Other Compilation of severe errors in famous textbooks

[andresB]

For the sake of helping student to avoid confusions, I wonder if we can make a compilation of known errors made in standard and commonly used textbooks. Not talking about some random typos, but more when like the entire treatment of a subject is fundamentally flawed.

 

[WWGD]

I think they may already have sites that do that. Have you tried a search for "[BookName] errata"? Edit: But maybe we can compile here as many of these links as possible, alpha by author.

 

[weirdoguy]

Erratas are for minor typos, and OP excluded this out of discussion.

 

[andresB]

I think they may already have sites that do that. Have you tried a search for "[BookName] errata"? Edit: But maybe we can compile here as many of these links as possible, alpha by author.

Reaging the STEM bible thread, I saw an argument about the Ballentine's treatment of several topics in QM, so I was thinking in things like that instead of things like "x" is missing a 1/2 that can be solved via an errata.

 

[DaveC426913]

"[BookName] fails".

 

[WWGD]

Ok, my bad, I did not read carefully. But isn't this partially a matter of taste, opinion? Edit: Unless there are factual mistakes?

 

[gleem]

... more when like the entire treatment of a subject is fundamentally flawed.

Isn't this more of a problem with elementary maybe high school science texts?

 

[WWGD]

"[BookName] fails".

Not clear what you mean. I suggested searches for errata for specific books.

 

[DaveC426913]

Not clear what you mean. I suggested searches for errata for specific books.

Note what's inside the "quotes" in my contribution.

Your suggestion for "[BookName] errata" was challenged (whether rightly or wrongly) by others.

I suggested an alternate title: "[BookName] fails".*

* in the 21st century, "fails" is a valid noun (as in: "epic fails"), not just a verb.

 

[WWGD]

Note what's inside the "quotes" in my contribution.

Your suggestion for "[BookName] errata" was challenged (whether rightly or wrongly) by others.

I suggested an alternate title: "[BookName] fails".*

* in the 21st century, "fails" is a valid noun, not just a verb.

Fair enough. Maybe we can have book reviews and author ( of review) can elaborate on the flaws they perceive in the book being reviewed.

 

[ZapperZ]

For the sake of helping student to avoid confusions, I wonder if we can make a compilation of known errors made in standard and commonly used textbooks. Not talking about some random typos, but more when like the entire treatment of a subject is fundamentally flawed.

Do you know of a specific example of even one of such type from such a resource?

Books like these are often reviewed by many people, and even when there are errors, big or small, these are usually corrected in subsequent editions.

On the other hand, Wikipedia...

Zz.

 

[WWGD]

Do you know of a specific example of even one of such type from such a resource?

Books like these are often reviewed by many people, and even when there are errors, big or small, these are usually corrected in subsequent editions.

On the other hand, Wikipedia...

Zz.

Let alone that it is not likely a student knows enough to give a cogent criticism of the book's treatment of a topic or the overall quality of the book. Don't get me wrong, it is a good idea to discuss the topic and address things you disagree with but it seems like overreaching to try to do so while an undergraduate.

 

[andresB]

Let alone that it is not likely a student knows enough to give a cogent criticism of the book's treatment of a topic or the overall quality of the book. Don't get me wrong, it is a good idea to discuss the topic and address things you disagree with but it seems like overreaching to try to do so while an undergraduate.

Of course an undergraduate can´'t do it. The idea I had with the thread is that people that have good knowledge made the warnings so students (or people reading about the topic for the first time) don't waste their time or, even worst, get a false knowledge.

Personally I don't have the confidence to pretend I can give an authoritative opinion, but, for example, I've heard really bad reviews of Sakurai's (revised edition) treatment of the Wigner-Eckart theorem. Also, I've seen harsh reviews on the treatment of the Quantum Zeno effect given in Ballentine.

 

[ZapperZ]

Personally I don't have the confidence to pretend I can give an authoritative opinion, but, for example, I've heard really bad reviews of Sakurai's (revised edition) treatment of the Wigner-Eckart theorem. Also, I've seen harsh reviews on the treatment of the Quantum Zeno effect given in Ballentine.

But this is different than saying these books have ERRORS! Errors mean that the content is faulty!

You are confusing personal preference with there being mistakes in the content. Those are two entirely different things!

Zz.

