Basic Perturbation Theorems
First, we give some definitions (see [293, p 162] for a more detailed discussion)
We denote by A and B, densely-defined linear operators in a Hilbert space H, and by D(A) and Q(A), the operator domain and form domain of A respectively
Definition 1.1 Let A be self-adjoint Then B is said to be A-bounded if and only if
(ii) there are constants a, b > 0 such that
IIBcpll S aiiAcpll + bllcpll for cpeD(A)
The infimum of all such a is called the A-bound (or relative-bound) of B
There is an analogous notion for quadratic forms:
Definition 1.2 Let A be self-adjoint and bounded from below Then a symmetric operator B is said to be A-form bounded if and only if
(ii) there are constants a, b > 0 such that l(cp,Bcp)l < a(cp,Acp) + b(cp,cp) for cpeQ(A)
The infimum of all such a is called the A-jorn1-bound (relative form-bound) of B
It is important to note that the operators mentioned in the definitions do not necessarily have to be self-adjoint or symmetric However, we emphasize the self-adjoint case here, as it simplifies the formulation and proof of subsequent propositions.
A subspace in His called a core for A if it is dense in D(A) in the graph norm
It is called a form core if it is dense in Q(A) in the form norm
There is an elementary criterion for relative boundedness
Proposition 1.3 (i) Assume A to be self-adjoint and D(A) c D(B) Then B is A-bounded if and only if B(A + i)- 1 is bounded The A-bound of B is equal to lim II B(A + iiã)- 1 II ltãl-x
In the context of self-adjoint operators, if A is self-adjoint and bounded from below, and if the quadratic form associated with A is contained within that of B (denoted as Q(A) ⊆ Q(B)), then B is said to be A-form-bounded This condition is equivalent to the boundedness of the expression (A + i)^{-1/2} B (A + i)^{-1/2} Furthermore, the A-form-bound of B can be characterized by the limit of the norm ||(A + ii')^{-1/2} B (A + ii')^{-1/2}|| as ii' approaches zero.
The assertion (i) is demonstrated by substituting cp with (A + iy)⁻¹ in equation (1.1) and noting that ||B(A + iy)⁻¹|| < [a + (b/|y|)] Similarly, assertion (ii) can be derived in a parallel manner Additionally, we occasionally refer to an extended concept where we state that B is
A-compact if and only if B(A + i)- 1 is compact Here i can be replaced by any point of the resolvent set
Now we will state the basic perturbation theorem which was proven by Kato over 30 years ago, and which works for most perturbations of practical interest
The Kato-Rellich theorem states that if A is a self-adjoint operator and B is a symmetric operator that is A-bounded with an A-bound less than 1, then the operator A + B, defined on the domain of A, is also self-adjoint Furthermore, any core for the operator A serves as a core for the operator A + B.
We give a sketch of the proof Note that self-adjointness of A is equivalent to Ran(A + iJI) = H for some Jl > 0 [292, Theorem VIII.3] Then, as above, we conclude from ( 1.1) that
Thus, for JIIarge enough C := B(A + iJI)- 1 has norm less than 1, and this implies that Ran( 1 + C) = H This, together with the equation
( 1 + C)(A + iJI)cp =(A + B + iJI)cp cp e D(A) and the self-adjointness of A, implies that Ran(A + B + iJI) = H The second part of the theorem is a simple consequence of ( 1.1 )
Kato and Wust have made significant advancements regarding the case where a equals 1, yet it is important to note that most perturbations encountered in the theory of Schrödinger operators exhibit a relative bound of 0.
There is also a form version of Theorem 1.4 (due to Kato, Lax, Lions, Milgram and Nelson):
Theorem 1.5 (KLMN) Suppose that A is self-adjoint and bounded from below and that B is symmetric and A-bounded with form-bound a< 1 Then
(i) the sum of the quadratic forms of A and B is a closed symmetric form on Q(A) which is bounded from below
(ii) There exists a unique self-adjoint operator associated with this form which we call the form sum of A and B
(iii) Any form core for A is also a form core for A + B
For a proof, see [293, Theorem X.l7] We will denote the form sum by A + B when we want to emphasize the form character of the sum otherwise we will write A +B
Despite the similarities between operators and forms, a key asymmetry exists; while symmetric operators can be closed yet not self-adjoint, a closed form that is bounded from below corresponds to a unique self-adjoint operator Furthermore, the concept of essential self-adjointness has a form analogue, where a suitable set is defined as a form core Defining a closed quadratic form guarantees that the associated operator is self-adjoint, although it does not provide information about the operator or form domains Notably, identifying a convenient set, such as C~, as a form core is a significant and nontrivial observation.
TheClassesSv andKv
In this book, we will study the sum -A + V in virtually all cases But occasion- ally we will also study (- iV + a) 2 + Vas operators or forms in the Hilbert space
In the context of L²(ℝⁿ), V represents a real-valued function that describes the electrostatic potential, while a denotes a vector-valued function for the magnetic potential The self-adjoint representation of -A in L²(ℝⁿ) is denoted as H₀ In many cases, V can be viewed as a perturbation of H₀, which is supported by the uncertainty principle that allows kinetic energy to mitigate certain singularities of V, provided they are not excessively severe This phenomenon lacks a classical counterpart and is practically significant, as the Laplacian features a well-defined eigenfunction expansion and integral kernel, providing comprehensive insights into operator cores and related aspects.
There are two classes of perturbations we will discuss here The class Sv, which is an (almost maximal) class of operator perturbations of H 0 and the class
K v which is the form analog of Sv Sv was introduced originally by Stummel [352], and has been discussed by several authors (see e.g [308] )
Definition 1.6 Let V be a real-valued, measurable function on ~v We say that
The conditions for V and Sv to be equivalent are as follows: a) The limit of the supremum of the integral of the squared difference between x and y, weighted by the function V(y), approaches zero as v exceeds 4 b) The limit of the supremum of the integral of the logarithmic difference between x and y, also weighted by V(y), approaches zero when v equals 4 c) The supremum of the integral of the squared function V(y) remains finite when v is less than or equal to 3.
For the reader who is disturbed by the lack of symmetry in the above definition, we remark that for v s 3, sup J I V(y)l 2 dvy < 00
We define a Svãnorm on Sv by
In the context of the discussed theorem, we define the kernel \( K \) in relation to the operator norm \( \| \cdot \|_{L^p} \) for operators mapping from \( L^p(\Omega) \) to \( L^4(\Omega) \) This theorem illustrates the natural emergence of these quantities within the framework of functional analysis.
