Epilogue

If one does not sometimes think the illogical, one will never discover new ideas in science.

Max Planck, 1945

Mathematics is not a deductive science-that's a cliche. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial-and-error, experimentation, and guesswork.

Paul Halmos, 1985

The most vitally characteristic fact about mathematics, in my opin­ion, is its quite peculiar relationship to the natural sciences, or more generally, to any science which interprets experience on a higher more than on a purely descriptive level. ...

I think that this is a relatively good approximation to truth­which is much too complicated to allow anything but approxima­tions -that mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure. But, once they are so con­ceived, the subject begins to live a peculiar life of its own and is better, compared to a creative one, governed by almost entirely aes­thetic motivations, than to anything else and, in particular, to an empirical science ....

But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will

426 Epilogue

separate into a multitude of insignificant tributaries, and that the discipline will become a disorganized mass of details and complexi­ties. In other words, at a great distance from its empirical sources, or after much "abstract" inbreeding, a mathematical object is in danger of degeneration. At the inception, the style is usually classical; when it shows signs of becoming baroque, then the danger signal is up ....

Whenever this stage is reached, then the only remedy seems to be a rejuvenating return to the source: the reinjection of more or less directly empirical ideas. I am convinced that this was a necessary condition to conserve the freshness and the vitality of the subject and that this will remain equally true in the future.

John von Neumann, 1947

Mathematics is an ancient art, and from the outset it has been both the most highly esoteric and the most intensely practical of human endeavors. As long ago as 1800 B.C., the Babylonians investigated the abstract properties of numbers; and in Athenian Greece, geome­try attained the highest intellectual status. Alongside this theoretical understanding, mathematics blossomed as a day-to-day tool for sur­veying lands, for navigation, and for the engineering of public works. The practical problems and the theoretical pursuits stimulated one another; it would be impossible to disentangle these two strands.

Much the same is true today. In the twentieth century, mathemat­ics has burgeoned in scope and in diversity and has been deepened in its complexity and abstraction. So profound has this explosion of research been that entire areas of mathematics may seem unin­telligible to laymen-and frequently to mathematicians working in other subfields. Despite this trend towards-indeed because of it­mathematics has become more concrete and vital than ever before.

In the past quarter of a century, mathematics and mathematical techniques have become an integral, pervasive, and essential compo­nent of science, technology, and business. In our technically oriented society, "innumeracy" has replaced illiteracy as our principal edu­cational gap. One could compare the contributions of mathematics to our society with the necessity of air and food for life. In fact, we could say that we live in the age of mathematics-that our culture has been "mathematized." No reflection of mathematics around us is more striking than the omnipresent computer ....

There is an exciting development taking place right now, reuni­fication of mathematics with theoretical physics ....

In the last ten or fifteen years mathematicians and physicists re­alized that modern geometry is in fact the natural framework for gauge theory (cf. Sections 2.20ff in AMS Vol. 109). The gauge po­tential or gauge theory is the connection of mathematics. The gauge

Epilogue 427

field is the mathematical curvature defined by the connection; cer­tain "charges" in physics are the topological invariants studied by mathematicians. While the mathematicians and physicists worked separately on similar ideas, they did not just duplicate each other's efforts. The mathematicians produced general, far-reaching theories and investigated their ramifications. Physicists worked out details of certain examples which turned out to describe nature beautifully and elegantly. When the two met again, the results are more powerful than either anticipated ....

In mathematics we now have a new motivation to use specific insights from the examples worked out by physicists. This signals the return to an ancient tradition ....

Mathematical research should be as broad and as original as pos­sible, with very long-range goals. We expect history to repeat itself: we expect that the most profound and useful future applications of mathematics cannot be predicted today, since they will arise from mathematics yet to be discovered.

Arthur M. Jaffe, 1984

Mathematics is an organ of knowledge and an infinite refinement of language. It grows from the usual language and world of intuition as does a plant from the soil, and its roots are the numbers and simple geometrical intuitions. We do not know which kind of content math­ematics (as the only adequate language) requires; we cannot imagine into what depths and distances this spiritual eye (mathematics) will lead us.

Erich Kahler, 1941

Appendix

Almost all concepts, which relate to the modern measure and inte­gration theory, go back to the works of Henri Lebesgue (1875-1941). The introduction of these concepts was the turning point in the tran­sition from mathematics of the nineteenth century to mathematics of the twentieth century.

Naum Jakovlevic Vilenkin, 1975

For the convenience of the reader we summarize a number of important results about the following topics:

the Lebesgue measure; the Lebesgue integral; ordered sets and Zorn's lemma.

The Lebesgue Measure

Let us consider the space JR,N for fixed N = 1,2, .... By an N -cuboid we understand the set

C:= {(6, ... '~N) E JR,N: aj < ~j < bj for j = 1, ... ,N},

where aj and bj are fixed real numbers with aj < bj for all j. The volume of C is defined through

N

vol(C) := II (b j - aj).

j=l

430 Appendix

The Lebesgue measure J1, generalizes the classical volume of sufficiently regular sets in]RN to certain "irregular" sets.

More precisely, we have the following quite natural situation. There exists a collection A of subsets of]RN which has the following properties:

(i) Each open or closed subset of]RN belongs to A.

(ii) If A, B E A, then

A u B E A, A n B E A, and A - B E A.

(iii) If An E A for all n = 1,2, ... , then

00 00

U An E A and n An E A. n=l n=l

(iv) To each set A in A there is assigned a number J1,(A), where

0::::: J1,(A) ::::: 00.

Here, J1,(A) is called the (N-dimensional) measure of A, and the sets A in A are called measurable (in ]RN).

(v) If A, B E A and An B = 0, then

J1,(A U B) = J1,(A) + J1,(B).

If An E A for all n = 1,2, ... and An n Am = 0 for all n, m with n =I=- m, then

Here, we use "00 + 0 = 00."

(vi) If 0 is an N-cuboid, then 0 E A and

J1,(0) = vol(O).

(vii) The subset A of]RN has the N-dimensional measure zero, i.e., A E A and

J1,(A) = 0

iff, for each c > 0, there is a countable number of N-cuboids 01, O2 ,

... such that

00 00

j=l j=l

Appendix 431

(viii) If the set A has the N-dimensional measure zero and B ~ A, then the set B also has the N-dimensional measure zero.

(ix) The collection A is minimal, i.e., if a collection A' satisfies conditions (i) through (viii), then A ~ A'.

Then, the following hold true:

The measure JL is unique on A.

JL is called the Lebesgue measure. As usual, we write "meas" instead of JL, i.e.,

meas(A) := JL(A) for all A E A.

Example. A finite or countable number of points in ]RN has the N­dimensional measure zero.

In particular, the set Q of rational numbers has the one-dimensional measure zero in ]R, and the set

has the N-dimensional measure in ]RN.

Convention. By definition, a property P holds true "almost everywhere" iff P holds true for all points of ]RN with the exception of a set of N­dimensional measure zero.

One also uses "almost all." For example, almost all real numbers are irrational. Let M ~ ]RN. We write

u(x) = lim un(x) for almost all x E M n--+oo

iff this limiting relation holds for all x E M - Z, where the set Z has the N-dimensional measure zero.

Approximation Property. The Lebesgue measure is regular, i.e., for each measurable set M in ]RN, we have

meas(M) = inf meas(G),

where the infimum is taken over all the open subsets G of]RN with M ~ G. In particular,

meas(]RN) = +00 and meas(0) = O.

432 Appendix

Step Functions

u . /. ---c

L-~r--------r----_X

a b

FIGURE A.I.