 

[Keith_McClary]

Some Physics textbooks will "prove" that QM bound states must have negative energy, but that is wrong:

Barry Simon writes:

One of the more intriguing questions concerns the
presence of discrete eigenvalues of positive energy (that is, square-integrable
eigenfunctions with positive eigenvalues) . There is a highly non-rigorous but
physically appealing argument which assures us that such positive energy “bound
states” cannot exist. On the other hand, there is an
ancient, explicit example due to von Neumann and Wigner which presents
a fairly reasonable potential $V$, with $V(r) \to 0$ as $r \to \infty$ and which possesses an
eigenfunction with $E = 1$.
The potential$$V(r)=\frac{-32 \sin r[g(r)^3 \cos r-3g(r)^2\sin^3r+g(r)\cos r+sin^3r]}{[1+g(r)^2]^2}$$
with $g(r)=2r-\sin2r$ has the eigenvalue +1 with eigenfunction
$$u(r)=\frac{\sin r}{r(1+g(r)^2)}$$

http://www.math.caltech.edu/SimonPapers/5.pd http://www.math.caltech.edu/SimonPapers/5.pd

Simon's paper is almost as "ancient" as von Neumann and Wigner's result was when Simon wrote that.

 

[Demystifier]

Landau and Lifshitz, Mechanics, Sec. 23 - Oscillations of systems with more than one degree of freedom.

It says that that $\omega^2$ must be positive because otherwise energy would not be conserved, which is wrong.
https://www.physicsforums.com/threads/error-in-landau-lifshitz-mechanics.901356/

 

[Demystifier]

Also, I've seen harsh reviews on the treatment of the Quantum Zeno effect given in Ballentine.

Yes. Ballentine misunderstands the meaning of collapse in quantum mechanics, i.e. thinks that it doesn't exist even in some FAPP effective sense. It culminates in his conclusion that the quantum Zeno effect (theoretically most easily described in terms of collapses) does not exist, contrary to experiments which show that it exists.

 

[atyy]

Ballentine's treatment of quantum mechanics is fundamentally flawed. The book presents his personal theory, rather than standard quantum mechanics.

Feynman's treatment of hidden variables in quantum mechanics in his famous lectures is fundamentally flawed, probably because Feynman did not understand the topic at that time. There are also minor physics errors (not typos) elsewhere in the lectures, probably due to momentary carelessness. The lectures as a whole are magnificent.

 

[martinbn]

Landau and Lifshitz, Mechanics, Sec. 23 - Oscillations of systems with more than one degree of freedom.

It says that that $\omega^2$ must be positive because otherwise energy would not be conserved, which is wrong.
https://www.physicsforums.com/threads/error-in-landau-lifshitz-mechanics.901356/

I don't get your objection. I might be wrong but at a first glance what they wrote seemed fine.

 

[martinbn]

Feynman's treatment of hidden variables in quantum mechanics in his famous lectures is fundamentally flawed, probably because Feynman did not understand the topic at that time.

Which pages? And why is it fundamentally flawed?

 

[Demystifier]

I don't get your objection. I might be wrong but at a first glance what they wrote seemed fine.

Well, energy is conserved for any sign of $\omega^2$. Indeed, energy is conserved whenever the Hamiltonian does not have an explicit dependence on time, which is the case for any sign of $\omega^2$, as long as $\omega$ does not have an explicit dependence on time.

 

[Andy Resnick]

Well, energy is conserved for any sign of $\omega^2$. Indeed, energy is conserved whenever the Hamiltonian does not have an explicit dependence on time, which is the case for any sign of $\omega^2$, as long as $\omega$ does not have an explicit dependence on time.

It's curious: the second paragraph right after eqn 23.8 (in my 3rd edition) does claim the roots must be 'real and positive' but only provides a counterexample for imaginary ω, not negative ω. I wonder if there is an underlying assumption that negative real frequencies are the same (except for a constant phase factor) as positive frequencies.

 

[Demystifier]

imaginary ω, not negative ω

Perhaps I am stating the obvious, but imaginary $\omega$ means negative $\omega^2$.

 

[atyy]

Which pages? And why is it fundamentally flawed?

http://www.feynmanlectures.caltech.edu/III_01.html#Ch1-S8
"We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. We cannot make the mystery go away by “explaining” how it works. We will just tell you how it works. In telling you how it works we will have told you about the basic peculiarities of all quantum mechanics."