Theorem 1.7 V e Sv if and only if lim II (Ho + E)- 2 1 Vl 2 II oo oo = 0
Proof As with all functions of H 0 , (H 0 + E)- 2 is a convolution operator with an explicit kernel Q(x - y, E) [293, Theorem IX.29] It has the following properties (see [308, Theorem 3.1, Chap 6] )
2 Q(x- y,E) = OC(Inlx- yl- 1 ) if v = 4 as lx- yl +0, if v ~ 3
3 sup elx-yiQ(x - y, E) + 0 as E + oo, for any ~ > 0 lx-yl>6
The condition that sup J I V(y)l² dy < ∞ and x |x - y| for any V ∈ S" implies that V ∈ S" if and only if supx J Q(x - y, E) |V(y)|² dy → 0 as E → ∞ This conclusion arises from the fact that Q(ã - y, E) |V(y)|² serves as a positive integral kernel, and the property ||A||∞, X > = ||A||:x:ã holds for any A with a positive integral kernel.
The above result has an L 2 consequence by a standard "duality and inter- polation" argument:
Proof Let V e S" Then it is enough to show that
(1.4) since ( 1.3) follows then by Theorem 1 7 Assume for a moment that Vis bounded, and consider the function
F(z) is an operator-valued function that is L1 and L-bounded, exhibiting analyticity within the interior of the strip defined by {z ∈ ℂ | Re z ∈ [0, 1]} According to the Stein interpolation theorem [293, Theorem IX.21] and utilizing duality principles, significant implications arise from this framework.
II (Ho + E)- 2 1 Vl 2 II oc oo = II I VI 2 (Ho + E)- 2 II 1.1 , we get
( 1.4) follows for bounded V"s and by an approximation argument, also for all
Remark Note that Corollary 1.8 implies that if V e Sv, then it is H 0 -bounded with
H 0 -bound 0 by Proposition 1.3 (Proposition 1.3 has to be slightly modified for the semibounded case we are considering here)
One might think that since Sv is telling us something about L 1 -bounds and
L x is "stronger" than L 2 , there would be no way going from L 2 -bounds to Sv
So the following theorem is interesting
Theorem 1.9 Suppose there are a, b > 0 and a ~ with 0 < ~ < 1 such that, for all
IIVcpll~ < E;IIHocpll~ + aexp(bE;- 6 )IIcpll~
Proof We just have to pick the right cp"s Fix y e ~v, t e ~+, and consider the integral kernel cp(x) := Jexp(- tH 0 )(x, y)
Then, noting that II cp 11 2 = 1 and (by scaling)
IIH 0 cpll 2 = ct- 2 for suitable c > 0 we have
Now, take E;:= (1 + llntl)-", where y:= 2/(1 + ~), and multiply (1.5) by t exp(-tE) forE> 0 Then the R.H.S of (1.5) is integrable in t and its integral goes to zero as E + ~-Now if we use the identity
0 we get (1.2), and therefore VeSv by Theorem 1.7 D
The second class of potentials under consideration is K V', an analog of Sv, first introduced by Kato For related classes, refer to Schechter K V has been studied in detail in various contexts.
Aize1unan and Simon [1], and Simon [334]
Definition 1.10 Let V be a real-valued measurable function on ~v We say that
V E Kv if and only if a) lim[sup J lx- yl 2 -viV(y)ldvy] = 0, if v > 2
1.2 The Classes Sw and K w 7 b) lim [sup J lnl(x - y)l- 1 1 V(y)l dvy] = 0, if v = 2 al,O x lx-yiSa c) sup J I V(y)ldvy < oo, if v = 1 x lx-}'1 S I
We also define a Kv-norm by
II vII" w := sup J K (x, y; v) I v ( y) I d v y x lx-)'1 S I where K is the kernel in the above definition of Kv Then virtually everything goes through as before
Theorem 1.11 [7] VeKv if and only if lim II ( H o + E) -I I VI II x :o = 0
The proof is the same as in Theorem 1 7
Theorem 1.12 [7] Suppose there are a, b > 0 and a ~ with 0 < ~ < 1 such that, for all 0 < e < 1 and all cp e Q(H 0 )
(cp, I VI cp) S e(cp, H 0 cp) + aexp(be- 6 ) llcplli
The proof mirrors that of Theorem 1.9 (refer to [7, Theorem 4.9] for additional context) Notably, both classes Sv and Kv exhibit beneficial properties: if J1 is a subset of Sv, then the inclusion K~J is contained within Kv, and S~J is contained within Sv This means that if we have K~J (or S~J), there exists a linear surjective map T: ~v +.
In the context of an N-body system, we define J and V(x) as W(T(x)), where V belongs to Kv (or Sv) Here, a point x in ~v represents an N-tuple of 11-dimensional vectors, expressed as x = (x1, , xN), with Tx defined as xi - xj for distinct indices i and j Additionally, there are LP-estimates available that provide criteria for determining when a potential falls within Kv.
L~nif C Sv if {p>~ for v>4 p=2 for v ~ for v>2 p=2 for vO,
The function F(z) is analytic in the complex variable z When the real part of z equals zero, the magnitude of F(z) is less than a certain exponential factor, as the imaginary part contributes only a phase factor, which can be demonstrated using the Trotter product formula Conversely, when the real part of z equals two, the expression simplifies to a phase factor.
Therefore, by Hadamard's three line theorem [293, p 33] we conclude, for Rez =I
< [ (exp(-tH 0 ) 1/1 2 , cp)] 112 { (exp[-t(H 0 + 2 V)] I, cp)} 112 •
Furthermore, since exp(-tH) is positivity preserving [293, Theorem X 55], we have le-'H/1 < e-rH 1/1 , and hence l(exp(-tH)f,cp)l 2 < (exp(-tH 0 )1/I 2 ,cp)(exp[ -t(H 0 + 2V)]I,cp) (2.4)
Now let x e IR", and choose cp(x') := 0, llt/111 1 =I Then (2.4) reads as lexp(-tH)f • 0, we define the set B as {y | ||x - y|| < r} contained in Q Under these conditions, the inequality ||u(x)|| < c ∫ ||u(y)|| dν holds, where c is a constant that depends on r and the K v-norm of V_X.s (with X.8 being the characteristic function of B) Notably, if V is in Kv and Q equals IRV, the constant c can be selected independently of x.
This estimate is particularly valuable because if \( u \) is an eigenfunction belonging to \( L^2 \), it indicates that \( u \) approaches zero pointwise at \( x \) Furthermore, when there is exponential decay observed in an average sense, it leads to pointwise exponential decay as well.
In the context of the equation H u = Eu, it is important to recognize that V can be substituted with V - E, ensuring that the assertions remain valid, albeit with constants that depend on £ The equation (2.5) is identified as a subsolution estimate; when u is greater than zero, it suffices to establish the distributional inequality H u < 0, indicating that u merely needs to be a subsolution for (2.5) to hold true, as demonstrated in [7, Theorem 6.1] Additionally, this result extends the well-established estimates pertaining to (sub-)harmonic functions when V equals zero.