Recall that lK = ]R or lK = C. A function

u: M ~ ]RN -+ lK

is called a step function iff u is piecewise constant. To be precise, we suppose that the set M is measurable and that there exists a finite number of pairwise disjoint measurable subsets Mj of M such that meas(Mj) < 00

for all j and

u(x) = { ~j

where aj E lK for all j.

for x E M j and all j otherwise,

The integral of a step function u is defined through

Example. Let u: [a, b] -+ ]R be a step function as pictured in Figure A.I. Then the integral of u defined above is equal to the classic integral.

Measurable Functions

The function

is called measurable iff the following hold:

(i) The domain of definition M is measurable.

(ii) There exists a sequence (un) of step functions Un: M -+ lK such that

u(x) = lim un(x) for almost all x E M. n--+oo

Appendix 433

Theorem of Luzin. Let M be a measurable subset of JRN. Then, the function

u:M -+ II(

is measurable iff it is continuous up to small sets, i.e., for each 8 > 0, there is an open subset Mli of JRN such that the function

u:M -Mli -+ II(

is continuous and meas( M li ) < 8.

Standard Example. The function f: M ~ JRN -+ II( is measurable if it is almost everywhere continuous on the measurable set M (e.g., M is open or closed).

Calculus. Linear combinations and limits of measurable functions are again measurable.

More precisely, we set

F(x) := a(x)u(x) + b(x)v(x), G(x):= lu(x)l, (L)

H(x):= lim un(x), n-+oo

and we assume that the functions

a, b,u,v, Un: M ~ JRN -+ II(

are measurable for all n and the limit (L) exists for all x EM. Then, the functions

F, G, H: M ~ JRN -+ II(

are also measurable.

Modification of Measurable Functions. If we change a measurable function at the points of a set of measure zero, then the modified function is again measurable.

For example, if the limit (L) exists only for almost all x E M, i.e., for all x E M -Z with meas(Z) = 0, and if we set H(x) := ° for all x E Z, then the function H: M -+ II( is measurable provided all the functions Un: M -+ II(

are measurable.

The Lebesgue Integral

The definition of the Lebesgue integral is based on the very natural formula

[ udx:= lim [undx 1M n-too 1M (A)

434 Appendix

together with the following two formulas

u(x) = lim un(x) n--->oo

for almost all x E M (B)

and

for all n, m 2: no(E). (C)

Definition of the Lebesgue Integral. Let M be a nonempty measurable set. The function u: M ~ ]RN -> JK is called integrable (over M) iff the following two conditions are satisfied:

(i) There is a sequence (un) of step functions Un: M ->JK such that (B) holds.

(ii) For each E > 0, there is a number nO(E) such that (C) holds.

If u is integrable, then we define the integral through (A).

This definition makes sense since the limit exists in (A), and this limit is independent of the choice of the sequence (un).

Obviously, each integrable function is measurable. For the empty set M = 0 we define f0 u dx = O. We also use synony­

mously the following terminology:

(a) fMudx exists;

(b) u is integrable (over M);

(c) IfMudxl < 00.

Standard Example 1. Let M be a bounded open or compact subset of ]RN, and suppose that the function

u: M ->JK

is bounded and continuous almost everywhere, i.e., there is a set Z ~ M with meas( Z) = 0 such that u is continuous on the set M - Z and

lu(x)1 ::; const for all x E M.

Then, u is integrable over M.

Standard Example 2. Let the function u: M ~ ]RN -> JK be almost everywhere continuous on the measurable set M (e.g., M = ]RN). Suppose that

const lu(x)1 ::; (1 + Ixl)a for all x E M (G)

Appendix 435

and fixed a > N. Then, u is integrable over M. Condition (G) controls the growth of the function u as Ixl --> 00.

Standard Example 3. Let the function f: M <::; ]RN --> ]R be almost every­where continuous on the bounded measurable set M (e.g., M is bounded and open or M is compact). Suppose that there is a point Xo in M such that

lu(x)l:S; const Ix - xol,6

for all x E M with Xo =I- x

and fixed (3: 0 :s; (3 < N. Then, u is integrable over M. Condition (H) controls the growth of the function u as x --> Xo.

(H)

Measure. Let M be a measurable subset of]RN with meas(M) < 00. Then

1M dx = meas(M),

where we write J M dx instead of J M u dx with u == 1.

Linearity. Let the functions u, v: M --> lK be integrable over M and let a, (3 E lK. Then, the function au + (3v is also integrable over M and

1M (au + (3v)dx = a 1M udx + (3 1M vdx.

Absolute Integrability. Let u: M <::; ]RN --> lK be a measurable function. Then

1M U dx exists iff 1M luldx exists.

In addition, if one of these two integrals exists, then we have the gener­alized triangle inequality

Transformation rule. Let the function u: M <::; ]RN --> ]R be integrable over the nonempty open set M. Suppose that the function f: K --> M is a C 1-diffeomorphism21 from the open subset K of]RN onto M. Then

1M u(x)dx = L u(f(y)) det f'(y)dy.

21That is, I is bijective and both I and I-I are C1 .

436 Appendix

Here, det l' (y) denotes the determinant of the first partial derivatives of the function f at the point y.

Majorant Criterion. Let the function u: M ~ ]RN ----+ lK be measurable, and suppose that there exists a function g: M ----+ R that is integrable over M such that

lu(x)1 :S g(x) for almost all x E M.

Then, the functions u and lui are also integrable over M and

Vanishing Integrals. Let u: M ~ ]RN ----+ ]R be a measurable function such that u(x) ::: 0 for all x E M. Then

1M udx=O iff u(x) = 0 for almost all x E M.

Let the function v: M ----+ lK be integrable. Then, the integral fM v dx remains unchanged if we change the function v at the points of a set of N-dimensional measure zero.

Additivity with respect to domains. Let M and K be two disjoint measurable subsets of ]RN, and suppose that the function u: M U K ----+ lK is integrable over M and K. Then, u is also integrable over K U M, and

r udx= r udx+ r udx. JKUM JK JM

Convergence with respect to domains. Let u: M C ]RN ----+ lK be a function. Suppose that

CXJ

and M= UMn. n=l

Then, u is integrable over M iff u is integrable over all sets Mn and sUPn fMn luldx < 00. In this case,

r udx = lim r udx. 1M n~ooJMn

Absolute Continuity. Let u: M ~ ]RN ----+ lK be integrable. Then, for each c > 0, there is a 8 > 0 such that

Ii udxl < c

Appendix 437

holds true for all subsets A of M with meas(A) < 8.

Reduction to Bounded Sets. Let M be a nonempty unbounded mea­surable subset of JRN, N = 1, 2, ... , and let the function u: M ---; OC be integrable.

Then, for each c > 0, there is an open ball B in JRN such that

I r udxl::; r luldx < c, JM-H JM-H where H:= M n B. Hence

Observe that the set H is bounded.

p-Mean Continuity. Let u: M ~ JRN ---; OC be a measurable function on the nonempty bounded measurable set M. Suppose that

for fixed p 2': 1. Set u(x) := 0 outside M. Then, for each c > 0, there is a 8(c) > 0 such that

1M lu(x + h) - u(x)IPdx < c for all h E JRN with Ihl < 8(c).

Limits of Functions and Integrals

Theorem on Dominated Convergence. We have

lim r undx = r lim un(x)dx, n~CXJ } M } M n---+oo

where all the integrals and limits exist, provided the following two conditions are satisfied:

(i) The functions Un: M ~ JRN ---; OC are measurable for all n and the limit

lim un(x) exists for almost all x E M. n ..... oo

(ii) There is an integrable function g: M ---; JR such that

for almost all x E M and all n.

438 Appendix

Theorem on Monotone Convergence. Let (un) be a sequence of inte­grable functions Un: M <:;; JR.N ----> JR. such that

and

1M undx::; C for all n and fixed C > o.

