bibkey: “teschl2009mathematical” authors: “Gerald Teschl” year: 2009 title: “Mathematical Methods in Quantum Mechanics: With Applications to Schrodinger Operators” doi: null url: “https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf” claim: “The maximal L2 domain of a real multiplication operator is exactly the set of vectors whose pointwise product is in L2; the strong derivative domain of its unitary exponential equals that operator domain.” strata_touched: [] license: “citation-only” triage: “anchor”

Maximal multiplication domains and physical modulation

Teschl, Graduate Studies in Mathematics 99, equation (2.21), printed pages 59-60, gives the maximal multiplication domain. Theorem 5.1(ii), printed pages 123-124, identifies the strong derivative domain of the unitary exponential with the domain of its self-adjoint generator. For the real multiplier A(x) = -(b dot x)/hbar, the convention U(t) = exp(-itA) gives positive modulation and derivative (i/hbar)(b dot x)f. These are known results, with this parameter correspondence.

Physical modulation

For every natural number d, let E = EuclideanSpace Real (Fin d), with Lebesgue measure volume, and H = Lp Complex 2 volume. For every positive hbar, b in E, and f,v in H, let

M_b(t)f=[x\mapsto e^{it\langle b,x\rangle/\hbar}f(x)].

The complete derivative graph is

\operatorname{HasDerivAt}(t\mapsto M_b(t)f,v,0)
\iff
\exists h:\operatorname{MemLp}(x\mapsto\langle b,x\rangle f(x),2,\mathrm{volume}),\quad
v=\frac{i}{\hbar}\,h.\operatorname{toLp}(x\mapsto\langle b,x\rangle f(x)).

All directions, including zero, and all finite dimensions, including zero, are included. There is no finite-volume, global L1, Schwartz, or product-integrability premise on the derivative side.

Set a(x) = (b dot x)/hbar, z(x) = (b dot x)f(x), c = i/hbar, and k = cz. The quotient representative q_t(x) = t^(-1)(exp(it a(x))-1)f(x) converges pointwise to k. The phase derivative has norm |a(x)|, so |q_t(x)| is at most |k(x)| and |q_t(x)-k(x)| is at most 2|k(x)|. If z is square integrable, dominated convergence for the squared error proves convergence in the actual L2 norm and hence the derivative formula.

Conversely, a strong derivative v makes the quotient classes at t_n = 1/(n+1) converge to v in L2. Convergence in measure gives a strictly increasing subsequence converging almost everywhere. The countably many representative equalities hold together outside one null set. The scalar derivative forces the same subsequence to converge to k, so v=k almost everywhere. Thus k is in L2; since c is nonzero, z is in L2. The condition and toLp value are invariant under changes on null sets.

Locator

https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf

Equation (2.21), printed pages 59-60; Theorem 5.1(ii), printed pages 123-124. The author-hosted file identifies the 2009 first edition, Graduate Studies in Mathematics volume 99. The online-use permission appears on its title page; this note cites the source and paraphrases the argument.

Finite-measure Fourier mean squares

In the same retained first-edition PDF, Theorem 5.4, Section 5.2, printed pp.126–127, equations (5.8)–(5.9), states Wiener’s theorem for every finite complex Borel measure $\mu$ on $\mathbb{R}$:

$$\hat{\mu }(t)=\int e^{-it\lambda }d\mu (\lambda ),\underset{T\to \infty }{\lim }\frac{1}{T}{\int }_0^T|\hat{\mu }(t){|}^{2}dt=\sum _{\lambda \in \mathbb{R}}|\mu (\{\lambda \}){|}^2.$$

The source uses the unnormalized angular transform, and the atomic sum is finite. The inspected PDF SHA-256 is 8dc8de0b58aa0a3fedfe594a345f9b5875322e5526ea581cb640a98d55b82818; its author-hosted title page dates the online text to 12 February 2009. The source theorem is reused, without a new proof or priority claim.