Theorem 2.5 (Harnack Inequality) [7, 334, 131 ] Suppose V e Kioc' let Q c !Rv be an open set, Hu = Eu (in the distributional sense), u =/= 0 and u > 0 on Q Let
K be a compact set K c Q Then the following estimate holds u(x) c- 1 < 0 independent of r Let lV, := '7,u/ II '7,u II 2 •
" for a suitable subsequence {r,.} c N This implies Oea(H), since we have a Weyl sequence (see [292, Theorem VII.12])
Since Hu = 0 and we get
Thus, since II Li'7, II XJ and II v,, II oo are uniformly bounded,
IIH'7,ull 2 < c' J (lul 2 + IP'ul 2 )d"x
Let M(r) := Jcr lul 2 d"x Then (2.12) implies
Now assume there is no subsequence {r,.} such that
Then there exists a R e N and an ~ > 0 such that
Thus by induction we get
But this means that M(R) has an exponential growth which is a contradiction of the hypothesis that u grows polynomially D
Renzark A direct consequence of our proof of this theorem is that if Hu = Eu, u is polynomially bounded and u ~ L 2 • then E e ac 55 (H)
A kind of converse of Theorem 2.9 can also be proven, using trace class-valued measures and eigenfunction expansions It can be found in Simon's review article
[334 Theorem C5.4] or in [219] See also [46] We state it without proof
An assertion A(E) is said to hold H-spectrally almost everywhere if and only if EA(H) equals zero Here, A represents the set of elements E in R such that A(E) does not hold, and E.i(H) denotes the spectral projection of H onto A.
Theorem 2.10 [334] If V e K", then H-spectrally a.e there exists a polynomially- bounded solution of Hu = Eu
Note that this does not imply that for any E e a(H) Hu = Eu has a poly- nomially-bounded solution! Combining these two theorems one gets
Corollary 2.11 If V e K", then a( H) is the closure of the set of all E for which
Hu = Eu has a polynomially-bounded solution
Proof If A c 1R is the set where Hu = Eu has polynomially-bounded solutions, then b~ Theorem 2.9, A c a(H) Suppose that A is not dense in a(H) Then a( H)\ A contains a open set S c a(H) with E 5 (H) > 0 But this contradicts Theorem 2.1 0 D
The Allegretto-Piepenbrink Theorem
This article explores a theorem initially demonstrated by Allegretto, Piepenbrink, and Moss, which asserts that eigenvalues situated below the spectrum possess positive eigensolutions We will provide a proof of this theorem under minimal regularity conditions.
Theorem 2.12 Let v_ E Kv and v+ E Kroaã Then Hu = Eu has a nonzero distribu- tional solution which is everywhere nonnegative if and only if inf a(H) > E
Proof Suppose inf a( H)> E Let { f, }ne ~ be a sequence of C0-functions which are nonzero and positive with suppf, c {xe IR"In < lxl < 2n}
Let un := cn(H - E + n- 1 )- 1 f,., where en is chosen such that un(O) = l un is every- where nonnegative since (H - E + n- 1 ) has a positivity preserving resolvent
In the region where |x| < n, the function un satisfies the equation un = (E - n - 1)un, indicating a clear relationship According to Harnack's inequality, for any radius R > 0, the function un(x) remains positive within this range, allowing us to normalize it so that un(0) = 1 Furthermore, Harnack's inequality ensures the existence of a constant CR > 0 that maintains this property.
By considering a subsequence, we can establish that the sequence \( u_n \) converges to a limit point \( u \) in the weak-star sense of \( L^\infty \) This implies that for all compactly supported \( \varphi \in L^1 \), the relationship \( (u_n \cdot \varphi) \) converges to \( (u \cdot \varphi) \) as \( n \) approaches infinity Furthermore, it is evident that \( u \) satisfies the distributional equation \( Hu = Eu \), confirming that \( u \) is nonnegative and not identically zero.
Conversely suppose Hu = Eu has a nonzero nonnegative solution By Harnack's inequality u is strictly positive, and by Theorem 2.7
We will prove that for cp e C 0
(cp.(H- E)cp) = !IIVcp- gcpll~ • (2.13) which implies that H - E > 0
We first prove (2.13) assuming u E ex Then, by direct calculation (as operators) so proving (2.13) in that case
Under the given assumptions, we establish that the function \( u \) is continuous and remains locally bounded away from zero, as supported by Theorems 2.4 and 2.5 We define \( u_d \) as the convolution of \( u \) with an approximate identity \( j_d \) Furthermore, we introduce \( V \) as \( u_i (Au) + E \) and \( g_d \) as \( u_i t V_u t J \) Consequently, we derive that \( (H_0 + \lambda) u_t J = E u_t J \), which aligns with our previous findings.
(cp.(Ho + ~- E)cp) = II Vcp - 0JJCfJII 2 •
But since utJ + u local uniformly (u is continuous!) and VutJ + Vu in Lroc• we have that OJJ + g in Lroc and ~ + V in Lloc as b + 0 This proves (2.13) in general D
Agmon [4] has made a deep and complete analysis of all positive solutions of Hu = Eu if V is periodic.
Integral Kernels for exp(- tH)
Some operators possess integral kernels, which is a reassuring insight A widely applicable theorem, known as the Dunford and Pettis theorem, guarantees the existence of such integral kernels (refer to Treves [358]).
Theorem 2.13 Let (M,J.l) be a separable measure space, and !I' a separable Banach space Let A be a bounded operator from !£ to L 00 (M, dp) Then there exists a unique (up to sets of p-measure zero) weakly measurable function K from
M to !I'* such that, for each f e !I' and a.e x eM
In particular, choosing !i' = LP(M,dp), 1 < p < oo, so that !I'* = L 4 (M,dp) with q- 1 + p- 1 = 1, and noting the trivial converse of Theorem 2.13, we have
Corollary 2.14 If A is bounded from LP to L oo, then there is a measurable function,
K, on M x M obeying sup[JIK(x,y)l 4 d"y] 114 = IIAIIp.OCJ < oo, (2.14) xeM M so that, for any f e LP
Conversely, if A: LP + LP has an integral kernel Kin the sense of (2.15) obeying (2.14), then A is a bounded map from LP to L 00 •
Research indicates that the semigroup exp(-tH) possesses a uniformly-bounded and jointly continuous integral kernel, as outlined in Theorem 87.1 However, the findings related to exp(-tH) are significantly less robust An examination of the "free" case reveals additional insights into this phenomenon.