Then, there exists an integrable function u: M ----> JR. such that

u(x) = lim un(x) for almost all x E M n-->oo

and

Lemma of Fatou. Let (un) be a sequence of integrable functions Un: M <:;; JR.N ----> lR.. Suppose that

(a) un(x) 20 for all x E M and all n.

(b) J M undx ::; C for all n.

Then

More precisely,

u(x):= lim un(x) is finite for almost all x E M. n-->oo

If we set u(x) := 0 for all the points x of M with limn--> 00 Un (x) = 00, then the function u: M ----> JR. is integrable and

r u dx::; lim r undx::; C. JM n-->oo JM

Iterated Integration

Our goal is the following fundamental formula:

1M u(x,y)dxdy = iN (iL U(X'Y)dYj dx

= r (r u(x, y)dx dy. JJRL JJRN

(I)

Appendix 439

Here, we set u(x, y) = 0 outside M. Furthermore, let x E ]R.N, Y E ]R.L, and M ~ ]R.N+L.

Theorem of Fubini. Let u: M ~ ]R.N+L ....... K be integrable. Then formula (I) holds true.

To be precise, the inner integrals exist for almost all x E ]R.N (resp., for almost all y E ]R.L), and the outer integrals exist.

Theorem of Tonelli. Let u: M ~ ]R.N+L ....... K be measurable. Then the following two conditions are equivalent:

(i) The function u is integrable over M.

(ii) There exists at least one of the iterated integrals from (I) if u is replaced by lui, i.e., J(1 luldy)dx exists or J(1 luldx)dy exists.

If condition (ii) is satisfied, then all the assertions of Fubini's theorem are valid.

Special Case. Let M := {(x,y) E ]R.2:a < x < b,c < y < d}, where -00 :::; a < b :::; 00 and -00 :::; c < d :::; 00.

Then, N = L = 1 and formula (I) reads as follows:

1M u(x,y)dxdy = lb (ld U(X,Y)dY) dx = ld (lb

U(X,Y)dX) dy.

Parameter Integrals

We consider the function

F(p):= 1M f(x,p)dx,

for all parameters pEP. We are- given the function

f:M x P ....... K,

where M is a measurable subset of]R.N and P is a subset of]R.L or CL .

Continuity. The function F: P ....... K is well-defined and continuous pro­vided the following three conditions are satisfied:

(i) The function x f-+ f(x,p) is measurable on M for all parameters pEP.

(ii) There exists an integrable function g: M ....... ]R. such that

If(x,p)1 :::; g(x) for all pEP and almost all x E M.

440 Appendix

(iii) The function p f-> f(x,p) is continuous on P for almost all x E M.

Differentiability. Let P be a nonempty open subset of IR or <C. Then, the function F: P ......, lK is differentiable and

for all PEP,

provided the following two conditions are satisfied:

(i) The integral fM f(x,p)dx exists for all parameters pEP.

(ii) There exists an integrable function g: M ......, IR such that

Ifp(x,p)1 ::::; g(x) for all pEP and almost all x E M.

This condition tacitly includes the existence of the partial derivative fp(x,p) for all pEP and almost all x E M.

Functions of Bounded Variation

Let -00 < a < b < 00. The function g: [a, bJ ......, <C is called of bounded variation iff

n

V(g) := sup L Ig(x~n)) - g(xt\)1 < 00,

'D k=l

(1)

where the infimum is taken over all the possible finite decompositions V of the interval [a, b], i.e.,

with n = 1,2, .... (2)

The number V (g) is called the total variation of the function 9 on the interval [a, bJ.

Theorem of Jordan. The function g: [a, bJ ......, C is of bounded variation iff there exist nondecreasing functions gj: [a, bJ ......, IR, j = 1,2,3,4, such that

for all x E [a, bJ. (3)

The Classic Stieltjes Integral

We are given the continuous function f: [a, bJ ......, C and the function g: [a, bJ ......, C of bounded variation, where -00 < a < b < 00. Then, there exists the limit

(4)

Appendix 441

which is independent of the decomposition of the interval [a, bl from (2). Hence

lIb f(x)dg(x) I ::; C~;~blf(X)I) V(g).

If the function f: lR ---- <C is continuous and the function g: lR ---- <C is of bounded variation on each compact interval, then we set

100 f(x)dg(x):= lim Ib f(x)dg(x), -00 b-->+oo a

a---+-oo

provided this limit exists.

The Lebesgue-Stieltjes Integral

If the function f is not continuous, then one introduces the so-called Lebes­gues-Stieltjes integral which is identical to the Lebesgue integral in the special case where g(x) := x for all x E R

A summary of important properties of the Lebesgue-Stieltjes integral in­cluding measure theory can be found in Zeidler (1986), Vol. 2B, Appendix.

Standard Example. Let -00 ::; a < b::; 00. Then, the formula

Ib f(x)dg(x) = Ib f(x)g'(x)dx (5)

holds true provided the following assumptions are satisfied:

(i) The functions f, h: la, b[---- <C are measurable, and the functions hand fh are integrable over la, b[, in the sense of the Lebesgue integral.

(ii) For all x E la, b[,

g(x) := IX h(y)dy.

More precisely, under the assumptions (i) and (ii), the left-hand integral from (5) exists in the sense of a Lebesgue-Stieltjes integral, whereas the right-hand integral from (5) exists in the sense of a Lebesgue integral with g' = h.

If, in addition, f is continuous on the closure of la, b[, then the left-hand integral from (5) exists in the sense of a classic Stieltjes integral.

Ordered Sets and Zorn's Lemma

The set C is called ordered iff there is a relation, written as

u ::; v,

among some pairs of elements of C such that the following hold:

442 Appendix

(i) u:S;uforalluEC.

(ii) If u :s; v and v :s; w, then u :s; w.

(iii) If u :s; v and v :s; u, then u = v.

By a maximal element m of C we understand an element of C such that

m:S; u and u E C imply m=u.

A nonempty subset T of C is called totally ordered iff, for all u, vET, we have

u:S;v or v:S; u.

Zorn's Lemma. Let C be a nonempty ordered set which has the property that each totally ordered subset T of C has an upper bound, i. e., there is an element b of C such that

for all u E T,

where b depends on T. Then, there exists a maximal element in C.

Example 1. Let S be a set, and let C be the collection of all the subsets of S. For u, v E C, we write

iff u t:;; v.

Then, C becomes an ordered set.

Example 2. The set ffi. of real numbers is totally ordered, but ffi. does not have any maximal element.

Zorn's lemma can be used in mathematics if the usual induction argu­ment fails, since the set under consideration is not countable. In Section 1.1 of AMS Vol. 109 we use Zorn's lemma in order to prove the Hahn-Banach theorem.

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Sunder, V. (1987): An Invitation to von Neumann Algebras. Springer­Verlag, New York.

Szabo, I. (1987): Geschichte der mechanischen Prinzipien und ihrer wich­tigsten Anwendungen. Birkhauser, Basel.

456 References

Taylor, M. (1996): Partial Differential Equations, Vols. 1-3, Springer­Verlag, New York.

Temam, R. (1988): Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer-Verlag, New York.

Thaller, B. (1992): The Dimc Equation. Springer-Verlag, Berlin, Heidel­berg.

Thirring, W. (1991): A Course in Mathematical Physics. Vol. 1: Classi­cal Dynamical Systems; Vol. 2: Classical Field Theory; Vol. 3: Quan­tum Mechanics of Atoms and Molecules; Vol. 4: Quantum Mechanics of Larye Systems. Springer-Verlag, New York.

Thorne, K. (1994): Black Holes and Time Warps: Einstein's Outmgeous Legacy. Norton, New York.

