Abstract
We present a simplified proof of the von Neumann’s Quantum Ergodic Theorem. This important result was initially published in German by von Neumann in 1929. We are interested here in the time evolution \(\psi _t\), \(t\ge 0\) (for large times) under the Schrodinger equation associated with a given fixed Hamiltonian \(H : \mathcal {H} \rightarrow \mathcal {H}\) and a general initial condition \(\psi _0\). The dimension of the Hilbert space \(\mathcal {H}\) is finite.
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1 Introduction
Consider a fixed Hamiltonian H (a complex self-adjoint operator) acting on a complex Hilbert space \(\mathcal {H}\) of dimension D, where \(D\ge 3\). Then, \(\mathcal {H}\) can be written as
where each \(\mathcal {V}_a\), \(a=1,2,...,K\), is the subspace of eigenvectors associated with the eigenvalue \(\lambda _a\), and \(\lambda _1< \lambda _2<...<\lambda _K.\)
We fixed an initial condition \(\psi _0\) for the dynamic Schrodinger evolution. We consider the time evolution \(\psi _t = e^{-i\,t\,H} (\psi _0)\), \(t \ge 0\), and we are interested in properties for most of the large times (not all large times).
Now, we consider another decomposition \(\mathcal {D}\) of \(\mathcal {H}\) (which has nothing to do with the previous one)
We can consider a natural probability on the set \(\Delta \) of possible decompositions \(\mathcal {D}\) and we are interested here in properties for most of the decompositions \(\mathcal {D}\). For small \(\delta >0\), we are interested in the concept of a \((1-\delta )\) generic decomposition \(\mathcal {D}\) (in the probabilistic sense).
For a given fixed subspace \(\mathcal {H}_\nu \) of \(\mathcal {H}\), \(\nu =1,...,N\), the observable \(P_{\mathcal {H}_\nu }\) (the orthogonal projection on \(\mathcal {H}_\nu \)) is such that the mean value of the state \(\psi _t\), \(t \ge 0\), is given by \(E_{\psi _t} (P_{\mathcal {H}_\nu }) =<P_{\mathcal {H}_\nu }(\psi _t), \psi _t>= |P_{\mathcal {H}_\nu }(\psi _t)\,|^2.\)
In the first part of the paper, following the basic guidelines of the original work by von Neumann, we present lower bound conditions (in terms of \(\delta \), etc) on the dimensions \(d_\nu \), \(\nu =1,2,..,N\), of the different values of \(\mathcal {H}_\nu \) of a \((1-\delta )\)-generic orthogonal decomposition \(\mathcal {D}\) of the form \(\mathcal {H}\,=\, \mathcal {H}_1\, \oplus ...\oplus \mathcal {H}_N\), in such way that the dynamic time evolution \(\psi _t\), \(t \ge 0\), of a given \(\psi _0\), for most of the large times t, has the property that the expected value \(E_{\psi _t} (P_{\mathcal {H}_\nu }) \) is almost \(\frac{d_\nu }{D}\). In this way, there is an approximately uniform spreading of \(\psi _t\) among the different values of \(\mathcal {H}_\nu \) of a generic decomposition \(\mathcal {D}\). In this part, the main result is Theorem 15. We point out that these estimates are for a fixed initial condition \(\psi _0\).
The von Neumann’s Quantum Ergodic Theorem provides uniform estimates for all \(\psi _0\). This result is presented in Theorem 19. This will be done in the second part of the paper which begins in Sect. 4. To get this theorem, it will be necessary to assume hypothesis on the eigenvalues of the Hamiltonian H (see hypothesis \(\mathfrak {N\,\,R}\) just after Lemma 16).
Suppose, for instance, that \(A: \mathcal {H} \rightarrow \mathcal {H}\) is an observable and this self-adjoint operator has spectral decomposition
where \( \mathcal {H}_p\), \(p=1,...,N\) is the subspace of eigenvectors associated with the eigenvalue \(\beta _p\) and \(\beta _1< \beta _2<...<\beta _N.\) The probability that the measurement of A on the state \(\psi _t\) is \(\beta _p\) is given by \(<P_{\mathcal {H}_p }(\psi _t), \psi _t>\). This shows the relevance of the result. The point of view here is not to look for generic observables but for generic decompositions.
We stress a point raised on [3]. What is proved is a property of the kind: for most \(\mathcal {D}\), something is true for all \(\psi _0\). In addition, not a property of the kind: for all \(\psi _0\), something is true for most \(\mathcal {D}.\)
Of course, the main result can also be stated in terms of limits, when \(T\rightarrow \infty \), of means \( \frac{1}{T} \int E_{\psi _t} (P_{\mathcal {H}_\nu }) \mathrm{d}t\), which is a more close expression to the one present in the classical Ergodic Theorem.
We present here a simplified proof (with less hypothesis in some parts) when dim \(\mathcal {H}\) is finite of this important result which was initially published in German by von Neumann in 1929 (see [6]). The paper [5] presents a translation from German to English of this work of von Neumann. This 1929 paper also considers the concept of Entropy for such setting. We will not consider this topic in our note.
Several papers with interesting discussions about this work appeared recently (see, for instance, [1,2,3, 5] and other papers which mention these four)
Consider a general connected compact Riemannian manifold X and its volume form. When properly normalized, this procedure defines a natural probability \(w_X\) over X.
Given a compact Lie group (real) G, one can consider the associated bi-invariant Riemannian metric. If H is a closed subset of G, this metric can be considered in the quotient space \(X= \frac{G}{H}\), and in this way, we get a probability on such manifold X. We will denote by \(\pi \) the projection.
When we consider expected values of a function f, this we will be taken with respect to the above-mentioned probability.
Lemma 1
Given a continuous function \(f:X \rightarrow \mathbb {C}\) and \(\pi : G\rightarrow X\) the canonical projection, then
for every Borel set \(S\subset X\), and
The first integral is taken with respect to the volume form \(w_X\) and the second with respect to the volume form \(w_G\).
Note that vol \((G)=\) vol \((X)\,\) vol (H).
The proof is left for the reader.
Suppose \(\mathcal {H}\) is a complex Hilbert space of finite dimension D with an inner product \(<,\,>\) and a norm \(|\,\,\,|\).
Suppose we fix a decomposition \( \mathcal {D}\), that is
\( N>1\), is a orthogonal direct sum, where dim \(\mathcal {H}_\nu =d_\nu >0\) for all \(\nu =1,2,...,N.\)
Denote \(P_\nu \) the orthogonal projection of \( \mathcal {H}\,\) over \(\mathcal {H}_\nu \).
Moreover, \(S=\{ \psi \in \mathcal {H}\,|\, | \psi |=1\} \) denotes the unitary sphere. S has a Riemannian structure with a metric induced by the norm in \(\mathcal {H}.\) In the same way as before, there is an associated probability \(w_S\) is S.