For the project’s real even finite signed prime-minus-continuum head, the atoms are exactly $\pm \log n$, with masses $\Lambda (n)/\sqrt{n}$. The negative continuous component has no atomic mass. The fixed-head scalar-budget application uses this same theorem to diagnose a frequency-envelope loss. It does not assert growth of the actual weighted operator norm, an obstruction to a growing arithmetic cutoff, or an RH/Robin conclusion. No compiled project application is claimed.

Schur criterion for the local frequency kernel

The same retained first-edition PDF gives Lemma 0.32, printed pp.28–29, the Schur criterion for a measurable integral kernel dominated by $K_1(x,y)K_2(x,y)$. For conjugate exponents, its separate row and column norm bounds $C_1,C_2$ give operator norm at most $C_1{C}_2$. The local signed-frequency note uses the $L^2$ case with a Lorentzian factorization of the actual Fourier kernel. It is a direct source-criterion application, with no new theorem or compiled specialization claimed.

In Section 7.2, equation (7.47), printed p.172, the one-dimensional free resolvent has kernel $e^{-\sqrt{-z}|x-y|}/(2\sqrt{-z})$. For $z=-q^2$, $q>0$, this is the inverse angular transform of $({\xi }^2+q^2{)}^{-1}$ with its $1/(2\pi )$ factor. The local note uses the resulting classical identity $\int q^2/(q^2+{\xi }^2)e^{-i\xi t}d\xi =\pi q{e}^{-q|t|}$. The source supplies the transform and generic kernel estimate; it supplies no value or sign for the project’s arithmetic form.

Equality in Schur’s test checks the unprojected theta-edge shortcut

The criterion above and the classical equality condition in Cauchy–Schwarz are reused here. The application concerns the original minimal even theta form and its known critical family, under those notes’ paper-level model and domain premises. It does not construct a transfer or supply a new numerical lower bound or Lean certification.

Keep the signed conductance and the actual core

Write $d\nu =2\Phi (x)\cosh (x/2)dx$, with $\nu (\mathbb{R})=1$, and $D(h)=\frac{1}{2}\iint |h(y)-h(x){|}^{2}J(dx,dy)$ on the actual minimal even form domain $\mathcal{F}$. Every prime power remains in $J_p$, and $J_{\Gamma }=\Phi (x)\Phi (y){\psi }_{\Gamma }(|x-y|)dxdy$, where ${\psi }_{\Gamma }(t)=e^{-t/2}/(1-e^{-2t})$ for $t>0$.

On $X=\{(x,y):x\ne y\}$ the half-slack $q(h)=D(h)-{\mathrm{Var}}_{\nu }(h)/2$ has signed conductance $J-(\nu \otimes \nu )/2$. Its positive and negative parts are

$$\begin{array}{rl}{K}_{+}=& J_p+\Phi (x)\Phi (y)[{\psi }_{\Gamma }(|x-y|)-2\cosh (x/2)\cosh (y/2){]}_{+}dxdy,\\ K_{-}=& \Phi (x)\Phi (y)[2\cosh (x/2)\cosh (y/2)-{\psi }_{\Gamma }(|x-y|){]}_{+}dxdy.\end{array}\text{\hspace{1em}(S1)}$$

The prime graphs are singular to $dxdy$, so all remain in $K_{+}$. Both edge measures are $\sigma$-finite. In particular, $K_{+}$ is not finite: ${\psi }_{\Gamma }(t)\sim 1/(2t)$ at zero gives logarithmically infinite mass near the diagonal. The increment cancels this singularity in the form, and Schur’s test below uses $\sigma$-finite spaces. Set ${\mathcal{E}}_{\pm }=L^2(K_{\pm })$ and $C_{\pm }h(x,y)=(h(y)-h(x))/\sqrt{2}$. Ordinary variance accounting gives

$$q(h)=\Vert C_{+}h{\Vert }^2-\Vert C_{-}h{\Vert }^2,\Vert C_{+}h{\Vert }^2\le D(h),\Vert C_{-}h{\Vert }^2\le \frac{1}{2}{\mathrm{Var}}_{\nu }(h).\text{\hspace{1em}(S2)}$$

These are the same original form and mean term, not a new energy model. Both maps are continuous for its form norm. For $v_k={\Phi }^{(2k)}/\Phi -4^{-k}$, the existing critical-family result gives $v_k\in \mathcal{F}$ and $\Vert C_{+}{v}_k\Vert =\Vert C_{-}{v}_k\Vert$ for every $k\ge 1$.