Toda, M. (1989): Nonlinear Waves and Solitons. Kluwer, Dordrecht. Triebel, H. (1972). H6here Analysis. Verlag der Wissenschaften, Berlin. Triebel, H. (1987): Analysis and Mathematical Physics. Kluwer, Dordrecht. Triebel, H. (1992): Theory of Function Spaces II. Birkhiiuser, Basel. Visintin, A. (1994): Differentiable Models of Hysteresis. Springer-Verlag,

Berlin, Heidelberg. Visintin, A. (1997): Models of Phase Transitions. Birkhiiuser, Basel. Wald, R. (1984): Geneml Relativity. The University of Chicago Press,

Chicago, IL. Weinberg, S. (1992): Dreams of a Final Theory. Pantheon Books, New

York. Weinberg, S. (1995/96): The Quantum Theory of Fields, Vols. 1, 2. Cam­

bridge University Press, Cambridge, UK. Wess, J. and Bagger, J. (1991): Supersymmetry and Superymvity. Second

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Wiegmann, P. (1996): Completely Solvable Models of Quantum Field The­ory. World Scientific, Singapore.

Wiggins, S. (1990): Introduction to Applied Dynamical Systems and Chaos. Springer-Verlag, New York.

Yosida, K. (1988): Functional Analysis. 5th edition. Springer-Verlag, New York.

Yosida, K. (1991): Lectures on Differential and Integml Equations. Dover, New York.

Zabczyk, J. (1992): Optimal Control Theory. Birkhauser, Basel. Zeidler, E. (1986): Nonlinear Functional Analysis and Its Applications. Vol.

1: Fixed-Point Theorems; Vol. 2A: Linear Monotone Opemtors; Vol. 2B: Nonlinear Monotone Opemtors; Vol. 3: Variational Methods and Opti­mization; Vols. 4, 5: Applications to Mathematical Physics. Springer­Verlag, New York. (Second enlarged edition of Vol. 1, 1992; second enlarged edition of Vol. 4, 1997, Vol. 5 in preparation.)

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References 457

Zeidler, E. (1996): Chapters 1-6 and 10-19 of Teubner-Taschenbuch der Mathematik, Vols. 1, 2. Teubner-Verlag, Stuttgart-Leipzig. (English edi­tion in preparation.)

Zinn-Justin, J. (1996): Quantum Field Theory and Critical Phenomena, 3rd edition. Clarendon Press, Oxford.

Zuily, C. (1988): Problems in Distributions and Partial Differential Equa­tions. North-Holland, Amsterdam.

Hints for Further Reading

Comprehensive collection of exercises: Kirillov and Gvishiani (1982).

History of functional analysis: Dieudonne (1981), Mackey (1992) (harmonic analysis).

International mathematical congresses: Albers, Alexanderson, and Reid (1987).

Biographies of Hilbert and Courant: Reid (1970), (1976).

A summary of important material from linear functional anal­ysis: appendices to Zeidler (1986), Vols. 1, 2B, and 3.

Comprehensive bibliographies: Zeidler (1986), Vols. 1-5.

Classical textbooks on linear functional analysis: Riesz and Nagy (1955), Schechter (1971), Rudin (1973), Kolmogorov and Fomin (1975), Kato (1976), Dunford and Schwartz (1988), Yosida (1988).

Nonlinear functional analysis: Berger (1977), Aubin and Ekeland (1983), Deimling (1985), Zeidler (1986ff), Vols. 1-5, Ambrosetti (1993).

Operator algebras: Kadison and Ringrose (1983), Vols. 1-4, Sunder (1987), Sakai (1991).

Generalized functions, pseudodifferential operators, and Four­ier integral operators: Hormander (1983), Vols. 1-4, Kanwal (1983).

Function spaces: Kufner, John, and Fucik (1980), Triebel (1992).

Applications to partial differential equations: Leis (1986), Zei­dler (1986), Vols. 1-5, Dautray and Lions (1990), Vols. 1-6, Alt (1992), Racke (1992), Giaquinta (1993), Renardy and Rogers (1993), Evans (1994), Smoller (1994), Amann (1995), Taylor (1996), Vols. 1-3.

Applications to the calculus of variations: Friedman (1982), Rabi­nowitz (1986), Zeidler (1986), Vol. 3, Mawhin and Willem (1987), Giaquinta and Hildebrandt (1995), Chang (1993), Struwe (1996), and Jost (1999).

Minimal surfaces: Dierkes, Hildebrandt, Kiister, and Wohlrab (1992).

Applications to integral equations: Kress (1989), Dautray and Lions (1990), Vol. 4.

458 References

Applications to optimization and mathematical economics: Lu­enberger (1969), Zeidler (1986), Vol. 3, Aubin (1993), Zabczyk (1993).

Numerical functional analysis: Zeidler (1986), Vols. 2A, 2B, and 3, Dautray and Lions (1990), Vols. 1-6, Louis (1989).

Scientific computing: Allgower and Georg (1990), LeVeque (1990), Golub and Ortega (1993), Deuflhard and Hohmann (1993), Deuflhard and Bornemann (1994), Quarteroni and Valli (1994), Hackbusch (1985), (1992), (1994), (1995), (1996), Stoer and Bulirsch (1993), Kornhuber (1997).

Applications to industrial problems: Friedman (1989/94), Vols. 1-8.

Applications to the natural sciences: Zeidler (1986), Vols. 4 and 5, Dautray and Lions (1990), Vols. 1-6, Grosche, Ziegler, and Zeidler (1995) (handbook).

Applications to mechanics: Marsden (1992).

Applications to celestial mechanics: Meyer and Hall (1992), Neutsch and Scherer (1992), Ambrosetti and Coti-Zelati (1993).

Applications to dynamical systems: Temam (1988), Amann (1990), Wiggins (1990), Hale and Ko<;ak (1991), Mielke (1991), Hofer and Zehnder (1994), Marsden and Ratiu (1994), Katok and Hasselblatt (1995).

Manifolds: Abraham, Marsden, and Ratiu (1983), Zeidler (1986), Vol. 4, Isham (1989).

Applications to mathematical biology: Murray (1989).

Applications to nonlinear elasticity: Ciarlet (1983), Zeidler (1986), Vol. 4, Antman (1994).

Applications to fluid mechanics: Zeidler (1986), Vol. 4, Galdi (1994), Marchioro and Pulvirenti (1994), Lions (1995), Vols. 1, 2.

Solitons: Novikov (1984), Toda (1989), Matveev (1994).

Applications to capillarity: Finn (1985).

Large scale dynamics of multi-particle systems: Spohn (1991), Cercigniani, Illner, and Pulvirenti (1995).

Hysteresis and phase transitions: Visentin (1994), (1997), Brokate and Sprekels (1996).

Semiconductors: Markowich (1990).

Plasma physics and fusion: Nishikawa and Wakatani (1993).

Symplectic techniques in physics: Guillemin and Sternberg (1990), Hofer and Zehnder (1994).

Applications to quantum mechanics: Reed and Simon (1972), Vols. 1-4, Prugovecki (1981), Schechter (1982), Berezin and Shubin (1991).

Quantum statistics: Bratelli and Robinson (1979), Diu et al. (1989), Haag (1993), Simon (1993), Grosse (1995), Petrina (1995), Honerkamp (1998).

References 459

Quantum field theory: Glimm and Jaffe (1981), Reed and Simon (1972), Vol. 2 (the Garding-Wightman axioms), Bogoljubov and Shirkov (1983), Mandl and Shaw (1989), Bogoljubov, et al. (1990), Chang (1990), Haag (1993), Kaku (1993), Weinberg (1995/96), Vols. 1,2, Scharf (1995), Zinn-Justin (1996), Greiner and Reinhardt (1996).

