Monotonicity of the Polaron Energy II: General Theory of Operator Monotonicity

Abstract

We construct a general theory of operator monotonicity and apply it to the Fröhlich polaron hamiltonian. This general theory provides a consistent viewpoint of the Fröhlich model.

Appendices

Appendix A: Preliminaries

In this section, we will review some preliminary results about the operator inequalities introduced in Sect. 2.1. Almost all of results here are taken from the author’s previous work [32–36].

1.1 A.1 Operator Monotonicity

Proposition A.1

(Monotonicity)

Let A and B be positive self-adjoint operators. We assume the following.

• (a) dom(A)⊆dom(B) or dom(A)⊇dom(B).

• (b) $(A+s)^{-1}\unrhd 0$ and $(B+s)^{-1}\unrhd 0$ w.r.t. $\mathfrak {p}$ for all s>0.

Then the following are equivalent to each other.

• (i) $B\unrhd A$ w.r.t. $\mathfrak {p}$.

• (ii) $(A+s)^{-1}\unrhd (B+s)^{-1}$ w.r.t. $\mathfrak {p}$ for all s>0.

• (iii) $\mathrm {e}^{-tA}\unrhd \mathrm {e}^{-tB}$ w.r.t. $\mathfrak {p}$ for all t≥0.

Proof

See [32, 33]. □

Corollary A.2

Let A be a positive self-adjoint operator and let B be a symmetric operator. Assume the following.

• (i) B is A-bounded with relative bound a<1, i.e., dom(A)⊆dom(B) and ∥Bx∥≤a∥Ax∥+b∥x∥ for all x∈dom(A).

• (ii) $0\unlhd \mathrm {e}^{-tA}$ w.r.t. $\mathfrak {p}$ for all t≥0.

• (iii) $0\unlhd -B$ w.r.t. $\mathfrak {p}$.

Then $\mathrm {e}^{-t(A+B)}\unrhd \mathrm {e}^{- tA}\unrhd 0$ w.r.t. $\mathfrak {p}$ for all t≥0.

Proof

See [32, 33]. □

1.2 A.2 Perron-Frobenius-Faris Theorem

Theorem A.3

(Perron-Frobenius-Faris)

Let A be a positive self-adjoint operator on $\mathfrak {h}$. Suppose that $0\unlhd \mathrm {e}^{-tA}$ w.r.t. $\mathfrak {p}$ for all t≥0 and infspec(A) is an eigenvalue. Let P _A be the orthogonal projection onto the closed subspace spanned by eigenvectors associated with infspec(A). Then the following are equivalent.

• (i) dimranP _A =1 and P _A ▷0 w.r.t. $\mathfrak {p}$.

• (ii) 0◁(A+s)⁻¹ for some s>0.

• (iii) For all $x,y\in \mathfrak {p}\backslash \{0\}$, there exists a t>0 such that 0<〈x,e^−tA y〉.

• (iv) 0◁(A+s)⁻¹ for all s>0.

• (v) 0◁e^−tA for all t>0.

Proof

See, e.g., [10, 32, 41]. □

Appendix B: A Remark on the Strongly Continuous Semigroup

A family of operators {T _s  | 0≤s<∞} on a Hilbert space $\mathfrak {h}$ is called a one-parameter semigroup if

• (a) ,

• (b) T _s T _t =T _s+t for all s,t≥0.

In addition if T _s satisfies

• (c) for each $x\in \mathfrak {h}$, s↦T _s x is strongly continuous,

then the family is called a strongly continuous semigroup.

The following lemma is well-known [40].

Lemma B.1

Let T _s be a strongly continuous semigroup on a Hilbert space $\mathfrak {h}$ and , where

(12.1)

Then A is closed and densely defined.

If we further add conditions on T _s , we can prove the self-adjointness of A as follows.

Proposition B.2

Let T _s be a strongly continuous semigroup on a Hilbert space $\mathfrak {h}$. Assume

• (d) T _s is self-adjoint for all s≥0.

• (e) There exists an M>0 such that ∥T _s ∥≤e^−sM for all s≥0.

Then A is self-adjoint, bounded from below by M and T _s =e^−sA.

Proof

By Lemma B.1 and (d), A is closed and symmetric. We will show ker[A ^∗±i]={0}. (This is equivalent to the self-adjointness of A by the general theorem.) Pick η∈ker[A ^∗−i]. For all $x\in \mathfrak {h}$, one sees

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}s}\langle T_s x, \eta \rangle = -\langle AT_s x, \eta \rangle =-\langle T_s x, A^*\eta \rangle = -\mathrm {i}\langle T_s x, \eta \rangle . \end{aligned}$$

(12.2)

Here we used the facts T _s dom(A)⊆dom(A) and $\frac{\mathrm {d}}{\mathrm {d}s} T_{s} x =-A T_{s} x$. Solving the differential equation, we obtain 〈T _s x,η〉=u ₀ e^−is. Since

$$\begin{aligned} \big| \langle T_s x, \eta \rangle \big|\le \mathrm {e}^{-s M}\|x\| \|\eta\| \to 0 \end{aligned}$$

(12.3)

as s→∞, u ₀ must be 0. I.e., 〈x,η〉=0. Since dom(A) is dense in $\mathfrak {h}$, this means η=0, namely, ker[A ^∗−i]={0}. Similarly we can show ker[A ^∗+i]={0}. Thus we have the assertions in the proposition. □