V = 0) shows we cannot hope that there are integral operators in the sense of (2.14) and (2.15) for p = 2, since this kernel has no decay Thus, we need a weaker notion of integral kernel
We say that an operator A has a \Veak integral kernel K(x, y) if and only if
2.6 Integral Kernels for exp(- tH) 25
K e L 1 1 0 c(IR" x IR"), and for all L ~-functions with compact support f, g we have
Then we have the following result
Theorem 2.15 Suppose Vis a C 00 -function obeying
I(D 2 V)(x)l < C2(1 + lxlftrl) forallmulti-indices~whereeitherk(~) = k 0 -l~lwithk0 < 1 ork(~) = O(Cô > 0 suitable) Then exp(- it H) has a weak integral kernel P(x, y, t) for all t #; 0, and it is jointly c~ on IR" X IR" X (IR\ { 0} )
This was proven by Fujiwara [ 120, 121] in a series of papers; see also Fujiwara
[122] and Kitada [209], Kitada and Kumanogo [211] See also Zelditch [380] for an alternative proof The restrictions on V are undoubtedly too strong Zelditch [380] has eased the conditions, e.g he has the following
Theorem 2.16 Let V(x) = L:'=t ~(~x) where xe IR", ~is a function on IR"• with
~(k)e L 2 (1R"~), and~ is a linear map of R" onto R"~ Then exp( -it H) has a weak integral kernel
We give here a proof different from that of Zelditch, due to Cycon, Leinfelder and Simon [73] (see also [266] )
Proof For simplicity, we assume for a moment m = 1 and V = V 2 • We have
We know that exp(- itH 0 ) is bounded from L 1 to L ~ So, by Corollary 2.14, it is enough to show that
U(t):= e-irHeirHo is bounded from L :X, to L oo Assume first that V e C0 Then
0 where V(s) := exp(isH 0 ) V exp(- isH 0 ) Note that V(s) has the integral kernel
Now we expand U(t) by the Dyson- Phillips expansion
U(t) = L Q,.(t) ' n=O where Q,.(t) are integral operators with kernels Q,.(x, y, t) defined by Q 0 (x, y, t) := 1 and n-1 n
Q,.(x, y, t) := J n dsi J n dxi V(xi-t, xi; si) ,
0 ~ S 1 ~ S 2 Sn ~ f j = 1 ~ ~ j = 1 And we estimate the operators
IIQ II X 7 < o~.L ~ã [( dsj J fl dxj(2ns)-'' v c~j-~s~ xj) t"
IIU(t)llx.x < L IIQ,.IIx.x O #a=2 such a constant C exists by compactness D
One can use a little geometry in place of compactness and obtain an explicit value for C; see Simon [323]
The following proposition states properties of the Ruelle-Simon partition of unity that are crucial for the proof of the HVZ-theorem
Proof (i) V Ja is continuous and homogeneous of degree - 1 near infinity, so it tends to zero at infinity Hence V J 11 (H 0 + 1 )- 1 is compact (see e.g Reed and Simon
(ii) We prove (ii) for fii e C~; the general case follows by a straightforward approximation argument For Iii e C 0, the function J 11 1 11 has compact support (while 1 11 itself does not!) D
HVZ-Theorem 3.7 For a cluster decomposition a, define E(a) := inf u(H(a)) and
The HVZ-theorem was proven by Zhislin [382], van Winter [359] and
More on the Essential Spectrum
The essential spectrum of Schrodinger operators is fundamentally influenced by behavior at infinity, a concept supported by two theorems outlined in this section Unlike the HVZ-theorem, these theorems provide a less explicit determination of the essential spectrum, yet they are applicable to a broader range of potentials, including those that do not decay at infinity, such as periodic, almost periodic, and random potentials, which are excluded from the HVZ-theorem.
The crucial property ofSchrodinger operators that makes the above "general wisdom" true is local compactness, a concept particularly emphasized in the work of Enss (see e.g [ 1 00] )
Definition 3.9 A Schrodinger operator H = H 0 + V is said to have the local compactness property if f(x)(H + i)- 1 is compact for any bounded function f with compact support
Most Schrödinger operators relevant in physics exhibit the local compactness property For instance, when the potential V is either operator bounded or form bounded relative to H0, the Hamiltonian H demonstrates this local compactness characteristic.
In this section, we will consider the assumption that the Vis operator is bounded, without imposing any decay conditions at infinity This leads us to a straightforward lemma regarding these operators.
Lemma 3.10 Suppose that V is operator bounded with respect to H 0 • Let f be a bounded function with compact support Then both f(x)(H + i)- 1 and f(x)V(H + i)- 1 are compact operators
The first term in the above expression is compact, the others are bounded The proof that f(x)(H + i)- 1 is compact is obvious from the above D
We now state and prove the first of the announced theorems We denote, by
B" := { xllxl ~ n }, the ball around the origin, of radius n
Theorem 3.11 Let V be operator bounded with respect to H 0 • H := H 0 + V Then
;, e ucss(H) if and only if there exists a sequence of functions cp" e C0 (IR"\Bn) with
II CfJn ll2 = 1 such that
Remark ( 1) By the Weyl criterion, we know that ; e t1c 55 (H) is equivalent to the existence of a sequence of trial functions { cp"} obeying II CfJn 11 2 = 1 and CfJn ~ 0 with
3.4 More on the Essential Spectrum 37
( 3.18) Thus, Theorem 3.11 tells us that the weak convergence of the cp, actually takes place in a particular way
(2) The theorem can be proven under much weaker conditions on V All we have to ensure is that the conclusions of Lemma 3.10 remain true and that c: is an operator core
Proof By remark (1), the "~"-direction is trivial Let us assume that ) e t1c 55(H)
By the Weyl criterion there exists a sequence 1/1, e C0 (R"), II 1/1,.11 2 = 1, 1/1, ~ 0 such that
For any n choose a function x, e CXJ, 0 < x,.(x) ~ 1 such that x,.(x) = 1 for lxl > n + 1, and x,.(x) = 0 for lxl < n We claim that, for any n, there exists an i = i(n) > n such that
Assuming this for the moment, we set
(3.20) (3.21) (3.22) which goes to zero by (3.19-22) Thus, it remains to prove (3.20-22) For this, let X be a C0 -function, then: llxt/1,.11 = llx(H + i)- 1 [(H - ; ) + (i + ) )]1/1,.11
The first term goes to zero because of (3.19); the second one goes to zero since t/1, ~ 0 and x(H + i)- 1 is compact by Lemma 3.10 This proves (3.20) and (3.21)
There are numerous related results, such as ucss(H) = n , u(H + nx:xllxl ~~~}) ã
We now prove a result due to Persson [282] that gives the infimum of the essential spectrum in terms of a ããmin-max''-type expression:
Theorem 3.12 (Persson) Let V be operator bounded Then infucss(H) = sup inf (cp,Hcp)
KcR• qH;C 0 (1Rw Kt compact l,(p ã =1
Remarks 1) Persson's theorem says that inf ucss is not effected by ããwhat happens" in any compact set
2) For any fixed K, the term inf (cp,Hcp)
The ground-state energy for the Hamiltonian H on L² (ℝⁿ) with Dirichlet boundary conditions at cK is denoted as pECo(ifi' Kt) According to Reed and Simon IV, the infimum of the essential spectrum of H (ucss(H)) represents the supremum of all these ground state energies.