Lemma 2
For any \(\nu =1,2...,N\),
Proof
Suppose \(\nu \) is fixed, then take \(\psi _1,\psi _2,...,\psi _D\), and orthogonal basis of \(\mathcal {H}\), such that \(\psi _1,\psi _2,...,\psi _{d_\nu }\) is an orthogonal basis of \(\mathcal {H}_\nu .\)
Given \(\phi = \sum _{j=1}^D x_j\, \psi _j \in S,\) where \(\sum _{j=1}^D |x_j|^2=1\), then
Note that the integral \(\int _S \, |x_j|^2 d\, w_S (x)\) is independent of j and
Therefore, for any j
Therefore, it follows that
\(\square \)
Lemma 3
For any \(\nu =1,2...,N\),
Proof
To simplify the notation we take \(\nu =1\). Then, we denote \(d=d_1\) and \(P=P_1\).
Take \(\psi _1,\psi _2,...,\psi _D\), and orthogonal basis of \(\mathcal {H}\), such that, \(\psi _1,\psi _2,...,\psi _{d}\) is an orthogonal basis of \(\mathcal {H}_1.\)
By last Lemma, we have
If \(\phi = \sum _{j=1}^D x_j\, \psi _j \in S,\) then \(P(\phi ) =\sum _{j=1}^d x_j\, \psi _j .\)
Therefore
The last equality follows from a standard computation (see “Appendix 1”).
From this follows the claim. \(\square \)
2 Changing the decomposition
\(\mathcal {H}\) is fixed for the rest of the paper.
Now, we change our point of view. We fix \(\phi \in \mathcal {H}\) and we consider different decompositions of \(\mathcal {H}\) in direct sum. More precisely, we fix \(D=\) dim \(\mathcal {H}\) and N and we consider fixed natural positive numbers \(d_\nu \), \(\nu =1,2,...,N\), such that \(d_1+d_2+...+d_N=D\), and then, all possible choices of orthogonal decompositions with this data.
We denote by \(\Delta (d_1,d_2,...,d_N, \mathcal {H} ) = \Delta \) the set of all possible \( \mathcal {D}\), that is, all possible orthogonal direct sum decompositions:
For fixed \(\nu =1,2,...,N\), then \( P_\nu (\mathcal {D})\) denotes the projection on \(\mathcal {H}_\nu \) associated with the decomposition \(\mathcal {D}\).
Each choice of orthogonal basis \(\psi _1,\psi _2,...,\psi _D\) of \(\mathcal {H}\) defines a possible choice of direct orthogonal sum decomposition:
and so on.
The set of all orthogonal basis is identified with the set of unitary operators U(D) which defines a compact Lie group and a Haar probability structure.
In this way,
In the same way as before, we get a probability \(w_\Delta \) over \(\Delta \). Therefore, it has a meaning the probability \(w_\Delta (B)\) of a Borel set \(B\subset \Delta \) of decompositions.
Lemma 4
Consider a continuous function \(f: \mathbb {R} \rightarrow \mathbb {R}\). Then, for fixed \(\nu =1,2...,N\), and fixed \(\tilde{\phi }\) and \(\tilde{\mathcal {D}}\)
This constant value is independent of \(\tilde{\phi }\) and \(\tilde{\mathcal {D}}\).
Proof
If \(U: \mathcal {H} \rightarrow \mathcal {H}, \) is unitary, then \(U\, \mathcal {D}\) denotes
Then, for fixed \(\phi \) and \(\mathcal {D}\), we have
We prove the claim for \(P_1\). Suppose \(\psi _1,\psi _2,...,\psi _D\), is an orthogonal basis of \(\mathcal {H}\), such that \(\psi _1,\psi _2,...,\psi _{d_1}\) is an orthogonal basis of \(\mathcal {H}_1.\)
We can express \(\phi = \sum _{j=1}^D x_j \, \psi _j\), and moreover, \(U(\phi )= \sum _{j=1}^D x_j \, U( \psi _j)\).
\(U(\psi _1),U(\psi _2),...,U(\psi _D)\) is an orthogonal basis of \(\mathcal {H}\) associated with \(U\, \mathcal {D}\) and \(U(\psi _1),U(\psi _2),...,U(\psi _{d_1})\) is an orthogonal basis of \(U(\mathcal {H}_1)\).
Then,
By the other hand
and this shows the claim.
Therefore, we get
Finally, for a fixed \(\mathcal {D}\) and a variable U
because \(w_S\) is invariant by the action of U.
Then, the above integral on the variable \(\phi \) is constant by the action of U in a given decomposition \(\mathcal {D}\).
Now, consider a fixed \(\phi _1\) and another general \(\phi _2 = U ( \phi _1)\), where U is unitary.
As \(w_\Delta \) is invariant by the action of U, the integral
is constant and independent of \(\phi \).
Remember that \( w_S \times w_\Delta \) is a probability.
Consider now
then by Fubini, we get the claim of the Lemma (since the unitary group acts transitively on S and on \(\Delta \)). \(\square \)
Corollary 5
Consider a fixed \(\phi \in \mathcal {H}\), such that \(|\phi |=1\).
Then, for \(\nu =1,2...,N\), we get that
and
where \(\,.\,\) denotes integration with respect to \(\mathcal {D}.\)
Proof
This is consequence of Lemmas 2, 3, and 4. \(\square \)
Definition 6
Given \(\delta >0\), a Hilbert space \(\mathcal {H}\) and natural positive numbers \(d_j, j=1,2,...,N\), such that \(d_1 + d_2 +...+ d_N=D=\) dim \(\mathcal {H}\), we say that a property is true for \(\mathcal {D}\in \Delta (d_1,..,d_N, \mathcal {H})\), in \((1-\delta )\) sense, if the property is not true only for elements \(\mathcal {D}\) in a set of probability \(w_\Delta \) smaller than \(\delta \).
Corollary 7
Suppose \(\epsilon >0\) and \(\delta >0\) are given. Consider natural positive numbers \(d_\nu , \nu =1,2,...,N\), such that \(d_1 + d_2 +...+ d_N=D=\) dim \(\mathcal {H}\), and moreover, assume that for all \(\nu =1,2...,N\)
Consider a fixed \( \phi \) such that \(|\phi |=1\). Then, for decompositions, \(\mathcal {D}\in \Delta (d_1,..,d_N, \mathcal {H})\) in the \((1-\delta )\) sense, and \(\nu =1,2...,N\), we have
Proof
By Corollary 5 and Markov inequality, we have
Then, the probability that all N inequalities do not happen is
by hypothesis. \(\square \)
The corollary above means that for a fixed \(\phi \), if the \(d_\nu \) are all not very small, then for a big part of the decompositions \( \mathcal {D}\), we have that
is close by the mean value \(\,\frac{d_\nu }{D}\).