Scattering theory: Reed and Simon (1972), Vol. 3, Newton (1988), Colton and Kress (1992), Iagolnitzer (1993).

Elementary particles: Rolnick (1994).

Standard model of elementary particles: Nachtmann (1990), Donoghue, Golowich, and Holstein (1992), Kaku (1993), Weinberg (1996), Vol. 2.

Noncommutative geometry and the standard model of elemen­tary particles: Connes (1994).

The Feynman path integral: Albeverio (1975) as well as Albeverio and Brezniak (1993) (rigorous theory), Das (1993), Dittrich and Reutter (1994), Weinberg (1995/96), Vols. 1,2, Greiner and Reinhardt (1996), Zinn­Justin (1996), Grosche and Steiner (1998).

Cosmology: Zeidler (1986), Vol. 4, Peebles (1993).

Quantum cosmology: Esposito (1993).

Supersymmetry: Berezin (1987), Wess and Bagger (1991).

Superstring theory: Green, Schwarz, and Witten (1987), Kaku (1987), Lust and Theissen (1989), Hatfield (1992).

Quantum groups: Lusztig (1993), Kassel (1995).

Conformal field theory: Kaku (1991), Di Francesco et al. (1997).

Topology and physics: Nakahara (1990), Marathe and Martucci (1992), Monastirsky (1993), Schwarz (1994).

Topology, partial differential equations, pseudo differential op­erators, and the Atiyah-Singer index theorem: Gilkey (1984).

Textbooks in physics: Feynman, Leighton, and Sands (1963), Schmut­zer (1989), Vols. 1, 2, Greiner (1993), Vols. 1-7, Honerkamp and Romer (1993).

A survey on modern physics: Davies (1989).

Seminal papers in physics: Stroke (1995).

Essays on modern physics: Kaku and Trainer (1987), Heisenberg (1989), Weinberg (1992), Gell-Mann (1994), Schweber (1994), Scott and Davidson (1994), Thorne (1994).

Essays on modern mathematics: Cascuberta and Castellet (1992) (viewpoints of seven Fields medalists), Penrose (1992), (1994), Kac, Rota, and Schwartz (1994).

List of Symbols

What's in a name? That which we call a rose By any other word would smell as sweet.

General Notation

A=}B iff A{:}B f(x) := 2x xES x¢S {x: ... } S~T SeT SUT

SnT

S-T

A implies B if and only if

William Shakespeare (1564-1616) Romeo and Juliet 2,2

A iff B (i.e., A =} Band B =} A) f(x) = 2x by definition x is an element of the set S x is not an element of the set S set of all elements x with the property ... the set S is contained in the set T S ~ T and S =I- T (the set S is properly contained in T) the union of the sets Sand T (the set of all elements that live in S or T) the intersection of the sets Sand T (the set of all elements that live in Sand T) the difference set (the set of all elements that live in S and not in T) empty set set of all subsets of S (the power set of S)

462 List of Symbols

SxT {p} N R,C,Q,Z ][{

RN

][{N

Re z, 1m z

z Izl

[a,bj ja,b[ ja,bj [a,b[ sgnr Ojk

inf S

supS

minS

maxS

lim an n-+oo

limn-+ooan

product set {(x, y): xES and yET} set of the single point p set of the natural numbers 1,2, ... set of the real, complex, rational, integer numbers RarC set of all real N-tupels x = (Xl, ... ,XN) (Le., Xj E R for all j) set of all complex N-tupcls (Xl"'" XN) (Le., Xj E C for all j) RN or CN

real part of the complex number z = X + yi, imaginary part of z (Le., Re z := x, 1m z := y) conjugate complex number z := X - yi, absolute value of the complex number z, Izl := JX2 + y2 closed interval (the set {x E R: a :::::; x :::::; b}) open interval (the set {x E R: a < x < b}) half-open interval (the set {x E R: a < x :::::; b}) half-open interval (the set {x E R: a :::::; x < b}) signum of the real number r Kronecker symbol, Ojk := 1 if j = k, and Ojk := 0 if j =f:. k infimum of the set S of real numbers (the largest lower bound of S) supremum of the set S of real numbers (the smallest upper bound of S) the minimum of the set S of real numbers (the smallest element of S) the maximum of the set S of real numbers (the largest element of S) lower limit of the real sequence (an)

upper limit of the real sequence (an)

The Landau Symbols

f(x) = O(g(x)), If(x)1 :::::; constlg(x)I for all x in a neighborhood x -+ a of the point a

f(x) = o(g(x)), lim f(x) = 0 x-+a g(x)

List of Symbols 463

Norms and Inner Products

IIxll lim Xn = x

n-+oo

(or Xn ---+ x as n ---+ 00)

00

n=1

(x I Y)

(x I Y)

Ixl

Ixloo

(u I vh

(u I Vh,2

lIu111,2

(. I ·)E II·IIE

Operators

A:S ~ X ---+ Y

D(A) (or dom A) R(A) (or im A) N(A) (or ker A)

I (or id) A(S)

norm of x the sequence (xn ) converges to the point x

7 9

infinite series in a Banach space 76

inner product 105 N

Euclidean inner product, (x I Y) := L xnYn 109 n=1

(fh conjugate complex number to Yj) 1

Euclidean norm, Ixl := (x I x)! = (t, Ixnl2) "2 109

special norm, Ixloo := sup Ixnl 11 n

inner product on the Lebesgue spaces L2(G) 114

and L~(G), (u I vh := fa u(x)v(x)dx

norm on the Lebesgue spaces L2(G) and L~(G), 114 1

lIull2 := (u I u)! = (fa lu(xWdX) "2

inner product on the Sobolev space Wi(G), 120

(u I Vh,2 := 1 (uv + t8jU8jV) dx G 3=1

norm on the Sobolev space Wi(G), 1

11"11>., ,~ C" I ")1" ~ (fa (u' + ~ca;")' ) <Ix) , energetic inner product 273 energetic norm 273

operator from the set S into the set Y, where S ~ Y domain of definition of the operator A range (or image) of the operator A null space (or kernel) of the operator A, N(A) := {x: Ax = O} identical operator, Ix := x for all x image of the set S, A(S) := {Ax: XES}

17

17 17 70

76

464 List of Symbols

A-I(T) pre image of the set T, A-I(T) := {x: Ax E T} 17 A-I inverse operator to A 17 G(A) graph of the operator A, 414

G(A) := {(x,Ax):x E D(A)} IIAII norm of the linear operator A 70

IIIII norm of the functional I 75 AB (or A 0 B) the product of the operators A and B, 28

(AB)(u) := A(Bu) A~B the operator B is an extension of 260

the operator A A* adjoint operator to the linear operator A 263 AT dual operator to the linear operator A

(see Section 3.10 of AMS Vol. 109) A closure of the linear operator A 415 a(A) spectrum of the linear operator A 83 p(A) resolvent set of the linear operator A 83 r(A) spectral radius of the linear operator A 94 rank A rank of the linear operator A,

rank A := dim R(A) (see Section 3.9 of AMS Vol. 109)

ind A index of the linear operator A, ind A := dim N(A) - co dim R(A) (see Section 5.4 of AMS Vol. 109)

det A determinant of the matrix A tr A trace of the (N x N)-matrix

A = (akm), tr A := all + ... + aNN tr A trace of the linear operator A 347

in a Hilbert space

Special Sets

S closure of the set S 31 int S interior of the set S 31 ext S exterior of the set S 31 as boundary of the set S 31 Ue(p) E-neighborhood of the point p in a 15

normed space, Ue(p) := {x E X: Ilx - pil < d U(p) neighborhood of the point P 15 dim X dimension of the linear space X 6 Xc complexification of the linear space X 98 X/L factor space (see Section 3.9 of AMS Vol. 109) codim L co dimension of the linear subspace L,

codim L:= dim(X/L) (see Section 3.9 of AMS Vol. 109)