3) Theorem 3.12 can be proven under weaker assumptions See the book of
Let \( \alpha_0 \) be the infimum of \( O'c 55 (H) \) According to Theorem 3.11, we can find a sequence \( \phi_n \in C_0(\mathbb{R} \setminus B_n) \) such that \( \|\phi_n\| = 1 \) and \( (H - \alpha_0) \phi_n \to 0 \) Consequently, we have \( \sup \inf (\phi, H\phi) = \lim \inf (\phi, H\phi) < \lim (\phi_n, H\phi_n) \).
K (p e Co ( iR' K t n (p e Co ( R • Bn) n compact (p,l =1 l'(p:: =1
Jln := sup inf (cp, Hcp) t/11 "'"- 1 (p e c-o
According to the min-max theorem, the nth eigenvalue of the operator H, denoted as J.ln, is determined by counting multiplicities from the lowest value If there are fewer than n eigenvalues below the infimum of the spectrum of H, then J.ln equals J.ln+1, which also equals the infimum Furthermore, if there are infinitely many eigenvalues below the infimum of the spectrum, then J.ln approaches the infimum of the spectrum of H.
; 0 := inf O'c 55 (H) < sup inf ( cp, H cp) =: \'o ,
K (pECo(R• Kt l(pt!=1 it suffices to show that \'o > J.ln for all n Since
J.l 1 = inf (cp,Hcp) , qJECo(ifi't
A Theorem of Klaus: Widely Separated Bumps
Suppose now n > 1 and v 0 > Jl,.- 1 • If JJ,.- 1 = ; 0 , we are done If JJ,.- 1 < ; 0 , then
Jl, = inf (cp, Hcp) , fPlPa• ã ã • •Pn -I
\vhere the Pi are normalized eigenfunctions corresponding to the eigenvalues JJ;, and moreover, p 1 , ••• , p,._ 1 span the eigenspaces of the JJiã Now choose e > 0 and
Define for any cp the function cP(X) := (1 - C't) inf - C~: fPE Co(ifi' Ko) < (/), qJ)
Since c was arbitrary, we have
3.5 A Theorem of Klaus: Widely Separated Bumps
Before we turn to applications of geometric ideas in atomic physics, we use geometric methods in a different context:
Theorem 3.13(Klaus) Assume VeC 0 (1R), V < 0 Let {x,.:,.ez be a sequence of real numbers satisfying x, < x,.+ 1 and lx,.- x,.+ 1 1 ~ oc as In I~ oc
Define W := Lnez V(x- x,.), H := -d 2 /dx 2 + W and H' := -d 2 /dx 2 + V
Remark The above theorem is a special case of a theorem due to Klaus [214], who proved it by Birman-Schwinger techniques
The negative eigenvalues of H' are isolated points within the essential spectrum of H, indicating they do not belong to the continuous spectrum Furthermore, due to the one-dimensional nature of the problem, these eigenvalues cannot exhibit infinite multiplicity Consequently, they must be considered accumulation points of the discrete spectrum of H.
The HVZ theorem indicates that a potential decaying at infinity prevents certain phenomena from occurring However, in Chapters 9 and 10, we will explore intriguing examples of unexpected spectral phenomena, including singular continuous spectra and dense point spectra.
Proof The direction "ae 55 (H) => a(H')" can be proven by a standard application of Weyl's criterion We only argue ''a(H') => t1e 55 (H)" Let us define V,.(x) =
V(x- x,.) and H, = -d 2 jdx 2 + V, For notational convenience, we will assume that supp V, n supp V'" = ¢J for n # m Under this assumption, we may choose a partition of unity { j,.} II e Z With the following properties:
(ii) j,.jm = 0 if In - ml > 2,
(iii) j, e C0 and I Vj,.lxã ~ 0, IAi,.l~ + 0 as In I + oo
It is not difficult to see that A(z) is bounded and analytic as a function of z on
D := C\a(H') (The reader may adjust the proof of the lemma below.) We will show that
The equation A(z) = (H - z)⁻¹ [1 + B(z)] describes the behavior of compact operators B(z) that are analytic on the domain D According to the analytic Fredholm theorem, the inverse of 1 + B(z) exists on D\D for a discrete set within D As z approaches negative infinity, the norm of B(z) approaches zero, indicating that 1 + B(z) is invertible for certain values of z Additionally, the residues at the poles are finite rank operators, allowing us to extend (H - z)⁻¹ to an analytic function on D\D, with finite rank operator residues at the points of D This leads to the conclusion that the intersection of D and the essential spectrum of H is empty, thereby achieving the desired result.
Here B,.(z) is compact and analytic on U We compute:
Applications to Atomic Physics: A Warm-Up
To prove that B(z) = L 8 11(z) is well defined and compact, we make use of the following lemma:
Lemma Let C11, n e 7L be bounded operators, and let f 11 , g 11 be bounded func- tions satisfying supp f, n supp f, = f/J, and supp g 11 n supp g, = ¢J for n # m If
II f,.C 11 g 11 II -+ 0 as In I -+ oo, then the series
Proof Denote by l 11 and '711 the characteristic functions of supp f, and supp g 11 , respectively Then
< sup llf C~~g~~ll L llx~~.PIIII'7~~cpll lni>M I11I>M
< t; II t/111 II cp II for M large enough D
Proof of the Theorem (continued) Now we write B(z) as
We apply the lemma to any of the four terms separately Since we have shown norm convergence, we conclude that B(z) is compact D
3.6 Applications to Atomic Physics: A Warm-Up
In this chapter, we will focus on questions related to atomic physics, starting with a simplified example that will provide valuable insights for more complex issues We will examine the Hamiltonian to better understand these concepts.
The operator acting on L2, denoted as (1R 2 3), describes the dynamics of two electrons influenced by an infinitely heavy nucleus, with the repulsion strength between the electrons represented by A Here, r1 is defined as |x1| and r12 as |x1 - x2|.
For large electron repulsion, specifically when A is significantly greater than 1, it is anticipated that H(A) will not exhibit bound states This assertion will be demonstrated using the localization formula According to Lieb's method discussed in Section 3.8, it can be established that bound states do not exist when A exceeds 2 Numerical analysis, as referenced in Reinhardt [296], indicates that the critical threshold appears to be approximately 1.03.
The HVZ-theorem tells us that O'c 55 (H(A)) = [ -!, x ), since inf a(- A 1 -
1/r 1 ) = -!.Thus, the expected result is equivalent to
Proposition 3.14 For A sufficiently large, we have H(A) > -!
In this proof, we select a partition of unity consisting of functions j0, j1, and j2, which possess specific properties: the support of j0 is contained within the set {x | |x| < 1}, while the supports of j1 and j2 are contained in the sets {x | |x| > 1 and |x2| > |x|} and {x | |x2| > |x|} respectively Additionally, j1 and j2 are homogeneous functions of degree zero outside the unit sphere.