Definition 8
Given a Hilbert space \(\mathcal {H}\) and a fixed decomposition \(\mathcal {D}\) (associated with natural positive numbers \(d_j, j=1,2,...,N\), such that \(d_1 + d_2 +...+ d_N=D=\) dim \(\mathcal {H}\), we define a semi-norm in such a way that for a linear operator \(\rho :\mathcal {H} \rightarrow \mathcal {H}, \) by
The above means that if \(|\,\rho \,|_\infty \) is small, then all expected values \(E_{P_\nu } (\rho ) \), \(\nu =1,2,...,N\), are small
\(|\,\phi >\,<\phi \,|\) will denote the orthogonal projection on the unitary vector \(\phi \) in the Hilbert space \(\mathcal {H}\).
Lemma 9
Consider a \(\phi \in \mathcal {H}=\mathcal {H}_1\, \oplus ...\oplus \mathcal {H}_N\), such that \( |\phi |=1\). Denote \(\rho _{mc} = \frac{1}{D} I_{\mathcal {H}}.\)
Then
Proof
Suppose \(\psi _1,\psi _2,...,\psi _D\) is orthogonal basis of \(\mathcal {H}\), such that \(\psi _1,\psi _2,...,\psi _{d_1}\) is an orthogonal basis of \(\mathcal {H}_1.\)
If \(\phi = \sum _{j=1}^D x_j \phi _j\), then for \(i=1,2,...,d_1\)
and
for \(i> d_1\).
Therefore
In an analogous way, we have that for any \(\nu \)
From this follows the claim. \(\square \)
From the above, it follows:
Corollary 10
Under the hypothesis of Corollary 7, we get that for decompositions \(\mathcal {D}\in \Delta (d_1,..,d_N, \mathcal {H})\) in the \((1-\delta )\) sense
\(\square \)
3 Estimations on time
Definition 11
Given \(\delta >0\), we say that a property for the parameters \(t\in \mathbb {R}\) is true for \((1-\delta )\)-most of the large times, if
where \(A_T\) is the set of \(t\in [0,T]\), where the property is verified and \(\mu \) is the Lebesge measure on \(\mathbb {R}.\)
Lemma 12
Suppose \(f: \mathbb {R} \rightarrow \mathbb {R}\) is continuous and non- negative. Consider a certain \(\gamma >0\).
Suppose \(\rho \) is such that
Then, \(f(t)< \gamma \) for \(1-\frac{\rho }{\gamma }\)-most of the large times.
Proof
Therefore
and finally
\(\square \)
Suppose \(\mathcal {H}\) is Hilbert space, and \(d_j, j=1,2,...,N\) are such that \(d_1 + d_2 +...+ d_N=D=\) dim \(\mathcal {H}\), and \(H : \mathcal {H} \rightarrow \mathcal {H}\) a self-adjoint operator. Consider a fixed \(\phi _0 \in \mathcal {H}\), with \(|\phi _0|=1\), and \(\psi _t = e^{-\,i\, t \, H}\, \phi _0\), \(t\ge 0\), a solution of the associated Schrodinger equation.
Lemma 13
For fixed T and \(\nu =1,2,...,N\), consider the function
given by
Then, \(f_{\nu ,T}\) converges uniformly on \((\mathcal {D},\phi ) \in \Delta (d_1,d_2,...,d_N, \mathcal {H} )\times S\) when \(T\rightarrow \infty \), for any \(\nu =1,2,...,N\).
Proof
Suppose \(\phi _1,\phi _2,...,\phi _D\) is a set of eigenvectors of H which is an orthonormal basis of \(\mathcal {H}\).
Assume that \(\phi _0= \sum _{j=1}^D x_j \phi _j\). Then
where \(E_j\), \(j=1,2,..,D\) are the corresponding eigenvalues.
Then, for a given \(\nu \)
Therefore
where \(M \in \mathbb {N}\), \(u_1,..,u_M\) are real constants and \( |\,L_{w,\nu }(\mathcal {D},\phi ) \,|\le 2\).
Then
Finally, we get
As M is fixed, the claim follows from this. \(\square \)
Corollary 14
for any \(\nu =1,2,..,N\).
Proof
By Lemma 13 and Corollary 5, we have that
\(\square \)
Theorem 15
Suppose \(\epsilon >0\), \(\delta >0\) and \(\delta \,'>0\) are given. Consider natural positive numbers \(d_\nu , \nu =1,2,...,N\), such that \(d_1 + d_2 +...+ d_N=D=\) dim \(\mathcal {H}\), and, moreover, assume that, for all \(\nu =1,2...,N\),
Suppose \(H: \mathcal {H} \rightarrow \mathcal {H}\) is self-adjoint, the unitary vector \(\psi _0\in \mathcal {H}\) is fixed, and \(\psi _t = e^{-\,i\,t\,H} (\psi _0),\) \(t \ge 0.\)
Then, for \((1-\delta )\)-most of the decompositions \(\mathcal {D} \in \Delta (d_1,d_2,...,d_N, \mathcal {H} )\), the inequalities
are true for \((1- \delta \,')\)-most of the large times.
The estimates depend on the initial condition \(\psi _0\).
Proof
We denote
From Corollary 14, for each \(\nu \)
Therefore, there exists a set \(S\subset \Delta \), such that
and, at the same time \( f_\nu (\mathcal {D} )< \frac{\epsilon ^2\, \delta \, '\, d_\nu }{D\, N^2}, \) for all \(\mathcal {D}\in S\) and all \(\nu =1,2...,N.\)
Now, taking in Lemma 12 \(\rho =\frac{\epsilon ^2\, \delta \, '\, d_\nu }{D\, N^2},\) and \(\gamma =\frac{\epsilon ^2\, d_\nu }{D\, N},\) we get for all \(\mathcal {D}\in S\) and all \(\nu =1,2,...,N\)
for \((1- \frac{\delta \,' }{N})\)-most of the large times.
Therefore, the above inequalities for all \(\nu =1,2,..,N\) are true for \((1- \delta \,' )\) most of the large times. \(\square \)
Note that the mean value \(f_{\nu } (\mathcal {D})\,\) depends of the Hamiltonian H but the bounds of last theorem does not depend on H.
4 Uniform estimates
In this section, we will refine the last result considering uniform estimates which are independent of the initial condition \(\psi _0\) (for the time evolution associated with the fixed Hamiltonian \(H: \mathcal {H} \rightarrow \mathcal {H}\)).
Suppose \(\epsilon >0\), \(\delta >0\), and \(\delta \,'>0\) are given. Consider natural positive numbers \(d_\nu , \nu =1,2,...,N\), such that \(d_1 + d_2 +...+ d_N=D=\) dim \(\mathcal {H}\)
We denote for each \(\psi _0 \in \mathcal {H}\), where \(|\psi _0|=1\), and \( \mathcal {D}\in \Delta =\Delta (d_1,...,d_N; \mathcal {H})\)
where \(\psi _t= e^{-i\,t\, H} (\psi _0)\) (see Lemma 13).