The precise shortcut being tested

Suppose a measurable complex kernel $s(a,b)$, from positive edges $b$ to negative edges $a$, defines the ordinary integral operator

$$(Sf)(a)=\int s(a,b)f(b)d{K}_{+}(b).$$

Let finite strictly positive measurable weights $w_{\pm }$ and constants $a_0,b_0>0$ satisfy

$$\begin{array}{rlrl}\int |s(a,b)|w_{+}(b)d{K}_{+}(b)& \le a_0{w}_{-}(a)& & (K_{-}\text{-a.e. }a),\\ \int |s(a,b)|w_{-}(a)d{K}_{-}(a)& \le b_0{w}_{+}(b)& & (K_{+}\text{-a.e. }b),\\ a_0{b}_0& \le 1.& & \end{array}\text{\hspace{1em}(S3)}$$

Schur’s test gives $\Vert S\Vert \le \sqrt{a_0{b}_0}$. For these two measure spaces its usual same-space formulation can be applied to their disjoint union, with $S$ as the off-diagonal block. The proposed raw reconstruction requirement is

$$S{C}_{+}h=C_{-}h\text{for every original compact smooth even core test }h.\text{\hspace{1em}(S4)}$$

Under the inherited theta premises, (S3) and (S4) cannot both hold. This conclusion includes complex signed kernels; it is not a positivity-preservation argument.

Theta jets distinguish the radial edge

For $r\ge 0$ define

$$H_r(t)=\frac{\Phi (r+t)+\Phi (r-t)}{2\Phi (r)}.$$

It is holomorphic on the existing physical strip $|\Im t|<\pi /4$, has $H_r(0)=1$, zero odd derivatives, and even derivatives ${\Phi }^{(2k)}(r)/\Phi (r)$ at zero. Thus, if $|x|\ne |y|$ and $\lambda \ne 0$ satisfy

$$v_k(v)-v_k(u)=\lambda (v_k(y)-v_k(x))\text{for every }k\ge 1,$$

the analytic identity theorem gives $H_{|v|}-H_{|u|}=\lambda (H_{|y|}-H_{|x|})$. These functions are integrable on the real line. Their angular Fourier transforms are $\Xi (z)\cos (zr)/\Phi (r)$. Canceling $\Xi$ only on its known nonzero neighborhood of zero gives

$$\frac{\cos (z|v|)}{\Phi (v)}-\frac{\cos (z|u|)}{\Phi (u)}=\lambda \left(\frac{\cos (z|y|)}{\Phi (y)}-\frac{\cos (z|x|)}{\Phi (x)}\right).\text{\hspace{1em}(S5)}$$

The finite entire cosine sum then vanishes identically. Independence of its distinct exponential frequencies identifies its nonzero signed atomic supports, so $\{|u|,|v|\}=\{|x|,|y|\}$. An endpoint at zero only combines the two equal zero-frequency atoms. The same argument shows that the full vector of critical increments at a nondegenerate radial edge is nonzero. Consequently its proportionality fiber consists of at most eight signed and oriented endpoint pairs. No assertion that $\Phi$ is entire in its physical coordinate, or that $\Xi$ has no zeros, is used.

Norm attainment forces the ordinary kernel into a null fiber

By (S2), boundedness of $S$ and actual core density extend (S4) to $\mathcal{F}$, including every $v_k$. Put $f_k=C_{+}{v}_k$ and $g_k=C_{-}{v}_k$. Then $S{f}_k=g_k$ and $\Vert S{f}_k\Vert =\Vert f_k\Vert$. If $a_0{b}_0<1$, all these vectors vanish. This is impossible: $K_{-}$ is a nonzero finite absolutely continuous measure, it gives zero mass to $|x|=|y|$, and the jet separation above makes the countable critical vector nonzero at every other edge. Hence $a_0{b}_0=1$.