Feynman refers to the double slit experiment. However, most people would nowadays take the Bell tests to be the mystery of QM, not the double slit. There is interesting commentary in section 1 of https://arxiv.org/abs/1301.3274. Whitaker comments that Feynman corrected himself in his later lectures on computation https://aapt.scitation.org/doi/full/10.1119/1.4948268 "In any case, since what Feynman describes is indeed Bell's Theorem, it is extremely interesting that he adds that he often entertained himself by squeezing the difficulty of quantum mechanics into a smaller and smaller place, and he finds this place precisely in this analysis. Thus, Feynman's view is apparently clear—the content of Bell's Theorem is the crucial point that distinguishes classical and quantum physics."

"We make now a few remarks on a suggestion that has sometimes been made to try to avoid the description we have given: “Perhaps the electron has some kind of internal works—some inner variables—that we do not yet know about. Perhaps that is why we cannot predict what will happen. If we could look more closely at the electron, we could be able to tell where it would end up.” So far as we know, that is impossible. We would still be in difficulty. Suppose we were to assume that inside the electron there is some kind of machinery that determines where it is going to end up. That machine must also determine which hole it is going to go through on its way. But we must not forget that what is inside the electron should not be dependent on what we do, and in particular upon whether we open or close one of the holes. So if an electron, before it starts, has already made up its mind (a) which hole it is going to use, and (b) where it is going to land, we should find P1 for those electrons that have chosen hole 1, P2 for those that have chosen hole 2, and necessarily the sum P1+P2 for those that arrive through the two holes. There seems to be no way around this. But we have verified experimentally that that is not the case. And no one has figured a way out of this puzzle. So at the present time we must limit ourselves to computing probabilities. We say “at the present time,” but we suspect very strongly that it is something that will be with us forever—that it is impossible to beat that puzzle—that this is the way nature really is."

Feynman says something similarly erroneous in this video around 51 minutes.

Hidden variables for the double slit are possible.

 

[DaveC426913]

The idea I had with the thread is that people that have good knowledge made the warnings so students (or people reading about the topic for the first time) don't waste their time or, even worst, get a false knowledge.

Which would be great, except how do we decide what explanation prevails amidst multiple opposing views? By discussion of course. But there's no clear winner.

So, instead of an authoritative list of errata, what we get is a discussion thread where the issues are debated back and forth, possibly endlessly. See posts 15 through 24 for examples.

It's a laudable idea, I just think there's an XKCD for that...

 

[WWGD]

Which would be great, except how do we decide what explanation prevails amidst multiple opposing views? By discussion of course. But there's no clear winner.

So, instead of an authoritative list of errata, what we get is a thread where the issues are debated back and forth possibly endlessly. See posts 15 through 24 for examples.

It's a laudable idea, I just think there's an XKCD for that...

View attachment 253052

Ditto for quibbles on definitions when writing papers. Maybe we should set up a committee to decide. How should we set up the committee...( rabbit hole).

 

[WWGD]

But sorry Andres, don't mean to minimize your goal. I guess we can have here a centralized section for reviews of Physics books and discuss them.

 

[Andy Resnick]

Perhaps I am stating the obvious, but imaginary $\omega$ means negative $\omega^2$.

Yes, that is stating the obvious :)

 

[andresB]

But sorry Andres, don't mean to minimize your goal. I guess we can have here a centralized section for reviews of Physics books and discuss them.

I guess I was too optimistic, but at least I learned something about positive eigenvalues of the Schrodinger operators.

 

[Demystifier]

Ballentine's treatment of quantum mechanics is fundamentally flawed. The book presents his personal theory, rather than standard quantum mechanics.

This, of course, is only true for the parts that are related to collapse. The other parts (which constitute more than 90% of the whole book) are fine.

 

[vanhees71]

Well, one severe conceptual mistake is in some textbooks (if I remember right even in the Feynman Lectures vol. II and in Berkeley Physics Course vol. II, which shows that also Nobel Laureates make mistakes ;-)) when treating the magnetostatics of a wire relativistically. The mistake lies in the assumption that the wire is uncharged in the rest frame of the wire. In fact it's uncharged in the rest frame of the electrons. The correct treatment has to take into account the "self-induced" Hall effect though it's academic for house-hold currents, where the drift velocities are of the order of 1mm/s, but if you want to treat it fully relativistically you must take into account the correct relativistic version of Ohm's Law, $\vec{j}=\sigma \gamma (\vec{E} + \vec{v} \times \vec{B}/c)$ with $\sigma$ the usual electric conductivity (a scalar as any transport coefficient), $\gamma=(1-v^2/c^2)^{-1/2}$ (using Heaviside-Lorentz units).