Lemma 16
Suppose are given \(\epsilon >0\) and \(\delta '>0.\) Assume that there exists non-negative continuous functions \(g_\nu : \Delta \rightarrow \mathbb {R}\), \(\nu =1,2,..,N\), and \(K>0\), such that
Suppose \(\delta \) is such that
Then, for \((1-\delta )\)-most of the \(\mathcal {D} \in \Delta \), we have
for \((1- \delta ')\)-most of the large times and for any \(\psi _0 \in \mathcal {H}\) with \(|\psi _0|=1\).
Proof
Note that
Therefore, there exists a subset \(E \subset \Delta \), such that \(w_\Delta (E) < 1 - \delta \) and \(g_\nu (\Delta )< \delta ' \, \epsilon ^2\, \frac{d_\nu }{N^2 \, D}\), for all \(\Delta \in E\) and all \(\nu =1,2...,N.\)
The conclusion is: if \(\Delta \in E\), then \( f_\nu (\psi _0,\mathcal {D})\, < \delta ' \, \epsilon ^2\, \frac{d_\nu }{N^2 \, D}\), for all \(\nu =1,2...,N,\) and all \(\psi _0 \) with norm 1.
The proof of the claim now follows from the reasoning of Theorem 15 and Lemma 12. \(\square \)
Note that to have \(\delta \) in expression (4) small, it is necessary that all \(d_\nu \) are large.
We assume now several hypothesis on H. Consider a certain orthogonal basis of eigenvectors \(\phi _1,\phi _2,...,\phi _D\) of H. We denote by \(E_j\), \(j=1,2,..,D\) the corresponding eigenvalues.
We assume hypothesis \(\mathfrak {N\,\,R}\) which says
-
a)
H is not degenerate, that is, \(E_\alpha \ne E_\beta \), for \(\alpha \ne \beta \), and
-
b)
H has no resonances, that is, \(E_\alpha -E_\beta \ne E_{\alpha '} - E_{\beta '}\), unless \(\alpha = \alpha '\) and \(\beta =\beta '\), or, \(\alpha = \beta \) and \(\alpha '=\beta '\).
Lemma 17
for all \(\psi _0\in \mathcal {H}\), such that \(|\psi _0|=1\), and for all \(\mathcal {D} \in \Delta (d_1,...,d_N; \mathcal {H})\) and all \(\nu =1,2...,N.\)
Proof
Suppose \(\psi _0= \sum _{\alpha =1}^D \,c_\alpha \, \phi _\alpha \). Then
and
Therefore
Using the above expression in the computation of integral \(f_{\nu } (\psi _0,\mathcal {D})\) will remain just the terms, where the coefficient of t is zero. By hypothesis, this will happen just when \(\alpha = \delta \) and \(\beta =\gamma \), or, \(\alpha =\beta \) and \(\gamma =\delta \).
Note that the case \(\alpha =\beta =\gamma =\delta \) is counted twice in the estimation.
Therefore
because \(\,< \phi _\gamma ,P_\nu ( \mathcal {D}) \phi _\delta>\,= \overline{< \phi _\delta ,P_\nu ( \mathcal {D}) \phi _\gamma >} \).
Finally, putting together the first and third terms:
By the other hand
because \(|\psi _0|=1\).
By the same reason
\(\square \)
Now, we define for each \(\nu =1,2,...,N\), the continuous function \(g_\nu (\mathcal {D}): \Delta (d_1,...,d_N; \mathcal {H}) = \Delta \rightarrow \mathbb {R} \) given by
We point out that for each \(\mathcal {D}\), the expression \(g_\nu (\mathcal {D})\) depends just on H because as \(E_\alpha \) are all different, the eigenvector basis is unique up to a changing in order and multiplication by scalar of modulus one.
Now, we need a fundamental technical Lemma.
Lemma 18
There exist a constant \(C_1>0\), such that
if, \(C_1 \log D \,< \, d_\nu \,<\, \frac{D}{C_1}.\)
Note that if D is large, there is a lot of room for the values \(d_\nu \) to be able to satisfy last inequality. We will prove this fundamental lemma in the next sections.
If we assume the Lemma is true, then:
Theorem 19
Given \( \epsilon , \delta >0\) and \(\delta '>0\), take \(d_1,d_2,...,d_N\), such that, if \(D=d_1+...+d_N\), \(N>0\), then the following inequalities are true
where \(C_1\) comes from Lemma 18.
Assume that \(\mathcal {H}\) is a Hilbert space of dimension D and \(H: \mathcal {H} \rightarrow \mathcal {H}\) is a self-adjoint Hamiltonian without resonances and degeneracies, then for \((1-\delta )\) most of the decompositions \(\mathcal {D} \in \Delta (d_1,...,d_N;\mathcal {H})\) the system of inequalities
are true for most of the \((1-\delta ')\) large times and for any initial condition, \(\psi _0\in \mathcal {H}\), \(|\psi _0|=1\).
Proof
By hypothesis and Lemma 18, we get
The claim follows from Lemma 16 by taking \(K = \frac{10 \log D}{D}.\) \(\square \)
Main conclusion:
As we said before, for a given fixed subspace \(\mathcal {H}_\nu \) of \(\mathcal {H}\), the observable \(P_{\mathcal {H}_\nu }\) (the orthogonal projection on \(\mathcal {H}_\nu \)) is such that the mean value \(E_{\psi _t} (P_{\mathcal {H}_\nu })\) of the state \(\psi _t\) is \(<P_{\mathcal {H}_\nu }(\psi _t), \psi _t>= |P_{\mathcal {H}_\nu }(\psi _t)\,|^2.\)
For a fixed Hamiltonian H acting on a Hilbert space \(\mathcal {H}\) of dimension D, the main theorem gives lower bound conditions on the dimensions \(d_\nu \), \(\nu =1,2,..,N\), of the different \(\mathcal {H}_\nu \) values of a \((1-\delta )\)-generic orthogonal decomposition \(\mathcal {D}\) of the form \(\mathcal {H}\,=\, \mathcal {H}_1\, \oplus ...\oplus \mathcal {H}_N\), in such a way that the dynamic time evolution \(\psi _t\), obtained from any fixed initial condition \(\psi _0\), for most of the large times t, has the property that the projected component \(P_\nu (\mathcal {D})\, (\psi _t)\,=\,P_{\mathcal {H}_\nu }(\psi _t)\) is almost uniformly distributed (in terms of expected value) with respect to the relative dimension size \(\frac{d_\nu }{D}\) of \(\mathcal {H}_\nu .\) In this way, there is an approximately uniform spreading of \(\psi _t\) among the different values of \(\mathcal {H}_\nu \) of the decomposition \(\mathcal {D}\).
5 Proof of Lemma 18
The Lemmas 22 and 23 will permit to reduce the integration problem from the unitary group to a problem in the real line.
We will need first an auxiliary lemma. We denote by \(S^k\) the unitary sphere in \(\mathbb {R}^{k+1}\) and \(S^k_r\) the sphere of radius \(r>0\) in \(\mathbb {R}^{k+1}.\) We consider the usual metric on them.