To dominate the localization error L I Vj;l 2 , we may choose A 0 sufficiently large such that
This choice of A 0 is possible because j 1 and j 2 are homogeneous of degree zero for r large, while j 0 has compact support By the IMS-Iocalization formula, we can write
Notice that it is the long-range nature of the Coulomb interaction that helps us to control the localization error
Next we observe that, for any c > 0 and sufficiently large A, we have
3.7 The Ruskai-Sigal Theorem 43 for all x with lxl < 1 and lx 1 l, lx 2 1 >c Thus, for large A
The characteristic function of the ball of radius \( t \) is denoted as \( X_t \) It is established that the Hamiltonian described in equation (3.28) lacks bound states when \( t \) is sufficiently small, as supported by various bounds on the number of bound states (refer to [295], Theorem XIII.10) Consequently, if \( A \) exceeds a certain threshold, specifically \( A_0 \), the inequality \( \| H(A - A_0) \| > 0 \) holds true Additionally, within the support of \( j_1 \), the relationship \( \| x_1 - x_2 \| < \| x_1 \| + \| x_2 \| < 3 |X_t| \) is valid Therefore, for \( A > A_0 + 3 \), we find that \( \| H(A - A_0) \| t > \| H(-A_1 - A_2) \| \).
21)it • and by symmetry i2H(A- Ao)j2 > i2( -At- A2 -l:tl)h ã
Since -A;- Ai- 1/lxil > -1/4, we conclude for sufficiently large A D
In this discussion, the importance of the region near zero is emphasized, as it prevents the localization error from becoming O(r^-2), a situation that cannot be managed by Coulomb potentials.
Now we come to an important application of geometric methods in atomic physics We consider the Hamiltonian of an atom with nucleus charge Z, and N electrons: ã'*'( z) 1
A nucleus with charge Z can only attract a finite number of electrons, as the attractive force of the nucleus will eventually be outweighed by the repulsive forces among the electrons.
Let us give a more mathematical formulation of this expectation Define
Then, by the HVZ-theorem
Thus, our expectation can be formulated as
E(N + 1,Z) = E(N,Z) (3.31) for large N We emphasize that we are dealing with the discrete spectrum exclusively Thus, we make no assertion on embedded eigenvalues
Theorem 3.15 (Ruskai-Sigal Theorem) For any Z, there exists Nmaa(Z) such that
E(N + 1,Z) = E(N,Z) for all N > Nmaa(Z) Moreover, for fermionic particles, we have
Remarks ( 1) Theorem 3.15 was proven by Ruskai [302, 303] and by Sigal
(2) There exists an improved version of the Ruskai-Sigal theorem due to Lieb
[231, 232] which gives Nmaa(Z) < 2Z for all integers Z We present this theorem, as well as Lieb's elegant proof, in Sect 3.8 Our proof below, however, follows
Sigal's proof, while more extensive than Lieb's, is presented for its deeper physical insights into the phenomena involved Additionally, it leads to an enhancement of the Ruskai-Sigal theorem by Lieb, Sigal, Simon, and Thirring, which establishes that the limit in equation (3.32) is indeed 1 The refinement of Sigal's proof by Lieb, Sigal, Simon, and Thirring will be detailed below.
In the second part of our proof, we will consider the fermionic nature of our particles by applying the Pauli exclusion principle It is essential to ensure that our discussion focuses on Hamiltonians that are limited to antisymmetric states For a more detailed examination of this topic, we recommend consulting Appendix 4 of Sigal's paper.
Proof Sketch of the Ideas We divide the configuration space into N + 1 pieces:
A 0 , A 1 , ••• , ANã The first part, A 0 , consists of the region where all the electrons are close to the nucleus, and Ai essentially consists of the region where the ith
3.7 The Ruskai-Sigal Theorem 45 particle has larger distance to the nucleus than any other electron We then construct a partition of unity, J;, with supp J; c A; and with good control on the localization error L~=o IV J;(x)l 2 • On A 0 , the strong repulsion between the elec- trons will dominate both the attraction by the nucleus and the localization error, provided N is sufficiently large On A; (i > 1), we split HN into an (N - I)-body operator HN-t corresponding to the electrons 1, 2, , i - 1, i + 1, , N and the additional terms due to the ith electron Since that one is further from the nucleus than any other electron, the distance between the electrons i and j is at most twice the distance of the ith electron from the nucleus Therefore, the repulsion between electron i and the other electrons dominates the attraction of the ith electron by the nucleus as well as the localization error if N is large enough
Details of the Proof Define i=t ,N
A; := { xllx;l > (I - c5)X00(X), X00(X) > ~} , where p and b < 1/2 are positive numbers that will be fixed later on We will eventually choose pin an N-dependent way
We will create a partition of unity, denoted as { Ji }~=0, where the support of each Ji is contained within A Additionally, we will highlight key estimates regarding the gradients of the Ji and will provide their proofs at the conclusion of this section.
Lemma 3.16 There exists a partition of unity, {J;}~=o' with suppJi c: Ai such that the following estimates hold:
L IV J;(xW ~ AN on Ai j > I (3.34) i=o x~(x)p for a suitable constant A
Proof of Theorem 3.15 (continued) We set L(x) = L~=o IV J;(x)l 2 • By the IMS localization formula, we have
HN = J 0 (HN - L(x))J 0 + L J;(HN - L(x))J; (3.35) i= I Using (3.33), we estimate:
Jo(HN - L)Jo > Jo ( t (-A, - ~~~) + t, lx, ~ xi I - A; 2 112 ) } 0
Observe that we used -A;- Z/lx;l > -iZ 2 , and lx;- xil < 2p on suppJo For i =F 0, we define
We used above that X;> (1 - b)x~(x) on A; Thus, we proved
HN > L J;EN- 1 J; > EN-l if N is sufficiently large i=l
To obtain the additional result for fermions, we choose p N-dependent: p := ,N-t;J
Then the estimate of J;(HN - L)J;, i # 0, reads
2 (I - b) - Z - '1 will be eventually positive if Z = !( 1 - 2J)N and N is sufficiently large
To enhance our estimate of J 0 (HN - L)J 0, we must refine the rough approximation provided by equation (3.36) for the asymptotics of Nmax By incorporating the Pauli exclusion principle into our calculations, we can achieve a more accurate estimation.
With this estimate, we obtain
Again, with the above choice of Z, the term in brackets is positive for appropriate
Thus, we have proved that
1m < - - z-x Z 1- 2() which gives the desired result since b > 0 was arbitrary D
Our boson proof establishes that Nmax(Z) scales as O(Z²) While Lemma 3.16 can be enhanced to demonstrate Nmax(Z) = O(Z¹⁺ᵗ), as referenced in Sigal [313], this appears to be the optimal outcome achievable with the current methodology Additionally, Lieb's approach indicates that for bosons, Nmax(Z) is constrained to less than 2Z + 1.