 

[vanhees71]

This, of course, is only true for the parts that are related to collapse. The other parts (which constitute more than 90% of the whole book) are fine.

It's also a matter of opinion. I also don't subscribe to the collapse hypothesis. It's neither needed nor consistent with local relativistic QFT, which is the most successful theory yet (at least from a phenomenological point of view).

 

[Demystifier]

It's also a matter of opinion. I also don't subscribe to the collapse hypothesis. It's neither needed nor consistent with local relativistic QFT, which is the most successful theory yet (at least from a phenomenological point of view).

But you don't make explicitly wrong statements about collapse, because you accept "collapse" at least in the sense of information update and you do not deny the quantum Zeno effect. In that sense you are not like Ballentine.

 

[forcefield]

http://www.feynmanlectures.caltech.edu/III_01.html#Ch1-S8...
"We make now a few remarks on a suggestion that has sometimes been made to try to avoid the description we have given: “Perhaps the electron has some kind of internal works—some inner variables—that we do not yet know about. Perhaps that is why we cannot predict what will happen. If we could look more closely at the electron, we could be able to tell where it would end up.” So far as we know, that is impossible. We would still be in difficulty. Suppose we were to assume that inside the electron there is some kind of machinery that determines where it is going to end up. That machine must also determine which hole it is going to go through on its way. But we must not forget that what is inside the electron should not be dependent on what we do, and in particular upon whether we open or close one of the holes. So if an electron, before it starts, has already made up its mind (a) which hole it is going to use, and (b) where it is going to land, we should find P1 for those electrons that have chosen hole 1, P2 for those that have chosen hole 2, and necessarily the sum P1+P2 for those that arrive through the two holes. There seems to be no way around this. But we have verified experimentally that that is not the case. And no one has figured a way out of this puzzle. So at the present time we must limit ourselves to computing probabilities. We say “at the present time,” but we suspect very strongly that it is something that will be with us forever—that it is impossible to beat that puzzle—that this is the way nature really is."
...
Hidden variables for the double slit are possible.

Note that Feynman is there not talking about Bell's hidden variables but about (electron's) inner variables. It's not the same thing.

 

[vanhees71]

But you don't make explicitly wrong statements about collapse, because you accept "collapse" at least in the sense of information update and you do not deny the quantum Zeno effect. In that sense you are not like Ballentine.

Interesting. I've to read Ballentine's book on the Zeno effect again, but the Zeno effect isn't about collapse but it's about "stabilizing" an unstable state by some interaction. There's no need for collapse to understand it. It can be well explained within the statistical interpretation, and afaik it has been demonstrated already experimentally. I don't remember the details, but I think it was done with some metastable atomic state using a laser.

 

[Demystifier]

Interesting. I've to read Ballentine's book on the Zeno effect again, but the Zeno effect isn't about collapse but it's about "stabilizing" an unstable state by some interaction. There's no need for collapse to understand it. It can be well explained within the statistical interpretation, and afaik it has been demonstrated already experimentally. I don't remember the details, but I think it was done with some metastable atomic state using a laser.

The collapse is certainly not necessary to explain the Zeno effect, but it is useful as a quick and dirty way to obtain it.

 

[martinbn]

Well, energy is conserved for any sign of $\omega^2$. Indeed, energy is conserved whenever the Hamiltonian does not have an explicit dependence on time, which is the case for any sign of $\omega^2$, as long as $\omega$ does not have an explicit dependence on time.

I still don't understand. In the case of one degree of freedom you are right. But this section is specifically for more than one degree of freedom. In that case it is not at all obvious to me. May be it is an easy calculation and I am just being silly, but I don't see. In the $s=1$ case the matrix is just a number, so the determinant is just that number and the equation for $\omega^2$ can easily be used in the expresion for the energy. In the $s>1$ case that doesn't seem so. You still have the exponentials that will decay or grow if $\omega$ isn't real. Even if it can be shown that the decay/grow isn't there, their argument is quite natural and convincing and easy. There is no error there.