The next lemma is a classical result on Integral Geometry (see [4]). We will provide a simple proof in “Appendix 2”.
Lemma 20
Suppose X is a Riemannian compact manifold, \(f:X \rightarrow \mathbb {R}\) a \(C^\infty \)-function and \(g: \mathbb {R} \rightarrow \mathbb {R}\) a continuous function. We define
where \(\lambda \) is the volume form on X. Suppose that \(a\in \mathbb {R}\) is a regular value of f. Then, G is differentiable at \(v=a\) and
where \(X_a\) is the level manifold \(f=a\) and \(\lambda _a\) is the induced volume form in \(X_a\).
Corollary 21
Given positive integers d, D, where \(1<d<D-1\), denote by S the unitary sphere on \(\mathbb {R}^{2\, D}\) with the usual metric. Define
Suppose
then G is of class \(C^1\) and
and \( \frac{\mathrm{d} G}{\mathrm{d}v} (v)\,=0\), if \(v<0\) or \(v>1\).
Proof
For \(x_1^2 +...+x^2_{2\,d }=v\), we have
Then, \(|\text {grad}\,f(x)|=2\, \sqrt{v\, (v-1)}\), which is constant over \(S_v=\{f=v\}\). Note that
From last Lemma and from the above expression, it follows that (remember that vol \((S_r^{2n-1})\,= \,\frac{\,2 \, \pi ^n}{(n-1)\, !} \,r^{2n-1}\))
In the case \(v<0\) or \(v>1\), we have that G is constant. Finally, as \(S_0\) and \(S_1\) are submanifolds of S, we have that G is continuous for \(v=0\) and \(v=1\). \(\square \)
From now on, we fix \(\nu \), where \(1\le \nu \le N\), and we define
where \(\phi _1,...,\phi _D\) is the orthonormal basis for \(\mathcal {H}\) which were fixed in Sect. 4.
Lemma 22
Suppose \(1< d_\nu <D-1\). Let \(a \ge 0\) be such \(\sqrt{a} < \frac{d_\nu }{D}\) and \(\sqrt{a}+ \frac{d_\nu }{D}<1.\) Then, the probability, such that \((e_{\alpha ,\beta }\,-\frac{d_\nu }{D} )^2\, \ge \alpha \) is
Lemma 23
Suppose \(1< d_\nu <D-1\). Let \(\alpha \ne \beta \) and \(0\le a\le 1/4\). Then, the probability such that \(|\,e_{\alpha ,\beta }\,|^2\, \ge a\) is
Proof of Lemma 22
We just have to consider the case \(\nu =1\). We write \(d=d_1\) and denote by P the orthogonal projection of \(\mathcal {H}\) over \(\mathbb {C} \phi _1+...+ \mathbb {C} \phi _d.\)
We denote by \(p:\mathbb {U} \rightarrow \Delta \) the projection defined in the beginning of Sect. 2, where \(\mathbb {U}\) denotes the group of unitary transformations of \(\mathcal {H}.\)
If \(U\in \mathbb {U}\), then
\( e_{\alpha ,\alpha }(p(U))=<\phi _\alpha ,\) orthogonal projection of \(\phi _\alpha \) in \(\mathbb {C} U(\phi _1)+...+ \mathbb {C}\,U( \phi _d)>=\)
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,<U^{-1} (\phi _\alpha ), P (U^{-1} \phi _\alpha )>.\)
Denote \(q:\mathbb {U} \rightarrow S,\) where \(q(U)= U(\phi _\alpha ),\,\,\,U \in \mathbb {U}\) and \(\sigma :S \rightarrow \mathbb {R},\) where \(\sigma (\phi )= <\phi , P(\phi )>,\) \(\phi \in S,\) and where S is the unitary sphere of \(\mathcal {H}.\)
Then, we get the following commutative diagram:
As the inverse preserves the metric, it follows from Lemma 1 a) that the probability of \( e_{\alpha ,\alpha } \le b\) is equal to the probability that \(\sigma \le b\). Note that the metric on S as quotient of \(\mathbb {U}\) is the same as the induced by \(\mathcal {H}\), because \(\mathbb {U}\) acts transitively on S.
It will be more easy to make the computations via the right hand side of the diagram.
We identify \(\mathcal {H}\) with \(\mathbb {C}^D= \mathbb {R}^{2\, D}\), via \(\phi _1,\phi _2,...,\phi _D\). Then, S is identified with the unitary sphere in \(\mathbb {R}^{2\, D}\), also denoted by S, and
Therefore, by Corollary 21 with \(g=1\), we get
and
if \(v<0\) or \(v>1\).
Now, we normalize dividing by vol \(S= \frac{\, 2\,\pi ^D}{(D-1)\,!}\) and we get
As \((e_{\alpha ,\alpha }\,-\frac{d}{D} )^2\, \ge a\) is equivalent to
we get that the probability of \((e_{\alpha ,\alpha }\,-\frac{d}{D} )^2\, \ge a\) is equal to the probability of \(\sigma \ge \frac{d}{D} \, + \sqrt{a}\) or \(\sigma \le \frac{d}{D} \, - \sqrt{a}\). From this follows that the probability of \((e_{\alpha ,\alpha }\,-\frac{d}{D} )^2\, \ge a\) is equal to
Observe that \(\sigma =\) constant is an analytic subset of S, and therefore, the associated probability is zero. The case \(a=0\) is trivial. \(\square \)
Proof of Lemma 23
We just have to consider the case \(\nu =1\). Take \(d=d_1\) and as before, we denote by P the orthogonal projection of \(\mathcal {H}\) over \(\mathbb {C} \phi _1+...+ \mathbb {C} \phi _d.\) Once more we denote by \(p:\mathbb {U} \rightarrow \Delta \) the projection defined in the beginning of Sect. 2.
If \(U\in \mathbb {U}\), then
\( e_{\alpha ,\beta }(p(U))=<\phi _\alpha ,\) orthogonal projection of \(\phi _\beta \) in \(\mathbb {C} U(\phi _1)+...+ \mathbb {C}\,U( \phi _d)>=\)
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,<U^{-1} (\phi _\alpha ), P (U^{-1} \phi _\beta )>.\)
Denote \(q_{\alpha ,\beta } :\mathbb {U} \rightarrow S\times S,\) where \(q_{\alpha ,\beta }(U)= (U(\phi _\alpha ), U(\phi _\beta )),\,\,\,U \in \mathbb {U}\), and S is the unitary sphere of \(\mathcal {H}.\)
Denote by \(M= q_{\alpha ,\beta }(\mathbb {U})= \{(\phi ,\psi )\in S \times S\,| \,\phi \) is orthogonal to \( \psi \,\}\).
Let \(H_{\alpha ,\beta } \subset \mathbb {U}\) the closed subgroup of the U, such that \(U(\phi _\alpha )= \phi _\alpha \) and \(U(\phi _\beta )= \phi _\beta \).
