanov Yu.Yu. et al. J. Comput. Appl. Math., 1990, v29, P51-60. 10.Lobanov Yu.Yu., Zhidkov E.P. JINR, E2-87-507, Dubna, 1987. ll.Zhidkov E.P., Lobanov Yu.Yu., Shnhbagian R.R. Math.Model., 1989, vl, No8, pl39-157 (in Russian). 12.Lobanov Yu.Yu..Sidorova O.v.,Zhidkov E.P. JINR Rapid Communications, No4-84, Dubna, 1984, p28-32(in Russian). 13.Lobanov Yu.Yu.,Sidorova 0.V.,Zhidkov E.P. JINR, Pll-84-775, Dubna, 1984 (in Russian). 14.Camiz P. et al. J. Math. Phys., 1971, vl2, No 10, p20l0. 15.Goovaerts M.J. J. Math. Phys., 1975, vl6, p720-723. 16.Grimm R.C., Storer R.G. J. Сотр. Phys., 1971, v7, No 1, P134-156. Received by Publishing Department on July 25, 1991. 26 Жидков Е.П.. Ливанов Ю.Ю., Шахбагян P.P. ЕП-91-353 Вычисление функции Грина дифференциальных уравнений в частных производных типа Шредингера методом приближенного континуального интегрирования Разработан новый метод численного решения краевой задачи для дифферен циальных уравнений типа Шредингера в R". Метод основан на представлении многомерной функции Грина в виде кратного континуального интеграла и на ис пользовании построенных нами приближенных формул для таких интегралов. Дока зана сходимость приближений к точному значению, оценен остаточный член фор мул. Данный метод сводит исходную дифференциальную задачу к квадратурам, он позволяет избежать наличия сингулярности в нуле и решать задачу во всем пространстве R" без замены граничных условий на бесконечности условиями max при некоторых больших x , k = l,...,n, x = (х,,..., х^,..., хп).Метод не требует предварительной дискретизации пространства и времени. Рассмотре ны некоторые применения метода в статистической механике. Работа выполнена в Лаборатории вычислительной техники и автоматизации ОИЯИ. Препринт Объединенного института ядерных исследований. Дубна 1991 Lobanov Yu.Yu., Shahbagian R.R., Zhidkov E.P. Ell-91-353 Computation of Green Function of the Schrbdinger-Like Partial Differential Equations by the Numerical Functional Integration A new method for numerical solution of the boundary problem for Schrb- dinger-like partial differential equations in Rn is elaborated. The method is based on representation of multidimensional Green function in the form of multiple functional integral and on the use of approximation formulas which we constructed for such integrals. The convergence of approximations to the exact value is proved, the remainder of the formulas is estimated. Our method reduces the initial differential problem to quadratures, it enab les one to avoid the singularity at the origin and to solve the problem in the whole space Rn without replacement of the boundary conditions at infini max ty by the conditions at some large x , k = l,...,n, x = (x,,... .x^,..,xn). This method does not need any preliminary discretisation of space and time' as well. Some applications of the method in statistical mechanics are con sidered. The investigation has been performed at the Laboratory of Computing Techniques and Automation, JINR. Preprint of the Joint Institute fur Nuclear Retcarch. Dunn* 1991 65 коп. Корректура и макет Т.Е.Попеко. Подписано в печать 31.07.91 . Формат 60x90/16. Офсетная печать. Уч.-издлистов 2,20. Тираж 445. Заказ 44545. Издательский отдел Объединенного института ядерных исследований. Дубна Московской области.