 

[martinbn]

http://www.feynmanlectures.caltech.edu/III_01.html#Ch1-S8
"We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. We cannot make the mystery go away by “explaining” how it works. We will just tell you how it works. In telling you how it works we will have told you about the basic peculiarities of all quantum mechanics."

Feynman refers to the double slit experiment. However, most people would nowadays take the Bell tests to be the mystery of QM, not the double slit. There is interesting commentary in section 1 of https://arxiv.org/abs/1301.3274. Whitaker comments that Feynman corrected himself in his later lectures on computation https://aapt.scitation.org/doi/full/10.1119/1.4948268 "In any case, since what Feynman describes is indeed Bell's Theorem, it is extremely interesting that he adds that he often entertained himself by squeezing the difficulty of quantum mechanics into a smaller and smaller place, and he finds this place precisely in this analysis. Thus, Feynman's view is apparently clear—the content of Bell's Theorem is the crucial point that distinguishes classical and quantum physics."

"We make now a few remarks on a suggestion that has sometimes been made to try to avoid the description we have given: “Perhaps the electron has some kind of internal works—some inner variables—that we do not yet know about. Perhaps that is why we cannot predict what will happen. If we could look more closely at the electron, we could be able to tell where it would end up.” So far as we know, that is impossible. We would still be in difficulty. Suppose we were to assume that inside the electron there is some kind of machinery that determines where it is going to end up. That machine must also determine which hole it is going to go through on its way. But we must not forget that what is inside the electron should not be dependent on what we do, and in particular upon whether we open or close one of the holes. So if an electron, before it starts, has already made up its mind (a) which hole it is going to use, and (b) where it is going to land, we should find P1 for those electrons that have chosen hole 1, P2 for those that have chosen hole 2, and necessarily the sum P1+P2 for those that arrive through the two holes. There seems to be no way around this. But we have verified experimentally that that is not the case. And no one has figured a way out of this puzzle. So at the present time we must limit ourselves to computing probabilities. We say “at the present time,” but we suspect very strongly that it is something that will be with us forever—that it is impossible to beat that puzzle—that this is the way nature really is."

Feynman says something similarly erroneous in this video around 51 minutes.

Hidden variables for the double slit are possible.

To be honest I don't see your point. Can you elaborate? Nothing in what Feynman writes looks erroneous, let alone fundamentally flawed.

 

[Demystifier]

You still have the exponentials that will decay or grow if $\omega$ isn't real.

So what? The exponential grow of positive kinetic energy is accompanied with the exponential grow of negative potential energy, so that the total energy, that is the sum of positive kinetic energy and negative potential energy, is constant. If you don't believe me, solve the equations explicitly with negative $\omega^2$ and convince yourself that the total energy is indeed conserved.

 

[martinbn]

So what? The exponential grow of positive kinetic energy is accompanied with the exponential grow of negative potential energy, so that the total energy, that is the sum of positive kinetic energy and negative potential energy, is constant.

No, the exponetials are the same and can be factored out. So you have an exponetial times a bounded term. So the whole expression will grow or decay. It looks like this $e^{\lambda t}\times\text{bounded stuff}$. The $\lambda$ is zero only of the $\omega$ is real, otherwise it is negatve or possitive, so the whole energy will change with time.

If you don't believe me, solve the equations explicitly with negative $\omega^2$ and convince yourself that the total energy is indeed conserved.

That's what I am doing, but I can see how it works only in the case of one degree of freedom. The case at hand doesn't seem that way.

 

[Demystifier]

It looks like this $e^{\lambda t}\times\text{bounded stuff}$.

Almost, but not quite. It looks like
$$E_0 +e^{\lambda t}(B_{\rm kinetic}+B_{\rm potential})$$
where the $B$-terms are bounded. But $B_{\rm potential}<0$ (because $\omega^2<0)$ and in fact $B_{\rm kinetic}+B_{\rm potential}=0$, so the full energy is $E_0$, which is a constant.

That's what I am doing, but I can see how it works only in the case of one degree of freedom.

Can you at least see that for one degree of freedom you get my form above?