Then, \(M = \mathbb {U}/H_{\alpha ,\beta }\) and \(q_{\alpha ,\beta }: \mathbb {U} \rightarrow M\) is the canonical projection.
The quotient metric on M is the induced by \(S\times S\), because \(\mathbb {U}\) acts transitively on M.
Let \(f:M \rightarrow \mathbb {C}\) given by \(f(\phi , \psi ) = <\,\phi , P( \psi )\,>.\) Then, we get the following commutative diagram:
As the inverse preserves the metric of \(\mathbb {U}\), it follows that the probability of \( |e_{\alpha ,\alpha }|^2 \le a\) is equal to the probability that \(|f|^2\le a\) by Lemma 1 a).
Now, consider \(\varphi :M \rightarrow S\), such that \(\varphi (\phi ,\psi )=\psi \). This defines a \(C^\infty \) locally trivial fiber bundle with fiber \(S^{2 \, D-3}\). Indeed, \(E_\psi =\varphi ^{-1} (\psi )\) is the unitary sphere of the subspace \(\mathcal {H}_\psi \) which is the orthogonal set to \(\psi \) in \(\mathcal {H}\).
Given \(u\in \mathbb {R}\) denote:
Then
For each \(\psi \), we get \(\psi '\in \mathcal {H}\) via
Note that \(\psi ' \in \mathcal {H}_\psi \). Then
and it follows that
There exist an isomorphism identifying \(\mathcal {H}_\psi = \mathbb {C}^{D-1}= \mathbb {R}^{2\, D -2}\) between Hilbert spaces which transform \(\psi '\) in \((|\psi '|,0,...,0)\). This isomorphism identifies \(E_\psi \) with the unitary sphere E on \(\mathbb {R}^{2\, D-2}\) and \(F_u (\psi )\) with the set:
Now, applying Corollary 21 with \(D-1\) instead of D, \(d=1\), \(g=1\), and \(v= \frac{u}{|\psi '|^2}\), we get
for all \(\psi \in S\) and \(0<u\le |\psi \,'|^2\), and
if \(|\psi \, '|^2 \le u\le 1\), for any \(\psi \in S\).
Then, we get that \( \frac{d\,\text {Vol}_{E_\psi } F_u(\psi )}{\mathrm{d}u}\) is a continuous function of \((u,\psi )\) for \(0<u\le 1\) and \(\psi \in S\). As S is compact, we can take derivative inside the integral and we get
for any \(0<u\le 1.\)
By the definition of \(\psi \,'\), it is easy to see that \(|\psi \,'|^2 = |P(\psi )|^2 \,(1- |P(\psi )|^2)\).
Now, we consider \(g_u: \mathbb {R} \rightarrow \mathbb {R} \), where
if \(u \le w\, (1-w),\) and \(g_u(w)=0\) in the other case.
\(g_u(w)\) is a continuous function of u and w when \(0<u\le 1\), \(0\le w \le 1\).
From this follows that
for any \(0<u\le 1.\)
Now, we normalize dividing by Vol \((M)= \frac{ 2\, \pi ^{D-1 }}{(D-2)\, !}\,\frac{ 2\, \pi ^{D }}{(D-1)\, !}\) and we get
for any \(0<u\le 1.\)
Denote
for any \(0<u\le 1\), \(0\le w\le 1.\)
By Corollary 21, we get
for any \(0<u\le 1.\)
Estimating \(\frac{\partial A}{\partial w}\) by Corollary 21 and substituting in (7), we finally get
for any \(0<u\le 1.\)
If \(u>1/4\), \(w\,(1-w)<u\) for all w and the integral is zero.
If \(0<u\le 1/4\), \(u\le w (1-w)\) is equivalent to
Then
if \(0<u\le 1/4\), and
if \(1/4\le u\le 1.\)
Finally, for \(0< a\le 1/4\)
Considering the double integral in the region \(a\le u\le w\, (1-w)\), we get
The case \(a=0\) is trivial. \(\square \)
Remark
Note that if \(g:\Delta \rightarrow \mathbb {R}\) is a continuous function such that \(0\le g( \mathcal {D})\le r\), for all \(\mathcal {D} \in \Delta ,\) then we get the estimate
for \(0\le a\le 1\).
Given positive integer numbers d, D and \(a \in \mathbb {R}\), such that
we define
In the following, we will use the estimate \(\theta =11/12.\)
Lemma 24
There exists a constant \(C>4\), such that if \(a \ge 0,\) \(d\ge 1\) \(C\, \log D< d <\frac{D}{C}\) and \(\frac{1}{D}< \sqrt{a} < \frac{d}{8\, D} \), then
Proof
Note that our hypothesis implies that \(1<d<D-1\), \(a^2 < \frac{d^2}{D^2}\) and \(\frac{d}{D} + \sqrt{a} <1\).
-
a)
By Stirling formula, when \(D\rightarrow \infty \), \(d \rightarrow \infty \), \(D/d\rightarrow \infty \), we get that
$$\begin{aligned} \frac{(D-1)\,!}{(d-1)\,!\, (D-d-1)\, ! }\sim \frac{1}{e} \, \sqrt{\frac{d}{2 \pi } } \left( \frac{d}{D}\right) ^{-d} \left( 1- \frac{d}{D}\right) ^{d-D}. \end{aligned}$$As \(\sqrt{\frac{1}{2 \pi }}\,<\,1\), there exists a constant A such that if \(D>A\), \(d>A\) and \(D/d>A\), we get
$$\begin{aligned} \frac{(D-1)\,!}{(d-1)\,!\, (D-d-1)\, ! }\,< \, \frac{\sqrt{d}}{2 } \left( \frac{d}{D}\right) ^{-d} \left( 1- \frac{d}{D}\right) ^{d-D}. \end{aligned}$$If we take \(C>A+1\), it follows from the hypothesis of the Lemma that \(D>d\,C >d\,A\), \(d>C \log D>C>A\) and \(D-d>d\, C-d=d (C-1)> d\, A>A\).