Proof of Lemma 3.16 Let t/1 be a ex-function on IR satisfying 0 < t/l(t) S 1 and t/l(t) = 1 fort> 1 - e, t/J(t) = 0 fort < 1 - b; 0 < e 1 - b, hence t.lfi(x)l 2 = 1 - xe~x)) 2 + x(x~~x)) 2 t x(x 1 :;~)) 2
Lieb's Improvement of the Ruskai-Sigal Theorem SO
In this section, we present Lieb's simple proof of an improved version of Theorem 3.15 We use the notations of Sect 3 7
Corollary If Z is an integer, then Nmax(Z) ~ 2Z
In particular, the Corollary tells us that the ion H 2 - has no bound states, i.e it is unstable To prove Theorem 3.17, we will use the following lemma:
Lemma 3.18 If qJ e L 2 (R 3 ) and qJ e D(- Lf) n D(lxl), then Re(qJ.Ixl (- Lf)qJ) > 0
Proof If the function f is sufficiently regular, one has
Choosing f(x) = lxl- 1 and multiplying (3.41) by lxl from both sides, we obtain formally
!
(1x111/1.( EN-1- EN- L11 -~~~+it, lx1 ~ xil)l/1)
= (lxllt/J,(EN-1- EN)t/1) + (lx1lt/J, -Alt/1)
J=2 whereweused(lxdt/J,(HN-l- EN_ 1)t/l) = Jlxd(t/lx 1 ,(HN-l- EN- 1)t/lx.)dx1 >
By symmetry of the above formulae, we obtain, replacing x 1 by X; and summing over i,
Since lx; - xil < lx;l + lxil' we get
Remarks ( 1) One can show if N > 2Z + 1, then EN is not an eigenvalue
(2) Lieh [232] treats multi-center problems and various other refinements.
N-Body Systems with Finitely Many Bound States
The HVZ-theorem tells us that the infimum E of the essential spectrum of H is always defined by two cluster decompositions, i.e
In contrast to that, the question whether adis(H) is finite or infinite depends, in part, on
In many instances, the quantity adis(H) is finite when E3 > E However, when E3 equals E, the operator H = -A + Li E, then the finiteness of adis(H) holds for short-range potentials V and for once negatively charged ions These findings are derived from an abstract theorem (Theorem 3.23), which will be proven first The results discussed trace back to Zhislin and his collaborators, with the formulation and proofs attributed to Sigal For further reading, additional references can be found in Sigal's work.
We first introduce an appropriate partition of unity
Definition 3.19 A partition of unity { ia}a indexed by all cluster decompositions a of { 1, 2, , N} is called a Deift-Agmon-Sigal partition of unity if
(i) each ia is homogeneous of degree zero outside the unit sphere,
(ii) {lxl > 1} n suppja c: {x = (x 1, xN}IIx;- x 1 1 > Clxl whenever (ij) ¢a}, with a suitable constant C,
(iii) for two distinct cluster decompositions a and a' with #a = #a' = 2, we have { lxl > 1} n suppja n SUPPiaã = l/J
Related partitions were first identified in Delft and Simon, highlighting their significance as noted by Sigal An existence proof for a DAS partition of unity can be established by slightly modifying the proof of Proposition 3.5, which addresses the existence of the Ruelle-Simon partition of unity Similar to the Ruelle-Simon partition, each iala is relatively compact if the V; 1 are, as indicated in Proposition 3.6 A key estimate for the localization error plays a vital role in proving the finiteness of adis.
Proposition 3.20 For any c > 0, there exists a C,_ such that outside the unit sphere
3.9 N-Body Systems with Finitely Many Bound States 53
It is established that a two-body potential W, which decays at infinity like |x|^{-2}, does not generate an infinite number of bound states when the constant a is sufficiently small, as noted by Reed and Simon This indicates that the localization error will not lead to an infinite number of bound states for two-cluster Hamiltonians Additionally, due to the condition E3 > E, Hamiltonians with three or more clusters will also have only a finite number of bound states below E, defined as inf ac:ss(H), as demonstrated in Lemma 3.22.
Proo.f By (i), Via is homogeneous of degree minus one; therefore it suffices to show (3.45) on the unit sphereS We consider the set
By Definition 3.19(iii), x e A implies ia(x) = I for exactly one a with #a = 2 and iaã(X) = 0 for any other decomposition a' Hence, I Vjb(x)l = 0 for any cluster decomposition, b Thus,
Consider now A 6 = { x e SIL #a=l j;(x) > I - b }; 0 < b < 1/2 Taking b = b(e) small enough, we can assure, by (3.46), that
For each cluster decomposition a let la denote the characteristic function of suppja The IMS localization formula tells us that
To demonstrate that H possesses a finite number of bound states, it is sufficient to establish that each term on the right-hand side of equation (3.47) has a finite number of bound states This conclusion follows from the relevant lemma and the understanding that the j's represent a partition of unity.
Lemma 3.21 Let A, B be self-adjoint operators (i) If A > B, then the number
N(A, E) of bound states of A below E = inf acss(B) satisfies N(A, E)< N(B, E) (ii) If both A and B have a finite number of bound states below 0, inf acss(A) >
0, inf acss(B) > 0, and A + B is essentially self-adjoint on D(A) n D(B), then A + B has finitely many bound states below 0
Proof (i) is easily proven using the min-max principle (see e.g Reed and Simon
Let P and Q represent the projections onto the eigenspaces corresponding to the negative eigenvalues of operators A and B Consequently, AP and BQ are finite rank operators Additionally, it follows that A + B is greater than AP + BQ, which is also a finite rank operator By applying the previous result, we achieve the desired conclusion.
Since we suppose E 3 > E, it is easy to see that the terms in (3.47) resulting from three and more clusters contribute only a finite number of bound states
Lemma 3.22 Fix c If E 3 > E, the number of bound states of Hb + lblb-
Cc( 1 + lxl 2 )- 1 below E is finite
Proof lbl.b- C£(1 + lxl 2 )- 1 is H(b)-compact, hence infac:ss(H(h) + lblb- Ct(1 + lxl 2 )- 1 ) = infac:ss(H(b)) > E3 > E
Hence the number of bound states below E is finite by the definition of a disã D
Using the above considerations, H has finitely many bound states below
E, if all the two cluster terms in (3.47) have (Actually, only those with inf ac:ss(H(a)) = E have to be considered.) We therefore investigate now those terms more carefully
Let a be a decomposition into two clusters We saw already in Sect.3.2 that the Hilbert space L 2 (X) splits into
L 2 (X) = L 2 (Xa) ® L 2 (Xa) , and moreover see (3.13, 14) Since #a = 2, we have X a ~ R~ Of course, ~za itself splits into two parts corresponding to the two clusters in a
The normalized ground state of ~za is denoted as pa, which is recognized for being nondegenerate, as established in Reed and Simon [295], Xl11.12 We represent the projection operator from L2(Xa) onto pa as P" Additionally, we define P(a) as 1 x a ®.