List of Symbols 465

L.L orthogonal complement to the linear 165 subspace L

as the product as := {ax: XES}, a E lR, C 7 S+T the sum S + T:= {x + y: xES and yET} 7 M ffi L orthogonal direct sum (M ffi L, 165

where L = M.L), X®Y tensor product 224 X* dual space 75 X E energetic space 273 span S linear hull of the set S 31 co S convex hull of the set S 31 co S closed convex hull of the set S 47 dist(p, S) distance of the point p from the set S diam S diameter of the set S 47 meas S measure of the set S 431 8(x) the Dirac delta function 158 8 the delta distribution 161

Derivatives

u' (t) derivative of an operator function 80 u = u(t) at time t

8j f partial derivative %1 J

8C1.f 8r'8~2 ... 8'ft f, where a = (al, ... , aN) 159 (the classical symbols are also used for the derivatives of generalized functions)

lal the sum al + ... + aN 159 8

derivative in the direction of 181 8n

the exterior normal N

!::.f Laplacian, !::.f := L 8} f 125 n=l

8F(x; h) variation of the functional F at the point x in direction of h (see Section 2.1 of AMS Vol. 109)

8n F(x; h) nth variation of the functional F at the point x in the direction of h (see Section 2.1 of AMS Vol. 109)

A'(x) (or dA(x)) Frechet-derivative of the operator A at the point x (see Section 4.2 of AMS Vol. 109)

dn A(x)(hl , ... , hn ) nth Frcchet-differential of the operator A at the point x in the directions of hI, ... ,hn (see Section 4.2 of AMS Vol. 109)

466 List of Symbols

Spaces of Continuous Functions

C[a, b], C(G) L(X, Y), Linv(X, Y)

Spaces of Holder Continuous Functions

C<>[a, b], Ck'<>[a, b], C<>(G), ck'<>(G) (C<>(G) = cO'<>(G))

Spaces of Smooth Functions

Ck[a, b], ck(G), Ck(G), COO(G), Ck(G)c (CO(G) := C(G)) CO'(G) (or V(G)), S

Spaces of Integrable Functions (Lebesgue Spaces)

L2(a, b), L2(G), L~(G) (L2(G) := L~(G) if lK = JR)

Sobolev Spaces

Spaces of Sequences

lKoo , l~, l~ (b := l~ if lK = JR)

Spaces of Distributions

V'(G), S'

14, 116 73,79

95ff

96, 116 116,214

130, 114

131, 132

95, 177

160, 219

List of Theorems

A good memory does not recall everything, but forgets the unimpor­tant.

Folklore

Theorem 1.A (The Banach fixed-point theorem) Theorem loB (The Brouwer fixed-point theorem) Theorem I.C (The Schauder fixed-point theorem) Theorem I.D (The Leray-Schauder principle) Theorem I.E (The method of sub- and supersolutions) Theorem 2.A (Main theorem on quadratic minimum problems) Theorem 2.B (The Dirichlet principle) Theorem 2.C (The Ritz method) Theorem 2.D (The perpendicular principle) Theorem 2.E (The Riesz theorem) Theorem 2.F (Dual quadratic variational problems) Theorem 2.G (Nonlinear monotone operators) Theorem 2.H (The nonlinear Lax-Milgram theorem) Theorem 3.A (Complete orthonormal systems) Theorem 4.A (Eigenvalues and eigenvectors of linear,

symmetric, compact operators) Theorem 4.B (The Fredholm alternative for linear, symmetric,

compact operators) Theorem 5.A (The Friedrichs extension of symmetric operators) Theorem 5.B (The abstract Dirichlet problem)

19 53 61 65 69

121 138 141 165 167 170 173 175 202

232

237 280 282

468 List of Theorems

Theorem 5.C (The eigenvalue problem) Theorem 5.D (The Fredholm alternative) Theorem 5.E (The abstract heat equation) Theorem 5.F (The abstract wave equation) Theorem 5.G (The abstract Schrodinger equation)

284 306 310 310 323

List of the Most Important Definitions

Intelligence consists of this; that we recognize the similarity of dif­ferent things and the difference between similar ones.

Spaces

linear space dimension linear subspace

Banach space norm separable

Baron de la Brede et de Montesquieu (1689-1755)

reflexive (see Section 2.8 of AMS Vol. 109) Hilbert space

inner product orthogonal elements orthogonal projection complete orthonormal system Fock space (bosons or fermions)

Lebesgue space Sobolev space

energetic space dual space

7 7

30 10 7

84

107 105 105 165 200 364 114 273 273

74

470 List of the Most Important Definitions

metric space and topological space (see Chapter 1 of AMS Vol. 109)

Convergence

norm convergence Cauchy sequence weak convergence (see Section 2.4 of AMS Vol. 109) sequentially continuous sequentially compact relatively sequentially compact

Operators

domain of definition range and preimage injective surjective bijective inverse operator linear symmetric

the Friedrichs extension adjoint

dual (cf. Section 3.10 of AMS Vol. 109) self-adjoint

Hamiltonian orthogonal projection operator

skew-adjoint unitary

Fourier transformation trace class

statistical state statistical operator Hilbert-Schmidt operator

continuous k-contraction Lipschitz continuous Holder continuous

homeomorphism diffeomorphism compact strongly monotone monotone or coercive (see Section 2.18 of AMS Vol. 109) semigroup

Green function (propagator)

8 10

27 33 33

17 17 17 17 17 17 70

264 280 263

264 328 270 264 212 216 347 348 350 347

26 19 27 97 28

436 39

273

298 386

List of the Most Important Definitions 471

one-parameter group 298 dynamics of a quantum system 328

Fredholm alternative 237 linear Fredholm operator and index

(see Section 5.4 of AMS Vol. 109) nonlinear Fredholm operator (see Section 5.15 of AMS Vol. 109)

m-linear bounded (see Section 4.1 of AMS Vol. 109)

Functional

nonlinear 17 linear 74 COnvex 29 bilinear form 120

bounded 120 symmetric 120

distribution (generalized function) 160 tempered distribution 219 Fourier transformation 220 generalized eigenfunction 344 Dirac delta distribution 161 Green function 160 fundamental solution 183

Palais~Smale condition (see Section 2.16 of AMS Vol. 109)

Embedding

continuous compact

Spectrum

eigenvalue and eigenvector generalized eigenvector resolvent set resolvent operator essential spectrum spectral family

Set

open

measurements in quantum systems

neighborhood interior

261 261

83 344

83 83 84

333 343

15 15 30

472 List of the Most Important Definitions

closed closure boundary

compact or relatively compact dense convex bounded countable

Point

fixed point critical point (see Section 2.1 of AMS Vol. 109) saddle point (see Section 2.2 of AMS Vol. 109) bifurcation point (see Section 5.12 of AMS Vol. 109)

Operator Algebras

Banach algebra von Neumann algebra C*-algebra observable state

pure mixed KMS-state (thermodynamic equilibrium)

* -automorphism dynamics of a quantum system

Derivative

time derivative generalized derivative of a function derivative of a distribution nth variation (see Section 2.1 of AMS Vol. 109) Frechet derivative (see Section 4.2 of AMS Vol. 109)

Integral

Lebesgue integral Lebesgue measure integration by parts Lebesgue-Stieltjes integral Feynman path integral