-
b)
The derivative of \(u^{d-1}\, (1-u)^{D-d-1} \) with respect to u in (0, 1) is zero only on the point \(u= \frac{d-1}{D-1}\) which is smaller than d / D. Moreover
$$\begin{aligned} \frac{d}{D}- \sqrt{a}<\frac{d}{D}- \frac{1}{D}=\frac{d-1}{D}< \frac{d-1}{D-1}. \end{aligned}$$Then, \(\frac{d-1}{D-1}\in ( \frac{d}{D}- \sqrt{a}, \frac{d}{D} )\subset ( \frac{d}{D}- \sqrt{a}, \frac{d}{D} + \sqrt{a}).\) From this it follows that \(u^{d-1}\, (1-u)^{D-d-1} \) takes its maximal values on the set \([0, \frac{d}{D}- \sqrt{a}]\,\cup \, [ \frac{d}{D}+ \sqrt{a},1] \) on the point \(\frac{d}{D}- \sqrt{a}\) or on the point \(\frac{d}{D}+ \sqrt{a}.\) Under our hypothesis, if \(C>A+1\), we get that for \(\epsilon =1\) or \(-1\):
$$\begin{aligned} I(d,D,a)< & {} \frac{\sqrt{d}}{2 } \left( \frac{d}{D}\right) ^{-d} \left( 1- \frac{d}{D}\right) ^{d-D} \left( \frac{d}{D} + \epsilon \sqrt{a}\right) ^{d-1} (1-\frac{d}{D} - \epsilon \sqrt{a})^{D-d-1}\\= & {} \frac{\sqrt{d}}{2 }\, \frac{(1 + \epsilon \frac{D}{d} \sqrt{a})^{d} \,(1 - \epsilon \frac{D}{D-d}\sqrt{a})^{D-d} }{ (\frac{d}{D} + \epsilon \sqrt{a})\, \left( 1-\,\frac{d}{D} - \epsilon \sqrt{a}\right) }. \end{aligned}$$ -
c)
If \(\epsilon =1\) with \(C>4\), \(C>A+1\), we get
$$\begin{aligned} (\frac{d}{D} + \epsilon \sqrt{a})\, \left( 1-\,\frac{d}{D} - \epsilon \sqrt{a}\right) = \frac{d}{D} + \sqrt{a}\, - \,\frac{d^2}{D^2} - 2\frac{d}{D} \sqrt{a} - a>\end{aligned}$$$$\begin{aligned} \frac{d}{D} - \, \frac{d^2}{D^2} - 2 \frac{d}{D} \sqrt{a}> \frac{d}{D} - \, \frac{d^2}{D^2} - 2 \frac{d^2}{8\, D^2} =\frac{d}{D} - \, \frac{5\,d^2}{4\,D^2}> \frac{d}{D} \left( 1 -\frac{5\,d}{4\,D}\right) >\frac{d}{2\,D}. \end{aligned}$$If \(\epsilon =1\) with \(C>4\), \(C>A+1\), one can show in the same way that
$$\begin{aligned} \left( \frac{d}{D} + \epsilon \sqrt{a}\right) \, \left( 1-\,\frac{d}{D} - \epsilon \sqrt{a}\right) > \frac{d}{2\, D}. \end{aligned}$$In this way, we finally get that for \(\epsilon =1\) or \(\epsilon =-1\)
$$\begin{aligned} I(d,D,a)< & {} \frac{\sqrt{d}}{2 } \frac{2 \, D}{d} \left( 1+ \epsilon \,\frac{D}{d} \sqrt{a}\right) ^{d} (1- \epsilon \frac{D}{D-d} \sqrt{a})^{D-d}\\= & {} \frac{ D}{\sqrt{d}} \left( 1+ \epsilon \,\frac{D}{d} \sqrt{a}\right) ^{d} \left( 1- \epsilon \frac{D}{D-d} \sqrt{a}\right) ^{D-d} . \end{aligned}$$
Note that
This is so because \(\log (1+x)=x- \frac{x^2}{2} + \frac{x^3}{3}+...\), for \(|x|<1\), \(\frac{D}{d} \sqrt{a}<1/8\), and \(\frac{D}{D-d} \sqrt{a}<\frac{1}{24}.\)
Therefore, if \(C>4\) and \(C>A+1\), then
for \(\epsilon =1\) or \(\epsilon =-1\).
Note that
Therefore, if \(C>4\) and \(C>A+1\), we finally get
\(\square \)
Motivated by the Remark before Lemma 24, we will choose a convenient choice of a.
Corollary 25
There exist \(C_0>4\), such that if d and D are such that \( C_0 \log D< d < \frac{D}{C_0}\), then
where \(a=\frac{8\, d\, \log D}{\theta \, D^2}.\)
Proof
Take \(C_0>C\) (of Lemma 24) and \(C_0> 24^{2}\). Then
because \(\frac{8}{\theta }<9.\)
Moreover, \(\sqrt{a}> \frac{ \sqrt{d\, \log D}}{ D}> \frac{1}{ D}.\)
By Lemma 24, we get that
\(\square \)
Lemma 26
Suppose \(C_0\) is the constant of Corollary 25. Given \(1\le \nu \le N,\) suppose that \(C_0\log D< d_\nu < \frac{D}{C_0},\) then
Proof
Suppose \(a= \frac{8\, d_\nu \, \log D}{\theta \, D^2}\).
By Corollary 25 and Lemma 22 (see also the beginning of the proof of Lemma 24), we get that the probability of the above integrand to be great or equal to a is smaller than \( D\, \frac{1}{D^3\, \sqrt{d_\nu }} = \frac{1}{D^2 \sqrt{d_\nu } }\).
As we point out in the Remark before Lemma 24, the integral is smaller than
Note that
because \(d_\nu ^{3/2} \,\log D> C_0^{3/2}\, (\log D)^{5/2}> C_0^{3/2}>8> \frac{11}{3}.\)
Therefore
\(\square \)
In Lemma 18, the function \(g_\nu \) is defined as the sum of two terms (see expression (6). The Lemma 26 takes care of the upper bound of the integral of the second term. Now, we will estimate the upper bound for the first term (using the Remark done before Lemma 24). First, we need two lemmas.
Lemma 27
Suppose \(\phi \) and \(\psi \) are orthonormal and \(E\subset \mathcal {H}\) is a subspace. Denote by P the orthogonal projection of \(\mathcal {H}\) over E.
Then, \( |\,<\phi ,\,P(\psi )\,>\,|^2\le 1/4\).
Proof
If \(\psi \) is orthogonal to E or \(\psi \in E\), we have that \(<\phi ,P(\psi )>=0.\)
Suppose \(\psi \) is not on E and is also not orthogonal to E. Suppose \(\psi =\psi _1+\psi _2\), where \(\psi _1 \) is orthogonal to E and \(\psi _2 \in E\).
Let \(\lambda =|\psi _1| \) and \(\mu =|\psi _2|\), then \(\psi _1=\lambda e_1\), \(\psi _2= \mu \, e_2\), where \(e_1\) and \(e_2\) are orthonormal.
Denote by \(\theta \) the orthogonal projection of \(\phi \) over \(\mathbb {C}\, e_1 + \mathbb {C}\, e_2\). Then
Now, \(<\phi ,\,\psi >=0\) implies that
Suppose \(\theta = a \, e_1 + b\, e_2\), then \(|a|^2 + |b|^2\le 1.\) By the other hand, \(1=|\psi |=|\psi _1 + \psi _2|= |\lambda |^2 + |\mu |^2\) and
From this, it follows that \(|\frac{-\alpha }{a}|^2 + |\frac{\alpha }{b}|^2=1\), that is, \(|\alpha |^2= \frac{|a|^2 \Vert b|^2}{|a|^2 +|b|^2}< \frac{1}{4}.\)
Note that if \(a\,b=0\), then \(\alpha =0\). \(\square \)
Lemma 28
Given positive integers d, D, where \(1<d\) and \(D> 2 d +2\) denote
then, f(t) is increasing on the interval (0, 1).