Equality now holds throughout the classical row/column Schur bound for each $f_k$. In particular, row Cauchy–Schwarz is equality almost everywhere. With $m(a)=\int |s(a,b)|w_{+}(b)d{K}_{+}(b)$, its equality condition and (S4) give

$$f_k(b)=w_{+}(b)e^{-i\arg s(a,b)}\frac{g_k(a)}{m(a)}\text{\hspace{1em}(S6)}$$

for $|s(a,\cdot )|d{K}_{+}$-almost every $b$ in any such row with $m(a)>0$. The row and column certificates justify these integrals by Cauchy–Schwarz and Tonelli. Only countably many $k$ occur, so one may choose a conull set of rows and, inside each row, a common conull set for all components. There is no uncountable intersection assertion.

At almost every negative edge $a$, some $g_k(a)$ is nonzero, which forces $m(a)>0$. Equation(S6) makes the complete critical increment vector at $b$ proportional to the vector at $a$. Equation(S5) confines that row to the finite fiber with the same unordered radial endpoints. But $K_{+}$ gives every point mass zero: its continuous part is a density, and each prime graph is parametrized by continuous $dx$. Atomic jump lengths are not point atoms in this edge-pair measure. Therefore the finite fiber has zero $K_{+}$-mass, forcing $m(a)=0$, a contradiction.

What remains available for the actual estimate

This tests an exact full-core reconstruction with an absolute Schur certificate of product at most one. It excludes neither an abstract contraction nor a kernel whose actual norm is better than its absolute certificate. Products greater than one tending to one, controlled approximate reconstruction, operators specified only on $\mathcal{R}$, measure/distribution kernels and other norm estimates remain separate possibilities. Finite atomic matrices are also outside the atomless fiber argument.

In particular, let $P_{\pm }$ project onto $\overline{C_{\pm }N}$ and set $e_{\pm }=(I-P_{\pm })C_{\pm }$. All known critical vectors have $e_{\pm }{v}_k=0$, so the preceding norm-attainment argument has no stated conclusion for a transfer between those projected increments. The existing mixed nullity gives an isometry $U(C_{+}n)=C_{-}n$ from $\overline{C_{+}N}$ onto $\overline{C_{-}N}$ and $P_{-}{C}_{-}h=U{P}_{+}{C}_{+}h$: test the latter identity against the dense family $C_{-}N$ and use $q(n,h)=0$. Orthogonal decomposition therefore gives

$$q(h)=\Vert e_{+}h{\Vert }^2-\Vert e_{-}h{\Vert }^2.\text{\hspace{1em}(S7)}$$

This is the ordinary simultaneous Hilbert-space cancellation applied to the already specified mixed null family. Bare existence of a contraction reconstructing $e_{-}$ from $e_{+}$ only reformulates the desired norm domination. Its reconstruction and norm estimates must still be proved jointly by independent estimates for the same actual transfer. The original half-bound, RH, Robin and the cofinal signed comparison remain unresolved. This section applies existing Schur equality and Fourier uniqueness to the specified theta family; it claims no new general theorem or Lean result.

One source correction for the two critical edge projections

The negative-edge Gram application uses the same signed edge measures to put an explicit positive symbolic lower bound on the negative-edge metric, via three common-neighbor anchors. On the centered critical source space this makes the common Gram operator boundedly invertible and both critical edge-image ranges closed. The two projections in (S7) then use the same correction $n_h=G^{-1}{P}_N{C}_{-}^{\ast }{C}_{-}h$; it generally differs from $P_{N}h$. The paired edge-error bound for a Neumann truncation uses the exact $P_N$ and $G$. It is not a finite algorithm, a pointwise error bound, or a lower bound for the original theta energy. Acquisition of $P_N$ and the projected transfer’s joint reconstruction/norm estimate remain unproved. The path, Gram projection and Neumann tools are classical applications, with no new general theorem or Lean certification.