15 30 31 33 83 29 33 84

18

76 359 357 359 358 358 358 360 358 359

80 129 162

434 429 118 441 385

Subject Index

a posteriori error estimate 19 a priori error estimate 19 a priori estimates 64 absolute continuity 437 absolute integrability 435 absolute temperature 352 absolutely convergent 76 abstract boundary-eigenvalue

problem 255 abstract boundary-value problem

254 abstract Dirichlet problem 282 abstract Fourier series 200 abstract heat equation 255, 306 abstract Schrodinger equation

256, 323 abstract setting for quantum

mechanics 327 abstract setting of quantum

statistics 348 abstract wave equation 256, 310 action 393 addition theorems 305 adjoint operator 263 admissible paths 392

admissible sequence 274 algebraic approach to quantum

statistics 357 almost all 431 almost everywhere 431 annihilation operator 366 anticommutation relations 367 antilinear 169 Appolonius' identity 178 Arzela-Ascoli theorem 35 asymptotically free 368 *-automorphism 358

balls 16,94 Banach algebra 76 Banach fixed-point theorem 18 Banach space 10 barycenter 47 barycentric subdivision 49 basic equation of quantum

statistics 351 basis 42 Bernstein polynomials 86 Bessel inequality 202 beauty of functional analysis 256

474 Subject Index

Big Bang 357 bijective 17 bilinear form 120 black holes 340 Bolzano-Weierstrass theorem 34 Born approximation, 403, 404 Bose-Einstein statistics 355 bosonic Fock space 364 bosons 363 bound states 372, 394, 396 boundary 31 boundary point 31 boundary-eigenvalue problem

245, 285, 316 boundary-value problem 125 bounded 33 bounded bilinear form 120 bounded orbits 421 bounded sequence 9 Brouwer fixed-point theorem 53 Brownian motion 381

C* -algebra 357 calculus of variations 117, 126 Cauchy sequence 10 Cayley transform 423 characteristic equation 94 charge density 180 chemical potential 352 chronological operator 395 classical function spaces 95, 97 classical Schwarz inequality 12 closed 15 closed balls 16 closed linear subspace 30 closure 31 closure of an operator 414 closed convex hull 31 commutation relations 365 commutative C* -algebra 358 compact 90 compact embedding 96, 261 compact operator 39 compact set 33 compactness 33

complete 10 complete orthonormal system

200, 209, 210, 223, 232, 247, 374

complete system of generalized eigenfunctions 346

completeness relation 374 completeness theorem 222 complex linear space 4 complex normed space 7 complexification 98 complexification of real Hilbert

spaces 178 composite states of elementary

particles 227 conservation of energy 336, 404 continuity 26 continuous 27 continuous Dirac calculus 375 continuous embedding 261 continuous spectrum 409 contraction principle 19 convergence 9 convex 29 convex hull 31 convexity 29 convolution 182 Coulomb force 180 Coulomb potential 420 countable 84 creation operators 364 critical point 321

defect indices 423 deflection of a string 157 degenerate kernels 251 degrees of freedom 229 dense 84 density 84, 189, 222 derivative 80 diagonal sequence 36 diameter 47 dielectricity constant 420 diffeomorphism 435 differential operator 267

diffusion 379, 386 diffusion equation 379 dimension 6 Dirac calculus 222, 373 Dirac 8-distribution 161 Dirac 8-function 158, 346 Dirac function 375 Dirichlet principle 101, 123, 138 Dirichlet problem 125 discrete Dirac calculus 374 discrete spectrum 409 disk 28 dispersion 328 dispersion of the energy 352 dispersion of the particle number

352 distance 7, 47 distributions 160 domain of definition 17 dominated convergence 437 dual maximum problem 169 dual space 75 duality map 169, 279 duality of quadratic variational

problems 169 duality theory 142 dynamics 359 dynamical systems 303 dynamics of the harmonic

oscillator 341 dynamics of quantum systems

328 dynamics of statistical systems

349 Dyson formula 395

eigenfunction 247, 340 eigenoscillations 198 eigenoscillations of the string 317 eigensolution 230, 241 eigenspace 230 eigenstate 329 eigenvalue 83, 230 eigenvalue problem 283 eigenvector 230

Subject Index 475

elastic energy 256 elasticity 145 electric field 180 electric field of a charged point

180 electrostatics 183 electrostatic potential 180 embedding theorems 262 energetic extension 279 energetic inner product 123, 144,

273 energetic norm 144, 273 energetic space 144, 154, 273 energy 309 energy of the harmonic oscillator

309 energy conservation 303, 310,

313 entropy 349, 353 equicontinuous 35 equivalent norms 42, 99 error estimate 19, 142 error estimates via duality 142 essential spectrum 84, 372 Euclidean norm 12, 109 Euclidean strategy in quantum

physics 379 Euler-Lagrange equation 125 expansion of our universe 340,

357 exponential function 78 extension 261 extension principle 213 exterior 31 exterior point 31

Fatou lemma 110, 438 Fermi-Dirac statistics 356 fermions 363 fermionic Fock space 366 Feynman diagrams 400 Feynman formula 393 Feynman path integral 381, 385,

393

476 Subject Index

Feynman relation for transition amplitudes 397

Feynman-Kac formula 391 finite-dimensional Banach spaces

42 finite-dimensional space 6 finite elements 151 finite €-net 38 finite multiplicity 230 fixed point 19 Fock space 363 force density 156 formal Hamiltonian 339 Fourier coefficients 195, 200 Fourier integral 198 Fourier method 315, 316 Fourier series 195, 203, 241, 247 Fourier transform 216, 376, 378,

413 Fourier transform of tempered

generalized functions 219

Fredholm alternative 237, 245, 249, 284, 287

Fredholm integral operator 95 free Hamiltonian 371 Friedrichs extension 258, 280 Friedrichs extension in complex

Hilbert spaces 419 Friedrichs' mollification 186 Fubini's theorem 439 function 17 functional calculus 293, 331 functions of bounded variation

440 functions of self-adjoint

operators 293 fundamental solution 179, 182,

183, 186, 413 fundamental theorem of calculus

119

Gauss method of least squares 197

Gauss theorem 119

Gaussian functions 223 Gelfand-Kostyuchenko theorem

424 Gelfand-Levitan-Marchenko

integral equation 410 general position 45 generalized boundary values 135,

138 generalized derivative 129 generalized diffusion equation

386 generalized Dirichlet problem

138 generalized eigenfunctions 343,

372,377 generalized eigenvectors 424 generalized Fourier series 195 generalized functions 156, 160 generalized functions in

mathematical physics 179

generalized initial-value problem 185

generalized plane wave 184 generalized problem 285, 306,

310 generalized solution 258 generalized triangle inequality 8 generator 298 geometric series 79 golden rule for the rate of

convergence 143 golden rule of numerical analysis

143 graph 414 graph closed operators 414 Green function 147, 157, 164,

246, 251, 387, 392 group property 300

half-numberly spin 357 Hamiltonian 328, 371, 418 Hamiltonian of the harmonic

oscillator 341

Hamiltonian of the hydrogen atom 420

harmonic oscillations 196, 303 harmonic oscillation in quantum

mechanics 338 heat equation 182 Heisenberg S-operator 402 Heisenberg uncertainty principle

331, 342 Hermitean functions 210, 339 Hermitean polynomials 210 Hilbert space 107 Hilbert-Schmidt operator 347 Hilbert-Schmidt theory 232 Holder continuous functions 96,

97 homeomorphic 28 homeomorphism 28 homogeneity in time 302 *-homomorphism 358 hydrogen atom and the

Friedrichs extension 420

idea of orthogonality 124 ideal elements 127 identical operator 77 infinite series 76 infinite-dimensional space 6 initial-value problem 24 injective 17 inner product 105 integer spin 357 integrable 434 integration by parts 117, 119,