 

[martinbn]

Almost, but not quite. It looks like
$$E_0 +e^{\lambda t}(B_{\rm kinetic}+B_{\rm potential})$$
where the $B$-terms are bounded. But $B_{\rm potential}<0$ (because $\omega^2<0)$ and in fact $B_{\rm kinetic}+B_{\rm potential}=0$, so the full energy is $E_0$, which is a constant.Can you at least see that for one degree of freedom you get my form above?

Yes, my point exactly. For one degree it is simple. For many, I don't see it.

 

[Demystifier]

Yes, my point exactly. For one degree it is simple. For many, I don't see it.

OK, so we agree that for one degree the energy is conserved, right? To solve the equations explicitly for many degrees, you have to diagonalize the Hamiltonian. For inspiration in the case of 2 degrees, see e.g. my http://de.arxiv.org/abs/1702.03291 Sec. 3 (and ignore the $y$-dependence).

 

[martinbn]

OK, so we agree that for one degree the energy is conserved, right? To solve the equations explicitly for many degrees, you have to diagonalize the Hamiltonian. For inspiration in the case of 2 degrees, see e.g. my http://de.arxiv.org/abs/1702.03291 Sec. 3 (and ignore the $y$-dependence).

Suppose it is diagonal to begin with. Then the matrices $k_{ij}$ and $m_{ij}$ are diagonal and have diagonal elements $k^i$ and $m^i$. Then the possible values for $\omega^2$ are $\frac{k^i}{m^i}$. But it is only one of them, because they (L&L) are looking for solutions of the form $x_l=A_le^{i\omega t}$. In this decoupled case the energy is the sum of the energies of the individual systems and only one of them will be zero. The others will not, so you still have the exponential decay or increase if $\omega$ is not real. My guess is that your mistake is that you assume that $x_l=A_le^{i\omega_l t}$ with different omegas so that all terms will cancel.

 

[Demystifier]

My guess is ...

Why don't you just solve the equations of motion explicitly and completely, instead of guessing? When you do that, and when you insert the solution into the expression for the full Hamiltonian, you will see that all the time-dependent terms cancel.

 

[martinbn]

Why don't you just solve the equations of motion explicitly and completely, instead of guessing? When you do that, and when you insert the solution into the expression for the full Hamiltonian, you will see that all the time-dependent terms cancel.

Erm, I did. And they don't cancel. Did you try it? Take two degrees of freedom with different k's and m's, then it is obvious that one term will cancel and the other will be left.

 

[martinbn]

Just add that in your paper sec3. you are looking at the case where the k and the m are the same for both subsystems. Then it all works out. In general though it will not.

 

[formodular]

If you are you referencing the argument after equation 23.8, there is no error - they prove the claim directly there, but the physical argument is also correct as the Lagrangian will then explicitly depend on time and so Noether does not let you even state conservation of energy, but the Lagrangian was assumed to be time-independent from the beginning, so that Noether then gives energy conservation.

 

[bhobba]

Simon's paper is almost as "ancient" as von Neumann and Wigner's result was when Simon wrote that.

Speaking of Von-Neumann I am surprised nobody mentioned his famous no-go theorem on Hidden Variables in his Mathematical Foundations of QM (I wish I could say I picked it up when I read it, but didn't). It was only universally picked up years later, due to Von-Neumann's well deserved reputation and not carefully checking its assumptions. Greta Hermann did, but she was dismissed. I hope it was because of the respect Von-Neumann had - not because she was a woman. Either way not one of sciences finest hours. Then their was his scathing rebuke of the Dirac Delta function. Rather than say we need further developments in math to make sense of it, which has now been done (admittedly requiring mathematicians like Grothendieck whose mathematical reputation is the equal of Von-Neumann himself) he simply dismissed it as a fiction. Von-Neumann is one of my heroes, being more that just a great mathematician, but that even rarer beast, a polymath, however like all human beings perfect he was not.

Thanks
Bill

 

[Keith_McClary]

It was only universally picked up years later

Simon (as a "predoctoral fellow") found and corrected an error of von Neumann and Wigner (see Examples and Remarks B). He also has:

Note added in proof: There is a minor technical flaw in the proof of Theorem 2 ... We have really only proven that w = 0 outside a sufficiently large sphere.

BTW, I messed up the link to his paper, this should work:
On Positive Eigenvalues of One-Body Schrodinger Operators

Nobody's perfect.