P", where 1xa denotes the projection on L 2 (Xa) onto the whole space and we set
3.9 N-Body Systems with Finitely Many Bound States 55
We will use below the brackets ( ã, ã) to denote the scalar product in any of the spaces L 2 (X), L 2 (Xa), L 2 (Xa) It should be clear from the context which one is meant
To state the ""abstract" theorem, we introduce the potential "Ya 6 on X a:
For any \( b > 0 \), the brackets \( ( ã, ã ) \) in equation (3.48) represent the scalar product in \( L^2(X_a) \) It is important to note that the operator \( H( "Ya 6 ) := -A + "Ya 6 \) functions as a one-body operator on \( L^2(X_a) \sim L^2(R^n) \) The subsequent theorem simplifies the issue of the finiteness of \( adis(H) \) by linking it to the analysis of \( H( "Ya 6 ) \).
Theorem 3.23 Suppose E 3 > E If, for any two cluster decomposition a, the (one-body) operator -(1 - 'I)Lf + "Ya 6 has finitely many bound states for all £5 > 0, and a suitable '1 > 0, then H has finitely many bound states
Remarks ( 1) It will become clear in the proof that the condition on - (I - 'I)Lf +
"Ya 6 need only be required for those a with inf a(Ha) = E
(2) For the treatment of Theorem 3.23 on the fermionic subspace, see Sigal
Verifying the conditions of the theorem for a specific potential can be challenging, which is why we refer to it as "abstract." In the following sections, we will introduce two significant classes of examples to illustrate this concept.
Proof Let ia be a Deift-Agmon-Sigal partition of unity By the IMS-localization formula and Proposition 3.20, we have
+ L ia(H(a) + lala- Ct:(l + lxl 2 )- 1 )ja
What remains to be proven is the finiteness of the discrete spectrum of each of the H(a) + lala- e(l + lxl 2 )- 1 • We set
The term Q(a)K(a)Q(a) contributes only a finite number of bound states, as indicated by the condition inf aess(Q(a)K(a)Q(a)) > E To assess the impact of the mixed terms in equation (3.49), we apply a specific decoupling inequality.
Lemma 3.24 (Combes-Simon Decoupling Inequality [323]) Let A be a self- adjoint operator, let P be a projection, and set Q = 1 - P Then for any b > 0
Before we prove Lemma 3.24, we continue the proof of the theorem Applying the lemma to K (a) and P(a), we get
The last term still has the infimum of its essential spectrum above E provided we take b small enough Furthermore, take
( 1/Ja ® (/)2, K (a)(t/Ja ® (/)2)) > ( (/)2, - d(/)2) + ( (/)2' ( 1/Ja, laX.at/Ja) (/)2)
- e(l + lxl~)- 1 ]({J 2 ) + E , where lxl 2 is meant on L 2 (X), while lxl~ is meant on L 2 (Xa) Therefore, we obtain
In a similar way we get
In total, we have shown that
The Hamiltonian represented as -(1 - '7)J + ~ 6 exhibits a finite discrete spectrum Additionally, the expression - '7A - 2c(1 + lxl~)-1 also possesses a finite discrete spectrum when the parameter e is sufficiently small, as noted by Reed and Simon Consequently, applying Lemma 3.24 leads us to the theorem's conclusion.
We now prove the Combes-Simon decoupling inequality
3.9 N-Body Systems with Finitely Many Bound States 57
We estimate the mixed terms by the Schwarz inequality: l(qJ,QAPqJ)I = 1(ô5t'2QqJ,ô5-t'2QAPqJ)I
< (ô5ti2QqJ, ô5t12QqJ) 112 (b-112QAPqJ, ô5-112QAPqJ) 112
< !< (ô5ti2QqJ, ô5t12QqJ) + (b-112QAPqJ, ô5-112QAPqJ))
Estimating Re(P AQ) in a similar way and inserting in (3.50), we obtain
We now present two applications of the above ""abstract" theorem:
Theorem 3.25 Assume dimension Jl > 3 If the potentials V; 1 belong to L~ 12 (R~) for p > 3, and to L 2 (R~) for p = 3, and if furthermore E 3 > E, then H = H 0 +
L V; 1 has only finitely many bound states
We demonstrate that the negative component of ~6 is included in L~12 (R~), indicating that - (I - '7)A + ~6 possesses only a finite number of bound states below zero, as established by the Cwickei-Lieb-Rosenbljum bound (refer to Reed and Simon IV, [295], XIII.12) This conclusion supports the assertion made in Theorem 3.23.
Let a= {A 1 ,A 2 } Define M" = LieA"m; We can write any xeX as (x 1 + y 1 , x2 + Y2, 'Xn + Yn) with (x., 'Xn)E XCI and
Since xeX, we have M 1 y 1 = -M 2 y 2 for ieA 1 , jeA 2 • Therefore X ; - x 1 =
Y; - y 1 + X; - x 1 = y + X; - x 1 with y = Y; - y 1 We see from this that
( 1/1 a, V;i 1/1 a ) = f 11/1 a ( x) 1 2 V; 1 ( y + x; - x 1) dx xca is a convolution It is well known that
(see Reed and Simon IV, [295], XIII.39) Thus, the Young inequality tells us that the convolution (1/Ja, V; 1 1/1a) E L~ 12 (~~) The term (1/Ja, 1; Xat/1°) can be handled by the estimate
As above, ( 1/Ja, V;fl/la > is a convolution Assume first JJ > 3 Since V;i e L~J 12 , we have V;f e L#J 14 • By the Young inequality this implies (1/Ja, V;fl/la) e L~J 12 (since
1/Ja e L 4 for any q) For JJ = 3, V;i e L 2, hence V;f e L 1 • Again we conclude that
Since the third term, (1/Ja, V;il/la) 2 is positive, we are done D
Finally, we state without proof
Theorem 3.26 Once negatively charged bosonic ions have only a finite number of bound states
Fermions can present a challenge due to the possibility of a degenerate ground state, which may introduce a dipole term in the effective potential However, if the ground state is nondegenerate, Theorem 3.26 remains applicable.
(2) We can apply Theorem 3.23 only to negative ions of charge I, since we do not know 1: 3 > E for higher charges For an ion of charge - k, E 3 > E means
The equation E = E(Z + k - l, Z) = F E(Z + k - 2, Z) = E 3 highlights a perplexing aspect of atomic physics: an ion with charge -k + I may lack bound states, while its counterpart with charge -k possesses infinitely many bound states This discrepancy underscores our limited mathematical understanding of atomic physics, revealing the complexities and contradictions that still exist in the field.
Appendix: The Stone-Weierstrass Gavotte
This book demonstrates that proving assertions for the resolvents (H - z)⁻¹ of an operator H is relatively straightforward, whereas providing a direct proof for the function f(H) for an arbitrary function f ∈ C₀(R) is significantly more challenging.
However, it is in many cases easy to deduce this seemingly stronger assertion from the knowledge that it holds for resolvents, i.e for the functions f(x) =