Proof
For any \(t\in (0,1)\), we have
Taking \(z=\frac{1+t}{1-t}>1\), we get
because \(z>1\), \(\square \)
Suppose \(0\le a < 1/4\) and d, D positive integers, such that \(1<d<D-1\). Define
Lemma 29
Suppose d, D are positive integers \(1<d,\, 2\,d+2 < D\). Then
Proof
Note that J(d, D, a) is positive.
In the integration, we divide the integral in two parts: \([1/2 - \sqrt{1/4-a},\,1/2]\) and \([1/2,\,1/2 + \sqrt{1/4-a}]\).
We make a change of variable \(w= 1/2\,-\sqrt{1/4-x}\) on the first interval and \(w= 1/2\,+\sqrt{1/4-x}\) on the second interval. On both cases, we get \(x=w\,(1-w)\) and \(a\le x\le 1/4.\)
From this, it follows
Now, we consider \(y=\frac{x-a}{ 1/4 -a}\). In this case \((1-4x)=(1-4a)(1-y)\).
Then
Note that just the expression under \([\,\,\) \(\,\,]\) depends on a. For each \(y\in (0,1)\), we have \(\sqrt{1-4a}\,\sqrt{1-y}\in (0,1) \) is an decreasing function of a. It follows from Lemma 28 that for each \(y\in (0,1)\), the integrand is a decreasing function of a.
Therefore, \(\frac{J(d,D,a)}{(1-4a)^{D-3/ 2}}\) is a decreasing function of a. As \(J(d,D,0)=1\) (see Lemma 23), it follows that
Finally, note that \((1- 4\, a)^{D-3/2}\le e^{ -4\, a\, (D-3/2) }\) \(\square \)
Corollary 30
If \(1<d, \,D>2\, d +2\) and \(\frac{\log D}{D}<\frac{1}{3}\), then
Proof
It follows from Lemma 29, because \( 0<\frac{3}{4} \frac{\log D}{D} < \frac{1}{4}.\) \(\square \)
Lemma 31
Suppose \(1\le \nu \le N \), \(3<d_\nu \), \(D>2 \, d_\nu +2\), and \( \frac{\log D}{D}<\frac{1}{5}.\)
Then
where \(\phi _1,..,\phi _D\) is an orthonormal basis of eigenvectors for H (without resonances).
Proof
By Lemma 23 and Corollary 30, the probability that the integrand is bigger than a is smaller than
because as \(e_{\alpha ,\beta } = \overline{e_{\beta ,\alpha }}\), we just have to take \(\alpha <\beta \).
By the Remark before Lemma 24, the integral is smaller than
because by Lemma 27 \(|e_{\alpha ,\beta } |<1/4 \).
As \(D-1<D\), we have
Now, as \(D\ge 9\), \(\log D\ge 2\), we get
Now, we put the two estimates together \(\frac{3}{4}\, \frac{\log D}{D} + \frac{1}{4}\, \frac{\log D}{D}\) and we get the claim of the Lemma. \(\square \)
The Lemma 18 follows from Lemmas 26 and 31. In this way, we get the claim of the Quantum Ergodic Theorem of von Neumann.
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A. O. Lopes was partially supported by CNPq and INCT.
Appendices
Appendix 1
In this Appendix, we will show that
First, we will show that when S is the unitary sphere in \(\mathbb {R}^n\), \(m\ge 1\), and \(n \ge m\), then
It is easy to see that (9) follows from (10).
-
1)
\(\int x_j^2 \mathrm{d}S(x)= \frac{\mathrm{vol}(S)}{n} \) for \(j=1,2,...,n\), because the integral does not depend of j.
-
2)
Suppose B is the unitary ball in \(\mathbb {R}^n\). Consider in polar coordinates
$$\begin{aligned} T: S \times [0,1] \rightarrow B, \end{aligned}$$where \(T(x,\rho )= \rho \,x.\) Then, \(T^* (\mathrm{d}x_1\wedge ...\wedge \mathrm{d} x_n)= \rho ^{n-1} d S (x) \wedge d \rho \). Therefore
$$\begin{aligned} \int _B (x_1^2 +..+x_n^2) \mathrm{d} x_1...\mathrm{d}x_n= \int _{S \times [0,1]} T^* ( (x_1^2 +..+x_n^2 ) \mathrm{d}x_1\wedge ...\wedge \mathrm{d}x_n)= \end{aligned}$$$$\begin{aligned} \int _{S \times [0,1]} \rho ^{n+1} d S(x) \wedge d \rho = \text {vol}(S) \, \int _0^1 \rho ^{n+1} d \rho = \frac{\text {vol} (S)}{n+2}. \end{aligned}$$Finally, \(\int _B x_j ^2 \mathrm{d}x_1...\mathrm{d}x_n= \frac{\text {vol} (S)}{n\,(n+2)}\), because it is independent of \(j=1,2...,n.\)
-
3)
For \(j=1,2...,n\), we have
$$\begin{aligned} \int _S x_j^4\, d S(x) = 3 \, \int _B x_j^2 \, \mathrm{d}x_1...\mathrm{d}x_n= \frac{3 \text {vol} (S)}{n\, (n+2)} \end{aligned}$$by the divergent theorem and by 2) above.
-
4)
If \(\le i< j\le n\), then
$$\begin{aligned} \int _S x_i^2\, x_j^2 \,d S(x) = \int _B x_j^2 \, \mathrm{d}x_1...\mathrm{d}x_n= \frac{ \text {vol} (S)}{n\, (n+2)}. \end{aligned}$$by the divergent theorem and by 2) above.
The integral
is a sum of terms of the kind \(\int _S x_i^2\, x_j^2 \,d S(x)\), \(i \ne j\), and \(\int _S x_j^4\, \,d S(x)\), \(j=1,2,..n\).
Just collecting the different terms and using the estimates above, we get the initial claim (10).
Appendix 2: Proof of Lemma 20
Suppose \(\epsilon >0\) is small enough, consider
Given \(h\in \mathbb {R}\), \(0<|h|<\epsilon \), then integrating \((g \circ f)\, \lambda \), we get
(where \(\lambda _v\) is the volume form on \(X_v =f^{-1}(v)\) for \(v \in (a-\epsilon ,\,a+ \epsilon ))\), because \(\mathrm{d}f (\text {grad} f)= |\,\text { grad}\, f\,|^2\). From this follows that for some \(0\le \theta \le 1\), we have
Now, we divide the above expression by h and we take the limit when \(h\rightarrow 0\) \(\square \)
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Lopes, A.O., Sebastiani, M. A detailed proof of the von Neumann’s Quantum Ergodic Theorem. Quantum Stud.: Math. Found. 4, 263–285 (2017). https://doi.org/10.1007/s40509-017-0100-7
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DOI: https://doi.org/10.1007/s40509-017-0100-7