159 integral equations 22, 62, 240 integral operator 18, 40, 265 integral of a step function 432 iterated integration 438 interior 31 interior point 31 inverse Fourier transformation

216, 378 inverse operator 17

Subject Index 477

inverse scattering theory 406, 409

irreversible process in nature 300, 303

isometric operators 422 *-isomorphism 358 iteration method 18, 68, 395, 404

k-contractive 19 Kato perturbation theorem 417 KdV equation 407 kinetic energy 321, 336 KMS-states 360 Knaster, Kuratowski, and

Mazur-kiewicz lemma 58

Korteweg-de Vries equation 406

lack of classic solution 127 Lagrange multiplier rule 354 Laguerre functions 222 language of physicists 221, 373 Laplacian 125, 277, 285 Lax pair 412 Lax-Milgram theorem 174 least-squares method 201 Lebesgue integral 434 Lebesgue measure 429 Lebesgue spaces 114 Lebesgue-Stieltjes integral 441 Legendre polynomials 209 Leray-Schauder principle 64 linear combinations 2 linear continuous functional 74 linear hull 31 linear integral equation 23 linear Lax-Milgram theorem 175 linear operator 70 linear orthogonality principle 172 linear space 3 linear subspace 30 linearly independent 5 Lipman-Schwinger integral

equation 404 Lipschitz continuous 27

478 Subject Index

Luzin's theorem 432

majorant criterion 436 Malgrange-Ehrenpreis theorem

186 mapping degree 55 mass conservation 379 matrix 71 maximally skew-symmetric 267 maximally symmetric 267 maximum 37 Maxwell-Boltzmann statistics

356 mean continuity 437 measure 430 measurable functions 432 measurable set 430 measurements 328, 349 method of finite elements 145 microscattering processes 400 mild (generalized) solution 307,

311, 320, 324 minimal sequence 176 minimum 37 Minkowski functional 50 mixed state 358 models of quantum field theory

368 momentum operator 342, 345,

376 momentum vector 336 monotone convergence 437 monotone operators 173 multiindex 159 multiplication operator 269, 334 multiplicity 230

Navier-Stokes equations 65 neighborhood 15 Neumann series 79 Newtonian equation 336 nonexpansive semigroup 299 nonlinear Fourier transformation

413

nonlinear Lax-Milgram theorem 175

nonlinear mathematical physics 173

nonlinear orthogonality principle 174

norm 7 normal order cone 67 normed space 7 nuclear spaces 424 null space 70

observables 328, 359 one-dimensional wave 323 one-parameter group 299 one-parameter unitary group

301, 328 open 15 open neighborhood 15 operator 16 operator functions 76, 77, 257 operator norm 70 order cone 66 ordered Banach space 67 ordered normed space 67 ordered sets 441 ordinary differential equations

24, 63 orthogonal 105 orthogonal complement 165, 178 orthogonal decomposition 165 orthogonal projection 165, 270 orthogonality 124 orthogonality principle 172, 175 orthonormal system 199

parallelogram identity 123, 178 parameter integrals 439 Parseval equation 203, 222 particle number 352 particle number operator 366 particle stream 344, 403 partial differential equations of

mathematical physics 256

partition function 353 path integral 390 Pauli principle 356, 363 Payley-Wiener theorem 186, 223 Peano theorem 63 perpendicular principle 123, 165 phase space 393 photon 340 physical interpretation of the

Green function 158 physical states 327 Picard-Lindelof theorem 24 Planck quantum action 329 Planck's radiation law 340, 357 plane wave 184 Poincare inequality 287 Poincare-Friedrichs inequality

136 Poisson equation 125, 258, 285 position operator 342, 346, 377 positive bilinear form 120 potential 182 potential barrier 403 potential energy 321, 336 potential equation 182 potential theory 95 pre-Hilbert space 105 preimage 17 principle of indistinguishability

363 principle of maximal entropy 353 principle of minimal potential

energy 256 principle of minimal potential

errors 142 principle of stationary action 321 principle of virtual power 142,

147 principle of virtual work 147 probability 336, 352 probability conservation of

quantum processes 303 probability of measuring the

position 343 product rule 107

Subject Index 479

propagation of probability 393 propagator 387 pure state 358 Pythagorean theorem 167

quadratic variational problems 121

quantization 336 quantization of action 393 quantum field theory 363 quantum hypothesis 340 quantum mechanics 327, 336 quantum statistics 348 quantum system 327

range 17 rate of convergence 19, 142 real normed space 7 real linear space 4 regularity theory 128 relatively compact 90 relatively sequentially compact

33 Rellich's compactness theorem

287 resolvent 83 resolvent set 83 resonance condition 249 restriction 259 retraction 55 reversibility 302 reversible processes in nature 302 Riesz theorem 167 Ritz equation 140, 152 Ritz method 140, 151, 179

S-matrix 402 scalar multiplication 3 scattering of a particle stream

399 scattering theory 368 Schauder fixed-point theorem 61 Schauder operator 41 Schmidt orthogonalization

method 207

480 Subject Index

Schrodinger equation 323, 336, 338, 381, 395

Schwarz inequality 105 self-adjoint operator 264, 416 semi-Fredholm 83 semigroup 298 separable 84, 116 separability 191 sequentially compact 33 sequentially continuous 27 sharp energy 341 simple eigenvalue 247 simplex 46 skew-adjoint operator 264 skew-symmetric operator 264 smoothing of functions 186 smoothing technique 117 Sobolev embedding theorems 192 Sobolev space 130, 260, 277 solitons 406 spectral family 333 spectral radius 94 spectral transformation 410 spectrum 83, 94, 238 spectrum of the hydrogen atom

421 Sperner lemma 56 Sperner simplex 57 standard model in statistical

physics 351, 360 states 327, 359 stationary particle states 337 stationary Schrodinger equation

337 statistical operator 350 statistical potential 353 statistical states 348 step function 432 Stieltjes integral 333, 440 string 158, 315 string energy 320 string equation 321 strong causality 302 strongly continuous semigroup

298

strongly monotone operators 173, 259, 273

strongly positive bilinear form 120

subsolution 69 superselection rules 328 superposition 196 superposition of eigenoscillations

317 supersolution 69 surjective 17 symmetric bilinear form 120 symmetric operator 230, 264,

415

temperature tempered delta distribution 221 tempered distributions 220 tensor product 224, 226 tensor product of functions 184 tensor product of generalized

functions 185 thermodynamic equilibrium 360 thermodynamical quantities 353 time evolution 329 time evolution of quantum

systems 330 time-dependent processes in

nature 298 time-dependent scattering theory

398 time-independent scattering

theory 404 Tonelli's theorem 439 total energy 321, 336 totally ordered 442 trace class operators 347, 421 transformation rule for integrals

435 transition amplitude 397 transition probabilities 397 Trefftz method 170 triangle inequality 7 triangulation 49

trigonometric polynomials 204, 206

two-soliton solutions 407

unbounded operator 300 unbounded orbits 421 uncertainty inequality 330 uncertainty principle 343 uniformly continuous 27 uniformly continuous one-para-

meter group 299 uniqueness implies existence 237 unitary operator 212, 219, 272,

330

variance 328 variational equation 140 variational lemma 117 variational problem 125, 140,

281,287

Subject Index 481

vertices 46 vibrating string 315 Volterra integral operator 95 volume potential 183 von Neumann algebra 359

waves 406 wave equation 182, 185, 309 wave operators 369 Weierstrass approximation

theorem 84 Weierstrass classical

counterexample 176 Weierstrass theorem 37 white dwarfs 357 Wiener path integral 381,386

Zorn's lemma 442